Disordered Systems Wiki - User contributions [en]

Main Page

2026-03-26T14:35:03Z

Rosso: /* Where and When */

Welcome to the WIKI page of the M2 ICFP course on the Statistical Physics of Disordered Systems, second semester 2026.

= Where and When =

* Each Monday at 2pm - 6 pm, from January 19th to March 23rd. No lecture on 23/02/26.
* Room 202 in Jussieu campus, Tours 54-55 until 16th February
* Room 209 in Jussieu campus, Tours 56-66 from 2nd March
* Room 107 in Jussieu campus, Tours 24-34 on 23 March
* Each session is a mixture of lectures and exercises.

== Exam ==

* '''D-Day:''' Monday, March 30 — 2:00 PM – 5:00 PM
* '''Where:''' Tower 24–25, Room 105

= The Team =

* [https://vale1925.wixsite.com/vros Valentina Ros] - vale1925@gmail.com
* [http://lptms.u-psud.fr/alberto_rosso/ Alberto Rosso] - alberto.rosso74@gmail.com

= Course description =

This course deals with systems in which the presence of impurities or amorphous structures (in other words, of disorder) influences radically the physics, generating novel phenomena. These phenomena involve the properties of the system at equilibrium (freezing and glass transitions), as well as their dynamical evolution out-of-equilibrium (pinning, avalanches), giving rise to ergodicity breaking both in absence and in presence of quantum fluctuations (classical metastability, quantum localization). We discuss the main statistical physics models that are able to capture the phenomenology of these systems, as well as the powerful theoretical tools (replica theory, large deviations, random matrix theory, scaling arguments, strong-disorder expansions) that have been developed to characterize quantitatively their physics. These theoretical tools nowadays have a huge impact in a variety of fields that go well-beyond statistical physics (computer science, probability, condensed matter, theoretical biology). Below is a list of topics discussed during the course.

'''Finite-dimensional disordered systems:'''

* Introduction to disordered systems and to the spin glass transition.
* Interface growth. Directed polymers in random media. Scenarios for the glass transition: the glass transition in KPZ in d>2.
* Depinning and avalanches. Bienaymé-Galton-Watson processes.
* Anderson localization: introduction. Localization in 1D: transfer matrix and Lyapunov.

'''Mean-field disordered systems:'''

* The simplest spin-glass: solution of the Random Energy Model.
* The replica method: the solution of the spherical p-spin model. Sketch of the solution of Sherrington Kirkpatrick model (full RSB).
* Towards glassy dynamics: rugged landscapes. Slow dynamics and aging: the trap model.
* The Anderson model on the Bethe lattice: the mobility edge.

=Lectures and tutorials=

'''If the layout of the formulas is bad, it might be because you are using Safari. Try opening the wiki with Firefox or Chrome.'''

{| class="wikitable" border="1"
|-
! width="100"|Date
! width="500"| 14h00-15h45
! width="500"| 16h00-17h45

|-valign="top"

| Week 1 (19/01)
|
* [[L1_ICTS| Spin Glass Transition (Alberto)]]

|
* [[T-I| A dictionary. The REM: energy landscape (Valentina)]]  [[Media:2025 P1 solutions.pdf| Sol Prob.1 ]]
|-valign=“top"

|-valign=“top"
| Week 2 (26/01)
|
* [[L2_ICFP| Stochastic Interfaces and growth (Alberto)]]
|
* [[T-2|The REM: freezing, condensation, glassiness (Valentina)]]   [[Media:2025 P2 solutions.pdf| Sol Prob.2 ]]
|-valign=“top"

|-valign=“top"
| Week 3 (02/02)
|
* [[L-3|Directed polymer in random media (Alberto)]]
|
* [[L-4| KPZ and glassiness in finite dimension (Alberto)]] [[https://colab.research.google.com/drive/1PTya42ZS2kU87A-BxQFFIUDTs_k47men?usp=sharing| notebook]]
|-valign=“top"
| Week 4 (9/02) and Week 5 (16/02)
|
* [[T-3| Spin glasses, equilibrium: replicas, the steps (Valentina)]]  [[Media:2025 P3 solutions.pdf| Sol Prob.3 ]]
|
* [[T-4| Spin glasses, equilibrium: replicas, the interpretation (Valentina)]]   [[Media:2025 P4 solutions.pdf| Sol Probs. 4 ]] 
[[Media:2025_Parisi_scheme.pdf| Notes: Probing states with replicas]]
|-valign=“top"
| Week 6 (02/03)
|
* [[LBan-IV| Driven Disordered Materials (Alberto)]] [[Media:DISSYTS.pdf| Slides ]]
|
* [[LBan-V| Avalanches in Disordered Materials (Alberto)]]
|-valign=“top"

| Week 7 (9/03)
|
* [[L-7| Anderson localization: introduction (Alberto)]]
|
* [[T-5| Rugged landscapes: counting metastable states (Valentina)]]  [[Media:2025 P5 solutions.pdf| Sol Prob.5 ]]
|-valign=“top"
| Week 8 (16/03)
|
* [[T-6| Rugged landscapes: stability of metastable states (Valentina)]] [[Media:2025 P666 solutions .pdf| Sol Prob.6 ]]
|
* [[T-7| Trap model and aging dynamics (Valentina)]] [[Media:2025 P7 solutions .pdf| Sol Prob.7 ]]
|-valign=“top"
| Week 9 (23/03)
|
* [[L-8| Localization in 1D, transfer matrix and Lyapunov exponent (Alberto)]] [[https://colab.research.google.com/drive/1ZJ0yvMrtflWNNmPfaRQ8KTteoWfm0bqk?usp=sharing| notebook]]
|
* [[L-9|Multifractality, tails (Alberto)]]
|-valign=“top"
| Extra (not in exam!)
|
* [[T-8| Localization on Bethe lattice: cavity & recursion (Valentina)]] [[Media:2025 P8 solutions.pdf| Sol Prob.8 ]]
|
* [[T-9| Localization on Bethe lattice: stability & mobility edge (Valentina)]] [[Media:2025 P9 solutions.pdf| Sol Prob.9 ]] 

|}



= Exercises =

<li> Week 1: [[Media:Exercises_1-3.pdf| Exercises 1-3 on extreme value statistics]]
</li>

<li> Week 2:
[[Media:Tutorial_and_Exercise_4.pdf| Tutorial and Exercise 4 on random matrices]] 
[[Media:Exercises_5-6.pdf| Exercises 5-6 on the random energy model]]
</li>

<li> Week 3:
[[Media:Exercises 7&8.pdf| Exercises 7-8 on interfaces]]
</li>

<li> Week 4:
[[Media:Exercises 9-10.pdf| Exercises 9-10 on glassiness]]
</li>

<li> Week 5 and 6:
[[Media:11-12_Exercises.pdf| Exercises 11-12 on dynamics]]
</li>

<li> Week 7:
[[Media:Exercises_13_15.pdf| Exercises 13-15 on branching and localization]]
</li>

<li> Week 8:
[[Media:Exercise_16-17.pdf| Exercises 16-17 on trap model and localization]]
</li>



= Evaluation and exam =

The exam will be on '''Monday, March 30th 2026, in room 105, Tours 24-25'''. It will be written, 3h long, from 2pm to 5pm. It consists of two parts:

Part 1: theory questions, see [[Sample questions|HERE]] for some examples.

Part 2: you will be asked to solve pieces of the 17 exercises given to you in advance.

'''You are not allowed to bring any material (printed notes, handwritten notes) nor to use any device during the exam.'''

All relevant formulas will be provided in the text of the exam. There will be one printed version of the WIKI pages available to you to consult.

L-9

2026-03-23T20:07:03Z

Rosso: /* Larkin model */

= Eigenstates =

Without disorder, the eigenstates are delocalized plane waves.

In the presence of disorder, three scenarios can arise:

* '''Delocalized eigenstates''', where the wavefunction remains extended over the whole system.
* '''Localized eigenstates''', where the wavefunction is exponentially confined to a finite region.
* '''Multifractal eigenstates''', occurring at the '''mobility edge''' of the Anderson transition, where the wavefunction exhibits a complex, scale–dependent structure.

In three dimensions, increasing the disorder strength leads to a transition between a metallic phase with delocalized eigenstates and an insulating phase with localized eigenstates. The critical energy separating these two regimes is called the mobility edge. Eigenstates at the mobility edge display multifractal statistics.

To characterize these different regimes it is useful to introduce the inverse participation ratio (IPR)

<math display="block">
\mathrm{IPR}(q)=\sum_n |\psi_n|^{2 q}.
</math>

For a system of linear size <math>L</math>, one typically observes a scaling

<math display="block">
\mathrm{IPR}(q) \sim L^{-\tau_q}.
</math>

The exponent <math>\tau_q</math> characterizes the spatial structure of the eigenstate.

== Delocalized eigenstates ==

In a delocalized state the probability is spread uniformly over the system:

<math display="block">
|\psi_n|^{2} \approx L^{-d}.
</math>

Therefore

<math display="block">
\mathrm{IPR}(q)
=
\sum_n |\psi_n|^{2q}
\sim L^d (L^{-d})^q
=
L^{d(1-q)}.
</math>

Thus

<math display="block">
\tau_q=d(q-1).
</math>

== Localized eigenstates ==

In a localized state the wavefunction is concentrated within a region of size <math>\xi_{\text{loc}}</math>.

Roughly

<math display="block">
|\psi_n|^2 \sim \xi_{\text{loc}}^{-d}
</math>

on <math>\xi_{\text{loc}}^d</math> sites and negligible elsewhere. Hence

<math display="block">
\mathrm{IPR}(q)\sim \text{const},
\qquad
\tau_q=0.
</math>

Thus localized states do not scale with system size.

== Multifractal eigenstates ==

At the mobility edge the eigenstates are neither extended nor localized. Instead, the amplitudes fluctuate strongly across the system.

The scaling of the IPR is characterized by a nonlinear function

<math display="block">
\tau_q.
</math>

The corresponding multifractal dimensions are defined as

<math display="block">
D_q=\frac{\tau_q}{q-1}.
</math>

It is useful to contrast fractal and multifractal scaling.

* In a '''fractal object''', a single exponent describes the scaling of the measure. If the wavefunction amplitudes scale with a single exponent <math>D</math>, then

<math display="block">
\mathrm{IPR}(q)\sim L^{-D(q-1)} .
</math>

All moments are controlled by the same dimension <math>D</math>.

* In a '''multifractal object''', different regions of the system scale with different exponents. In this case there is no single fractal dimension: different moments probe different effective dimensions of the wavefunction.

== Multifractal spectrum ==

To describe this structure it is useful to introduce the '''multifractal spectrum''' <math>f(\alpha)</math>.

We assume that the wavefunction amplitudes scale as

<math display="block">
|\psi_n|^2 \sim L^{-\alpha}
</math>

on approximately

<math display="block">
L^{f(\alpha)}
</math>

sites.

The function <math>f(\alpha)</math> therefore describes the fractal dimension of the set of sites where the wavefunction has exponent <math>\alpha</math>.

Using this representation,

<math display="block">
\mathrm{IPR}(q)
=
\sum_n |\psi_n|^{2q}
\sim
\int d\alpha \,
L^{-\alpha q} L^{f(\alpha)} .
</math>

For large <math>L</math>, the integral is dominated by the saddle point, giving

<math display="block">
\tau(q)=\min_{\alpha}\left(\alpha q-f(\alpha)\right).
</math>

Thus <math>\tau(q)</math> and <math>f(\alpha)</math> are related by a Legendre transform.

If <math>\alpha^*(q)</math> satisfies

<math display="block">
f'(\alpha^*(q))=q,
</math>

then

<math display="block">
\tau(q)=\alpha^*(q)q-f(\alpha^*(q)).
</math>

Multifractal wavefunctions exhibit a smooth spectrum with maximum

<math display="block">
f(\alpha_0)=d,
</math>

which corresponds to the most typical amplitude of the wavefunction.

It is useful to contrast this situation with the case of a simple fractal.

In a fractal object, the measure has a single scaling exponent. In our notation this means that the wavefunction amplitudes scale with a single value <math>\alpha_0</math> on a set of fractal dimension <math>D</math>:

<math display="block">
|\psi_n|^2 \sim L^{-\alpha_0}
</math>

on approximately

<math display="block">
L^{D}
</math>

sites, and are essentially negligible elsewhere.

In terms of the multifractal spectrum this corresponds to a trivial spectrum

<math display="block">
f(\alpha_0)=D,
\qquad
f(\alpha\neq\alpha_0)=-\infty.
</math>

By contrast, in a multifractal state the amplitudes are broadly distributed and many values of <math>\alpha</math> contribute. The function <math>f(\alpha)</math> then becomes a smooth curve describing the distribution of scaling exponents of the wavefunction.

Main Page

2026-03-23T12:26:32Z

Rosso: /* Where and When */

Welcome to the WIKI page of the M2 ICFP course on the Statistical Physics of Disordered Systems, second semester 2026.

= Where and When =

* Each Monday at 2pm - 6 pm, from January 19th to March 23rd. No lecture on 23/02/26.
* Room 202 in Jussieu campus, Tours 54-55 until 16th February
* Room 209 in Jussieu campus, Tours 56-66 from 2nd March
* Room 107 in Jussieu campus, Tours 24-34 on 23 March '''Attention: ROOM CHANGE LAST LECTURE!'''
* Each session is a mixture of lectures and exercises.

= The Team =

* [https://vale1925.wixsite.com/vros Valentina Ros] - vale1925@gmail.com
* [http://lptms.u-psud.fr/alberto_rosso/ Alberto Rosso] - alberto.rosso74@gmail.com

= Course description =

This course deals with systems in which the presence of impurities or amorphous structures (in other words, of disorder) influences radically the physics, generating novel phenomena. These phenomena involve the properties of the system at equilibrium (freezing and glass transitions), as well as their dynamical evolution out-of-equilibrium (pinning, avalanches), giving rise to ergodicity breaking both in absence and in presence of quantum fluctuations (classical metastability, quantum localization). We discuss the main statistical physics models that are able to capture the phenomenology of these systems, as well as the powerful theoretical tools (replica theory, large deviations, random matrix theory, scaling arguments, strong-disorder expansions) that have been developed to characterize quantitatively their physics. These theoretical tools nowadays have a huge impact in a variety of fields that go well-beyond statistical physics (computer science, probability, condensed matter, theoretical biology). Below is a list of topics discussed during the course.

'''Finite-dimensional disordered systems:'''

* Introduction to disordered systems and to the spin glass transition.
* Interface growth. Directed polymers in random media. Scenarios for the glass transition: the glass transition in KPZ in d>2.
* Depinning and avalanches. Bienaymé-Galton-Watson processes.
* Anderson localization: introduction. Localization in 1D: transfer matrix and Lyapunov.

'''Mean-field disordered systems:'''

* The simplest spin-glass: solution of the Random Energy Model.
* The replica method: the solution of the spherical p-spin model. Sketch of the solution of Sherrington Kirkpatrick model (full RSB).
* Towards glassy dynamics: rugged landscapes. Slow dynamics and aging: the trap model.
* The Anderson model on the Bethe lattice: the mobility edge.

=Lectures and tutorials=

'''If the layout of the formulas is bad, it might be because you are using Safari. Try opening the wiki with Firefox or Chrome.'''

{| class="wikitable" border="1"
|-
! width="100"|Date
! width="500"| 14h00-15h45
! width="500"| 16h00-17h45

|-valign="top"

| Week 1 (19/01)
|
* [[L1_ICTS| Spin Glass Transition (Alberto)]]

|
* [[T-I| A dictionary. The REM: energy landscape (Valentina)]]  [[Media:2025 P1 solutions.pdf| Sol Prob.1 ]]
|-valign=“top"

|-valign=“top"
| Week 2 (26/01)
|
* [[L2_ICFP| Stochastic Interfaces and growth (Alberto)]]
|
* [[T-2|The REM: freezing, condensation, glassiness (Valentina)]]   [[Media:2025 P2 solutions.pdf| Sol Prob.2 ]]
|-valign=“top"

|-valign=“top"
| Week 3 (02/02)
|
* [[L-3|Directed polymer in random media (Alberto)]]
|
* [[L-4| KPZ and glassiness in finite dimension (Alberto)]] [[https://colab.research.google.com/drive/1PTya42ZS2kU87A-BxQFFIUDTs_k47men?usp=sharing| notebook]]
|-valign=“top"
| Week 4 (9/02) and Week 5 (16/02)
|
* [[T-3| Spin glasses, equilibrium: replicas, the steps (Valentina)]]  [[Media:2025 P3 solutions.pdf| Sol Prob.3 ]]
|
* [[T-4| Spin glasses, equilibrium: replicas, the interpretation (Valentina)]]   [[Media:2025 P4 solutions.pdf| Sol Probs. 4 ]] 
[[Media:2025_Parisi_scheme.pdf| Notes: Probing states with replicas]]
|-valign=“top"
| Week 6 (02/03)
|
* [[LBan-IV| Driven Disordered Materials (Alberto)]] [[Media:DISSYTS.pdf| Slides ]]
|
* [[LBan-V| Avalanches in Disordered Materials (Alberto)]]
|-valign=“top"

| Week 7 (9/03)
|
* [[L-7| Anderson localization: introduction (Alberto)]]
|
* [[T-5| Rugged landscapes: counting metastable states (Valentina)]]  [[Media:2025 P5 solutions.pdf| Sol Prob.5 ]]
|-valign=“top"
| Week 8 (16/03)
|
* [[T-6| Rugged landscapes: stability of metastable states (Valentina)]] [[Media:2025 P666 solutions .pdf| Sol Prob.6 ]]
|
* [[T-7| Trap model and aging dynamics (Valentina)]] [[Media:2025 P7 solutions .pdf| Sol Prob.7 ]]
|-valign=“top"
| Week 9 (23/03)
|
* [[L-8| Localization in 1D, transfer matrix and Lyapunov exponent (Alberto)]] [[https://colab.research.google.com/drive/1ZJ0yvMrtflWNNmPfaRQ8KTteoWfm0bqk?usp=sharing| notebook]]
|
* [[L-9|Multifractality, tails (Alberto)]]
|-valign=“top"
| Extra (not in exam!)
|
* [[T-8| Localization on Bethe lattice: cavity & recursion (Valentina)]] [[Media:2025 P8 solutions.pdf| Sol Prob.8 ]]
|
* [[T-9| Localization on Bethe lattice: stability & mobility edge (Valentina)]] [[Media:2025 P9 solutions.pdf| Sol Prob.9 ]] 

|}



= Exercises =

<li> Week 1: [[Media:Exercises_1-3.pdf| Exercises 1-3 on extreme value statistics]]
</li>

<li> Week 2:
[[Media:Tutorial_and_Exercise_4.pdf| Tutorial and Exercise 4 on random matrices]] 
[[Media:Exercises_5-6.pdf| Exercises 5-6 on the random energy model]]
</li>

<li> Week 3:
[[Media:Exercises 7&8.pdf| Exercises 7-8 on interfaces]]
</li>

<li> Week 4:
[[Media:Exercises 9-10.pdf| Exercises 9-10 on glassiness]]
</li>

<li> Week 5 and 6:
[[Media:11-12_Exercises.pdf| Exercises 11-12 on dynamics]]
</li>

<li> Week 7:
[[Media:Exercises_13_15.pdf| Exercises 13-15 on branching and localization]]
</li>

<li> Week 8:
[[Media:Exercise_16-17.pdf| Exercises 16-17 on trap model and localization]]
</li>



= Evaluation and exam =

The exam will be on '''Monday, March 30th 2026'''. It will be written, 3h long. It consists of two parts:

Part 1: theory questions, see [[Sample questions|HERE]] for some examples.

Part 2: you will be asked to solve pieces of the 17 exercises given to you in advance.

'''You are not allowed to bring any material (printed notes, handwritten notes) nor to use any device during the exam.'''

All relevant formulas will be provided in the text of the exam. There will be one printed version of the WIKI pages available to you to consult.

L-8

2026-03-23T11:36:36Z

Rosso: /* Verification of Furstenberg's hypotheses for the Anderson model */

'''Goal.''' We introduce the Anderson model and study the statistical properties of its eigenstates.
In one dimension disorder leads to localization of all eigenstates, which can be understood using products of random matrices.

= Anderson model (tight-binding model) =

We consider non-interacting particles hopping between nearest-neighbour sites of a lattice in the presence of disorder.

The Hamiltonian reads

<math display="block">
H =
- t \sum_{\langle i,j\rangle} (c_i^\dagger c_j + c_j^\dagger c_i)
+
\sum_i V_i c_i^\dagger c_i .
</math>

The random variables <math>V_i</math> represent on-site disorder.
For simplicity we set <math>t=1</math>.

We assume that the disorder variables are independent and identically distributed, uniformly in the interval

<math display="block">
V_i \in \left(-\frac{W}{2},\frac{W}{2}\right).
</math>

In one dimension, the single-particle Hamiltonian is represented by the tridiagonal matrix

<math display="block">
H =
\begin{pmatrix}
V_1 & -1 & 0 & 0 & \dots \\
-1 & V_2 & -1 & 0 & \dots \\
0 & -1 & V_3 & -1 & \dots \\
0 & 0 & -1 & \ddots & -1 \\
\dots & \dots & \dots & -1 & V_L
\end{pmatrix}.
</math>

We study the statistical properties of the eigenvalue problem

<math display="block">
H\psi = \epsilon \psi ,
\qquad
\sum_{n=1}^L |\psi_n|^2 = 1 .
</math>

In the one-particle sector, this becomes a discrete Schrödinger equation.

---

== Density of states ==

Without disorder the dispersion relation is

<math display="block">
\epsilon(k) = -2\cos k,
\qquad
k\in(-\pi,\pi).
</math>

The energy band is therefore

<math display="block">
-2 < \epsilon < 2 .
</math>

The density of states is

<math display="block">
\rho(\epsilon)
=
\int_{-\pi}^{\pi}
\frac{dk}{2\pi}
\delta(\epsilon-\epsilon(k))
=
\frac{1}{\pi\sqrt{4-\epsilon^2}}
\qquad
(\epsilon\in(-2,2)).
</math>

In the presence of disorder the density of states broadens and becomes sample dependent.

== Transfer matrices ==

The discrete Schrödinger equation reads

<math display="block">
-\psi_{n+1} - \psi_{n-1} + V_n \psi_n = \epsilon \psi_n .
</math>

It can be rewritten as

<math display="block">
\begin{pmatrix}
\psi_{n+1} \\
\psi_n
\end{pmatrix}
=
T_n
\begin{pmatrix}
\psi_n \\
\psi_{n-1}
\end{pmatrix}
</math>

with

<math display="block">
T_n =
\begin{pmatrix}
V_n-\epsilon & -1 \\
1 & 0
\end{pmatrix}.
</math>

Iterating gives

<math display="block">
\begin{pmatrix}
\psi_{n+1} \\
\psi_n
\end{pmatrix}
=
\Pi_n
\begin{pmatrix}
\psi_1 \\
\psi_0
\end{pmatrix},
\qquad
\Pi_n = T_n T_{n-1} \cdots T_1 .
</math>

Thus the wavefunction is controlled by a product of random matrices.

=== Remark ===

The transfer matrix formulation rewrites the problem as a recursion, starting from initial data and propagating the solution along the chain.

This corresponds to a Cauchy problem rather than a boundary value problem. Nevertheless, the growth properties of the solutions contain direct information about the nature of the eigenstates: exponential growth is associated with localization, while bounded solutions correspond to extended states.

== Furstenberg theorem (physical formulation) ==

The exponential growth of the product of random matrices
<math display="block">
\Pi_n = T_n T_{n-1} \cdots T_1
</math>
can be understood as a matrix analogue of the law of large numbers. A set of simple sufficient conditions for this behavior is provided by a theorem of Harry Furstenberg.

To quantify this growth, we introduce the norm of a matrix <math>A</math> as
<math display="block">
\|A\|^2 = \frac{a_{11}^2 + a_{12}^2 + a_{21}^2 + a_{22}^2}{2}.
</math>

In physical terms, the mechanism behind the theorem relies on two key ingredients:

=== (i) Stretching ===

The transfer matrices must be able, with nonzero probability, to expand vectors.

More precisely, there exist realizations of <math>T_n</math> such that for some direction <math>v</math>,
<math display="block">
\|T_n v\| = \sigma_{\max}(T_n)\, \|v\|,
\qquad \sigma_{\max}(T_n) > 1.
</math>

Equivalently, <math>\sigma_{\max}^2(T_n)</math> is the largest eigenvalue of
<math display="block">
T_n^T T_n.
</math>

This ensures that typical products contain episodes of exponential amplification.

=== (ii) Mixing of directions ===

At each step, the transfer matrix changes the direction of the vector <math>(\psi_n,\psi_{n-1})</math>.

In the clean case, the same matrix is applied at every step, so the angular dynamics is deterministic and can be written as
<math display="block">
\theta_{n+1}=F(\theta_n).
</math>
Once the initial direction is fixed, the whole sequence is fixed. The system retains a perfect memory of its initial orientation.

In the disordered case, the transfer matrix varies from step to step, and the angular dynamics becomes
<math display="block">
\theta_{n+1}=F(\theta_n,T_n).
</math>
The direction is then continuously reshuffled and progressively loses memory of its initial value.

More importantly, there is no finite set of directions that is invariant under all transfer matrices. As a consequence, the dynamics does not get trapped into special directions and effectively explores the projective space.

This absence of invariant directions is what is meant by '''mixing of directions'''.

=== Consequence ===

Under these conditions, the norm of the product grows exponentially with probability one:
<math display="block">
\lim_{n\to\infty} \frac{1}{n} \log \|\Pi_n\| = \gamma > 0,
</math>
where <math>\gamma</math> is the Lyapunov exponent.

This result is the matrix analogue of the fact that the logarithm of a product of independent random variables becomes self-averaging (exercise 15).

== Verification of Furstenberg's hypotheses for the Anderson model ==

We now check explicitly that the transfer matrices of the one-dimensional Anderson model satisfy the two conditions stated above.

The transfer matrices are
<math display="block">
T_n =
\begin{pmatrix}
V_n-\epsilon & -1 \\
1 & 0
\end{pmatrix}.
</math>

=== (i) Stretching ===

We first verify that the matrices expand at least one direction with nonzero probability.

For a generic <math>2\times2</math> matrix, the maximal stretching factor is controlled by the largest eigenvalue of <math>T_n^T T_n</math>. Here
<math display="block">
T_n^T T_n =
\begin{pmatrix}
(V_n-\epsilon)^2+1 & -(V_n-\epsilon) \\
-(V_n-\epsilon) & 1
\end{pmatrix}.
</math>

The trace is
<math display="block">
\mathrm{Tr}(T_n^T T_n) = (V_n-\epsilon)^2 + 2,
</math>
and the determinant is
<math display="block">
\det(T_n^T T_n) = \det(T_n)^2 = 1.
</math>

Therefore the eigenvalues satisfy
<math display="block">
\lambda_+ \lambda_- = 1,
</math>
and the largest one obeys
<math display="block">
\lambda_+ \ge 1.
</math>

More precisely, as soon as <math>V_n-\epsilon \neq 0</math>, one has
<math display="block">
\lambda_+ > 1.
</math>

Thus the matrices expand at least one direction except for the fine-tuned case <math>V_n=\epsilon</math>, which has zero probability for a continuous disorder distribution. The stretching condition is therefore satisfied almost surely.

=== (ii) Mixing of directions ===

We now verify that the angular dynamics is not confined to a finite set of directions.

Define the angle
<math display="block">
\theta_n = \arctan\!\left(\frac{\psi_{n-1}}{\psi_n}\right),
</math>
which represents the direction of the vector <math>(\psi_n,\psi_{n-1})</math>.

The transfer matrix induces the map
<math display="block">
\theta_{n+1} =
\arctan\!\left(\frac{1}{V_n-\epsilon-\tan\theta_n}\right).
</math>

For fixed <math>\theta_n</math>, this expression depends continuously on the random variable <math>V_n</math>. If the disorder has a continuous distribution (for instance Gaussian), the image of a given angle is not restricted to a finite set.

As a consequence, the sequence <math>\theta_n</math> is not confined to a finite set of directions. More precisely, there is no finite collection of directions that is invariant under all transfer matrices.

The randomness of <math>V_n</math> continuously reshuffles the direction and prevents the dynamics from being trapped into special directions. This ensures that the projective dynamics effectively explores the angular space.

This is what is meant by '''mixing of directions''' in the present context.

=== Conclusion ===

The transfer matrices of the one-dimensional Anderson model satisfy both conditions:

# they expand at least one direction with nonzero probability;
# they do not confine the angular dynamics to a finite set.

As a consequence,
<math display="block">
\lim_{n\to\infty} \frac{1}{n}\log \|\Pi_n\| = \gamma > 0,
</math>
so the Lyapunov exponent is positive for generic disorder.

---

== Localization length ==

The transfer-matrix recursion corresponds to fixing the wavefunction at one boundary and propagating it through the system.

Typical solutions grow exponentially

<math display="block">
|\psi_n|\sim e^{\gamma n}.
</math>

However a physical eigenstate must satisfy boundary conditions at both ends of the system. Matching two such solutions leads to exponentially localized eigenstates

<math display="block">
|\psi_n|\sim e^{-|n-n_0|/\xi_{\text{loc}}}.
</math>

The localization length is

<math display="block">
\xi_{\text{loc}}(\epsilon)=\frac{1}{\gamma(\epsilon)}.
</math>

Thus in one dimension arbitrarily weak disorder localizes all eigenstates.
This result is consistent with the scaling theory of localization discussed earlier, which predicts that for <math>d\le2</math> disorder inevitably drives the system toward the insulating regime.

---

== Fluctuations ==

Quantities such as

<math>
|\psi_n|,\quad \|\Pi_n\|,\quad G
</math>

show strong sample-to-sample fluctuations, while their logarithm is self-averaging.

For instance

<math display="block">
\ln|\psi_n|
\sim
\gamma n + O(\sqrt n)
</math>

so that the logarithm of the wavefunction performs a random walk with a positive drift.

L-8

2026-03-23T11:34:14Z

Rosso: /* Furstenberg theorem (physical formulation) */

'''Goal.''' We introduce the Anderson model and study the statistical properties of its eigenstates.
In one dimension disorder leads to localization of all eigenstates, which can be understood using products of random matrices.

= Anderson model (tight-binding model) =

We consider non-interacting particles hopping between nearest-neighbour sites of a lattice in the presence of disorder.

The Hamiltonian reads

<math display="block">
H =
- t \sum_{\langle i,j\rangle} (c_i^\dagger c_j + c_j^\dagger c_i)
+
\sum_i V_i c_i^\dagger c_i .
</math>

The random variables <math>V_i</math> represent on-site disorder.
For simplicity we set <math>t=1</math>.

We assume that the disorder variables are independent and identically distributed, uniformly in the interval

<math display="block">
V_i \in \left(-\frac{W}{2},\frac{W}{2}\right).
</math>

In one dimension, the single-particle Hamiltonian is represented by the tridiagonal matrix

<math display="block">
H =
\begin{pmatrix}
V_1 & -1 & 0 & 0 & \dots \\
-1 & V_2 & -1 & 0 & \dots \\
0 & -1 & V_3 & -1 & \dots \\
0 & 0 & -1 & \ddots & -1 \\
\dots & \dots & \dots & -1 & V_L
\end{pmatrix}.
</math>

We study the statistical properties of the eigenvalue problem

<math display="block">
H\psi = \epsilon \psi ,
\qquad
\sum_{n=1}^L |\psi_n|^2 = 1 .
</math>

In the one-particle sector, this becomes a discrete Schrödinger equation.

---

== Density of states ==

Without disorder the dispersion relation is

<math display="block">
\epsilon(k) = -2\cos k,
\qquad
k\in(-\pi,\pi).
</math>

The energy band is therefore

<math display="block">
-2 < \epsilon < 2 .
</math>

The density of states is

<math display="block">
\rho(\epsilon)
=
\int_{-\pi}^{\pi}
\frac{dk}{2\pi}
\delta(\epsilon-\epsilon(k))
=
\frac{1}{\pi\sqrt{4-\epsilon^2}}
\qquad
(\epsilon\in(-2,2)).
</math>

In the presence of disorder the density of states broadens and becomes sample dependent.

== Transfer matrices ==

The discrete Schrödinger equation reads

<math display="block">
-\psi_{n+1} - \psi_{n-1} + V_n \psi_n = \epsilon \psi_n .
</math>

It can be rewritten as

<math display="block">
\begin{pmatrix}
\psi_{n+1} \\
\psi_n
\end{pmatrix}
=
T_n
\begin{pmatrix}
\psi_n \\
\psi_{n-1}
\end{pmatrix}
</math>

with

<math display="block">
T_n =
\begin{pmatrix}
V_n-\epsilon & -1 \\
1 & 0
\end{pmatrix}.
</math>

Iterating gives

<math display="block">
\begin{pmatrix}
\psi_{n+1} \\
\psi_n
\end{pmatrix}
=
\Pi_n
\begin{pmatrix}
\psi_1 \\
\psi_0
\end{pmatrix},
\qquad
\Pi_n = T_n T_{n-1} \cdots T_1 .
</math>

Thus the wavefunction is controlled by a product of random matrices.

=== Remark ===

The transfer matrix formulation rewrites the problem as a recursion, starting from initial data and propagating the solution along the chain.

This corresponds to a Cauchy problem rather than a boundary value problem. Nevertheless, the growth properties of the solutions contain direct information about the nature of the eigenstates: exponential growth is associated with localization, while bounded solutions correspond to extended states.

== Furstenberg theorem (physical formulation) ==

The exponential growth of the product of random matrices
<math display="block">
\Pi_n = T_n T_{n-1} \cdots T_1
</math>
can be understood as a matrix analogue of the law of large numbers. A set of simple sufficient conditions for this behavior is provided by a theorem of Harry Furstenberg.

To quantify this growth, we introduce the norm of a matrix <math>A</math> as
<math display="block">
\|A\|^2 = \frac{a_{11}^2 + a_{12}^2 + a_{21}^2 + a_{22}^2}{2}.
</math>

In physical terms, the mechanism behind the theorem relies on two key ingredients:

=== (i) Stretching ===

The transfer matrices must be able, with nonzero probability, to expand vectors.

More precisely, there exist realizations of <math>T_n</math> such that for some direction <math>v</math>,
<math display="block">
\|T_n v\| = \sigma_{\max}(T_n)\, \|v\|,
\qquad \sigma_{\max}(T_n) > 1.
</math>

Equivalently, <math>\sigma_{\max}^2(T_n)</math> is the largest eigenvalue of
<math display="block">
T_n^T T_n.
</math>

This ensures that typical products contain episodes of exponential amplification.

=== (ii) Mixing of directions ===

At each step, the transfer matrix changes the direction of the vector <math>(\psi_n,\psi_{n-1})</math>.

In the clean case, the same matrix is applied at every step, so the angular dynamics is deterministic and can be written as
<math display="block">
\theta_{n+1}=F(\theta_n).
</math>
Once the initial direction is fixed, the whole sequence is fixed. The system retains a perfect memory of its initial orientation.

In the disordered case, the transfer matrix varies from step to step, and the angular dynamics becomes
<math display="block">
\theta_{n+1}=F(\theta_n,T_n).
</math>
The direction is then continuously reshuffled and progressively loses memory of its initial value.

More importantly, there is no finite set of directions that is invariant under all transfer matrices. As a consequence, the dynamics does not get trapped into special directions and effectively explores the projective space.

This absence of invariant directions is what is meant by '''mixing of directions'''.

=== Consequence ===

Under these conditions, the norm of the product grows exponentially with probability one:
<math display="block">
\lim_{n\to\infty} \frac{1}{n} \log \|\Pi_n\| = \gamma > 0,
</math>
where <math>\gamma</math> is the Lyapunov exponent.

This result is the matrix analogue of the fact that the logarithm of a product of independent random variables becomes self-averaging (exercise 15).

== Verification of Furstenberg's hypotheses for the Anderson model ==

We now check explicitly that the transfer matrices of the one-dimensional Anderson model satisfy the two conditions stated above.

The transfer matrices are
<math display="block">
T_n =
\begin{pmatrix}
V_n-\epsilon & -1 \\
1 & 0
\end{pmatrix}.
</math>

=== (i) Stretching ===

We first verify that each matrix expands at least one direction.

For a generic <math>2\times2</math> matrix, the maximal stretching factor is controlled by the largest eigenvalue of <math>T_n^T T_n</math>. Here
<math display="block">
T_n^T T_n =
\begin{pmatrix}
(V_n-\epsilon)^2+1 & -(V_n-\epsilon) \\
-(V_n-\epsilon) & 1
\end{pmatrix}.
</math>

The trace is
<math display="block">
\mathrm{Tr}(T_n^T T_n) = (V_n-\epsilon)^2 + 2,
</math>
and the determinant is
<math display="block">
\det(T_n^T T_n) = \det(T_n)^2 = 1.
</math>

Therefore the eigenvalues satisfy
<math display="block">
\lambda_+ \lambda_- = 1,
</math>
and the largest one obeys
<math display="block">
\lambda_+ \ge 1.
</math>

More precisely, as soon as <math>V_n-\epsilon \neq 0</math>, one has
<math display="block">
\lambda_+ > 1.
</math>

Thus each matrix expands at least one direction, except for the fine-tuned case <math>V_n=\epsilon</math>, which has zero probability for a continuous disorder distribution. The stretching condition is therefore satisfied almost surely.

=== (ii) Mixing of directions ===

We now verify that the angular dynamics is not confined to a finite set of directions.

Define the angle
<math display="block">
\theta_n = \arctan\!\left(\frac{\psi_{n-1}}{\psi_n}\right),
</math>
which represents the direction of the vector <math>(\psi_n,\psi_{n-1})</math>.

The transfer matrix induces the map
<math display="block">
\theta_{n+1} =
\arctan\!\left(\frac{1}{V_n-\epsilon-\tan\theta_n}\right).
</math>

For fixed <math>\theta_n</math>, this expression depends continuously on the random variable <math>V_n</math>. If the disorder has a continuous distribution (for instance Gaussian), the image of a given angle is not restricted to a finite set.

As a consequence, the sequence <math>\theta_n</math> is not confined to a finite set of directions: under iteration of the product
<math display="block">
T_n T_{n-1} \cdots T_1,
</math>
the angular dynamics can access a continuum of directions.

This exploration is not required to be uniform; it is sufficient that all directions are accessible. This is precisely the mixing condition required by Furstenberg's theorem.

=== Conclusion ===

The transfer matrices of the one-dimensional Anderson model satisfy both conditions:

# they expand at least one direction;
# they do not confine the angular dynamics to a finite set.

As a consequence,
<math display="block">
\lim_{n\to\infty} \frac{1}{n}\log \|\Pi_n\| = \gamma > 0,
</math>
so the Lyapunov exponent is positive for generic disorder.

---

== Localization length ==

The transfer-matrix recursion corresponds to fixing the wavefunction at one boundary and propagating it through the system.

Typical solutions grow exponentially

<math display="block">
|\psi_n|\sim e^{\gamma n}.
</math>

However a physical eigenstate must satisfy boundary conditions at both ends of the system. Matching two such solutions leads to exponentially localized eigenstates

<math display="block">
|\psi_n|\sim e^{-|n-n_0|/\xi_{\text{loc}}}.
</math>

The localization length is

<math display="block">
\xi_{\text{loc}}(\epsilon)=\frac{1}{\gamma(\epsilon)}.
</math>

Thus in one dimension arbitrarily weak disorder localizes all eigenstates.
This result is consistent with the scaling theory of localization discussed earlier, which predicts that for <math>d\le2</math> disorder inevitably drives the system toward the insulating regime.

---

== Fluctuations ==

Quantities such as

<math>
|\psi_n|,\quad \|\Pi_n\|,\quad G
</math>

show strong sample-to-sample fluctuations, while their logarithm is self-averaging.

For instance

<math display="block">
\ln|\psi_n|
\sim
\gamma n + O(\sqrt n)
</math>

so that the logarithm of the wavefunction performs a random walk with a positive drift.

L-8

2026-03-23T06:16:40Z

Rosso: /* (ii) Mixing of directions */

'''Goal.''' We introduce the Anderson model and study the statistical properties of its eigenstates.
In one dimension disorder leads to localization of all eigenstates, which can be understood using products of random matrices.

= Anderson model (tight-binding model) =

We consider non-interacting particles hopping between nearest-neighbour sites of a lattice in the presence of disorder.

The Hamiltonian reads

<math display="block">
H =
- t \sum_{\langle i,j\rangle} (c_i^\dagger c_j + c_j^\dagger c_i)
+
\sum_i V_i c_i^\dagger c_i .
</math>

The random variables <math>V_i</math> represent on-site disorder.
For simplicity we set <math>t=1</math>.

We assume that the disorder variables are independent and identically distributed, uniformly in the interval

<math display="block">
V_i \in \left(-\frac{W}{2},\frac{W}{2}\right).
</math>

In one dimension, the single-particle Hamiltonian is represented by the tridiagonal matrix

<math display="block">
H =
\begin{pmatrix}
V_1 & -1 & 0 & 0 & \dots \\
-1 & V_2 & -1 & 0 & \dots \\
0 & -1 & V_3 & -1 & \dots \\
0 & 0 & -1 & \ddots & -1 \\
\dots & \dots & \dots & -1 & V_L
\end{pmatrix}.
</math>

We study the statistical properties of the eigenvalue problem

<math display="block">
H\psi = \epsilon \psi ,
\qquad
\sum_{n=1}^L |\psi_n|^2 = 1 .
</math>

In the one-particle sector, this becomes a discrete Schrödinger equation.

---

== Density of states ==

Without disorder the dispersion relation is

<math display="block">
\epsilon(k) = -2\cos k,
\qquad
k\in(-\pi,\pi).
</math>

The energy band is therefore

<math display="block">
-2 < \epsilon < 2 .
</math>

The density of states is

<math display="block">
\rho(\epsilon)
=
\int_{-\pi}^{\pi}
\frac{dk}{2\pi}
\delta(\epsilon-\epsilon(k))
=
\frac{1}{\pi\sqrt{4-\epsilon^2}}
\qquad
(\epsilon\in(-2,2)).
</math>

In the presence of disorder the density of states broadens and becomes sample dependent.

== Transfer matrices ==

The discrete Schrödinger equation reads

<math display="block">
-\psi_{n+1} - \psi_{n-1} + V_n \psi_n = \epsilon \psi_n .
</math>

It can be rewritten as

<math display="block">
\begin{pmatrix}
\psi_{n+1} \\
\psi_n
\end{pmatrix}
=
T_n
\begin{pmatrix}
\psi_n \\
\psi_{n-1}
\end{pmatrix}
</math>

with

<math display="block">
T_n =
\begin{pmatrix}
V_n-\epsilon & -1 \\
1 & 0
\end{pmatrix}.
</math>

Iterating gives

<math display="block">
\begin{pmatrix}
\psi_{n+1} \\
\psi_n
\end{pmatrix}
=
\Pi_n
\begin{pmatrix}
\psi_1 \\
\psi_0
\end{pmatrix},
\qquad
\Pi_n = T_n T_{n-1} \cdots T_1 .
</math>

Thus the wavefunction is controlled by a product of random matrices.

=== Remark ===

The transfer matrix formulation rewrites the problem as a recursion, starting from initial data and propagating the solution along the chain.

This corresponds to a Cauchy problem rather than a boundary value problem. Nevertheless, the growth properties of the solutions contain direct information about the nature of the eigenstates: exponential growth is associated with localization, while bounded solutions correspond to extended states.

== Furstenberg theorem (physical formulation) ==

The exponential growth of the product of random matrices
<math display="block">
\Pi_n = T_n T_{n-1} \cdots T_1
</math>
can be understood as a matrix analogue of the law of large numbers. A sufficient condition for this behavior is provided by a theorem of Harry Furstenberg.

To quantify this growth, we introduce the norm of a matrix <math>A</math> as
<math display="block">
\|A\|^2 = \frac{a_{11}^2 + a_{12}^2 + a_{21}^2 + a_{22}^2}{2}.
</math>

In physical terms, the theorem relies on two key ingredients:

=== (i) Stretching ===

Each matrix must be able to expand at least one direction. This means that for each <math>n</math> there exists a vector <math>v</math> such that
<math display="block">
\|T_n v\| = \sigma_{\max}(T_n)\, \|v\|,
\qquad \sigma_{\max}(T_n) > 1.
</math>

Equivalently, <math>\sigma_{\max}^2(T_n)</math> is the largest eigenvalue of the symmetric matrix
<math display="block">
T_n^T T_n.
</math>

=== (ii) Mixing of directions ===

At each step, the transfer matrix changes the direction of the vector <math>(\psi_n,\psi_{n-1})</math>.

In the clean case, the same matrix is applied at every step, so the angular dynamics is deterministic and can be written as
<math display="block">
\theta_{n+1}=F(\theta_n).
</math>
Once the initial direction is fixed, the whole sequence is fixed. The direction may change, but it keeps a perfect memory of its initial value.

In the disordered case, the transfer matrix varies from step to step, and the angular dynamics becomes
<math display="block">
\theta_{n+1}=F(\theta_n,T_n).
</math>
The direction is then continuously reshuffled and progressively loses memory of its initial orientation.

More precisely, there is no finite set of directions that is invariant under all transfer matrices. The dynamics therefore cannot be reduced to a finite set of preferred directions.

This is what is meant by '''mixing of directions''', and it is the key condition required by Furstenberg's theorem.

=== Consequence ===

Under these conditions, the norm of the product grows exponentially with probability one:
<math display="block">
\lim_{n\to\infty} \frac{1}{n} \log \|\Pi_n\| = \gamma > 0,
</math>
where <math>\gamma</math> is the Lyapunov exponent.

This result is the matrix analogue of the fact that the logarithm of a product of independent random variables becomes self-averaging (see Exercise 15).

== Verification of Furstenberg's hypotheses for the Anderson model ==

We now check explicitly that the transfer matrices of the one-dimensional Anderson model satisfy the two conditions stated above.

The transfer matrices are
<math display="block">
T_n =
\begin{pmatrix}
V_n-\epsilon & -1 \\
1 & 0
\end{pmatrix}.
</math>

=== (i) Stretching ===

We first verify that each matrix expands at least one direction.

For a generic <math>2\times2</math> matrix, the maximal stretching factor is controlled by the largest eigenvalue of <math>T_n^T T_n</math>. Here
<math display="block">
T_n^T T_n =
\begin{pmatrix}
(V_n-\epsilon)^2+1 & -(V_n-\epsilon) \\
-(V_n-\epsilon) & 1
\end{pmatrix}.
</math>

The trace is
<math display="block">
\mathrm{Tr}(T_n^T T_n) = (V_n-\epsilon)^2 + 2,
</math>
and the determinant is
<math display="block">
\det(T_n^T T_n) = \det(T_n)^2 = 1.
</math>

Therefore the eigenvalues satisfy
<math display="block">
\lambda_+ \lambda_- = 1,
</math>
and the largest one obeys
<math display="block">
\lambda_+ \ge 1.
</math>

More precisely, as soon as <math>V_n-\epsilon \neq 0</math>, one has
<math display="block">
\lambda_+ > 1.
</math>

Thus each matrix expands at least one direction, except for the fine-tuned case <math>V_n=\epsilon</math>, which has zero probability for a continuous disorder distribution. The stretching condition is therefore satisfied almost surely.

=== (ii) Mixing of directions ===

We now verify that the angular dynamics is not confined to a finite set of directions.

Define the angle
<math display="block">
\theta_n = \arctan\!\left(\frac{\psi_{n-1}}{\psi_n}\right),
</math>
which represents the direction of the vector <math>(\psi_n,\psi_{n-1})</math>.

The transfer matrix induces the map
<math display="block">
\theta_{n+1} =
\arctan\!\left(\frac{1}{V_n-\epsilon-\tan\theta_n}\right).
</math>

For fixed <math>\theta_n</math>, this expression depends continuously on the random variable <math>V_n</math>. If the disorder has a continuous distribution (for instance Gaussian), the image of a given angle is not restricted to a finite set.

As a consequence, the sequence <math>\theta_n</math> is not confined to a finite set of directions: under iteration of the product
<math display="block">
T_n T_{n-1} \cdots T_1,
</math>
the angular dynamics can access a continuum of directions.

This exploration is not required to be uniform; it is sufficient that all directions are accessible. This is precisely the mixing condition required by Furstenberg's theorem.

=== Conclusion ===

The transfer matrices of the one-dimensional Anderson model satisfy both conditions:

# they expand at least one direction;
# they do not confine the angular dynamics to a finite set.

As a consequence,
<math display="block">
\lim_{n\to\infty} \frac{1}{n}\log \|\Pi_n\| = \gamma > 0,
</math>
so the Lyapunov exponent is positive for generic disorder.

---

== Localization length ==

The transfer-matrix recursion corresponds to fixing the wavefunction at one boundary and propagating it through the system.

Typical solutions grow exponentially

<math display="block">
|\psi_n|\sim e^{\gamma n}.
</math>

However a physical eigenstate must satisfy boundary conditions at both ends of the system. Matching two such solutions leads to exponentially localized eigenstates

<math display="block">
|\psi_n|\sim e^{-|n-n_0|/\xi_{\text{loc}}}.
</math>

The localization length is

<math display="block">
\xi_{\text{loc}}(\epsilon)=\frac{1}{\gamma(\epsilon)}.
</math>

Thus in one dimension arbitrarily weak disorder localizes all eigenstates.
This result is consistent with the scaling theory of localization discussed earlier, which predicts that for <math>d\le2</math> disorder inevitably drives the system toward the insulating regime.

---

== Fluctuations ==

Quantities such as

<math>
|\psi_n|,\quad \|\Pi_n\|,\quad G
</math>

show strong sample-to-sample fluctuations, while their logarithm is self-averaging.

For instance

<math display="block">
\ln|\psi_n|
\sim
\gamma n + O(\sqrt n)
</math>

so that the logarithm of the wavefunction performs a random walk with a positive drift.

L-8

2026-03-22T19:10:26Z

Rosso: /* (ii) Mixing of directions */

'''Goal.''' We introduce the Anderson model and study the statistical properties of its eigenstates.
In one dimension disorder leads to localization of all eigenstates, which can be understood using products of random matrices.

= Anderson model (tight-binding model) =

We consider non-interacting particles hopping between nearest-neighbour sites of a lattice in the presence of disorder.

The Hamiltonian reads

<math display="block">
H =
- t \sum_{\langle i,j\rangle} (c_i^\dagger c_j + c_j^\dagger c_i)
+
\sum_i V_i c_i^\dagger c_i .
</math>

The random variables <math>V_i</math> represent on-site disorder.
For simplicity we set <math>t=1</math>.

We assume that the disorder variables are independent and identically distributed, uniformly in the interval

<math display="block">
V_i \in \left(-\frac{W}{2},\frac{W}{2}\right).
</math>

In one dimension, the single-particle Hamiltonian is represented by the tridiagonal matrix

<math display="block">
H =
\begin{pmatrix}
V_1 & -1 & 0 & 0 & \dots \\
-1 & V_2 & -1 & 0 & \dots \\
0 & -1 & V_3 & -1 & \dots \\
0 & 0 & -1 & \ddots & -1 \\
\dots & \dots & \dots & -1 & V_L
\end{pmatrix}.
</math>

We study the statistical properties of the eigenvalue problem

<math display="block">
H\psi = \epsilon \psi ,
\qquad
\sum_{n=1}^L |\psi_n|^2 = 1 .
</math>

In the one-particle sector, this becomes a discrete Schrödinger equation.

---

== Density of states ==

Without disorder the dispersion relation is

<math display="block">
\epsilon(k) = -2\cos k,
\qquad
k\in(-\pi,\pi).
</math>

The energy band is therefore

<math display="block">
-2 < \epsilon < 2 .
</math>

The density of states is

<math display="block">
\rho(\epsilon)
=
\int_{-\pi}^{\pi}
\frac{dk}{2\pi}
\delta(\epsilon-\epsilon(k))
=
\frac{1}{\pi\sqrt{4-\epsilon^2}}
\qquad
(\epsilon\in(-2,2)).
</math>

In the presence of disorder the density of states broadens and becomes sample dependent.

== Transfer matrices ==

The discrete Schrödinger equation reads

<math display="block">
-\psi_{n+1} - \psi_{n-1} + V_n \psi_n = \epsilon \psi_n .
</math>

It can be rewritten as

<math display="block">
\begin{pmatrix}
\psi_{n+1} \\
\psi_n
\end{pmatrix}
=
T_n
\begin{pmatrix}
\psi_n \\
\psi_{n-1}
\end{pmatrix}
</math>

with

<math display="block">
T_n =
\begin{pmatrix}
V_n-\epsilon & -1 \\
1 & 0
\end{pmatrix}.
</math>

Iterating gives

<math display="block">
\begin{pmatrix}
\psi_{n+1} \\
\psi_n
\end{pmatrix}
=
\Pi_n
\begin{pmatrix}
\psi_1 \\
\psi_0
\end{pmatrix},
\qquad
\Pi_n = T_n T_{n-1} \cdots T_1 .
</math>

Thus the wavefunction is controlled by a product of random matrices.

=== Remark ===

The transfer matrix formulation rewrites the problem as a recursion, starting from initial data and propagating the solution along the chain.

This corresponds to a Cauchy problem rather than a boundary value problem. Nevertheless, the growth properties of the solutions contain direct information about the nature of the eigenstates: exponential growth is associated with localization, while bounded solutions correspond to extended states.

== Furstenberg theorem (physical formulation) ==

The exponential growth of the product of random matrices
<math display="block">
\Pi_n = T_n T_{n-1} \cdots T_1
</math>
can be understood as a matrix analogue of the law of large numbers. A sufficient condition for this behavior is provided by a theorem of Harry Furstenberg.

To quantify this growth, we introduce the norm of a matrix <math>A</math> as
<math display="block">
\|A\|^2 = \frac{a_{11}^2 + a_{12}^2 + a_{21}^2 + a_{22}^2}{2}.
</math>

In physical terms, the theorem relies on two key ingredients:

=== (i) Stretching ===

Each matrix must be able to expand at least one direction. This means that for each <math>n</math> there exists a vector <math>v</math> such that
<math display="block">
\|T_n v\| = \sigma_{\max}(T_n)\, \|v\|,
\qquad \sigma_{\max}(T_n) > 1.
</math>

Equivalently, <math>\sigma_{\max}^2(T_n)</math> is the largest eigenvalue of the symmetric matrix
<math display="block">
T_n^T T_n.
</math>

=== (ii) Mixing of directions ===

The angular dynamics is not confined to a finite set of directions: under iteration of the product
<math display="block">
T_n T_{n-1} \cdots T_1,
</math>
the direction of a vector does not preserve any finite set of directions

=== Consequence ===

Under these conditions, the norm of the product grows exponentially with probability one:
<math display="block">
\lim_{n\to\infty} \frac{1}{n} \log \|\Pi_n\| = \gamma > 0,
</math>
where <math>\gamma</math> is the Lyapunov exponent.

This result is the matrix analogue of the fact that the logarithm of a product of independent random variables becomes self-averaging (see Exercise 15).

== Verification of Furstenberg's hypotheses for the Anderson model ==

We now check explicitly that the transfer matrices of the one-dimensional Anderson model satisfy the two conditions stated above.

The transfer matrices are
<math display="block">
T_n =
\begin{pmatrix}
V_n-\epsilon & -1 \\
1 & 0
\end{pmatrix}.
</math>

=== (i) Stretching ===

We first verify that each matrix expands at least one direction.

For a generic <math>2\times2</math> matrix, the maximal stretching factor is controlled by the largest eigenvalue of <math>T_n^T T_n</math>. Here
<math display="block">
T_n^T T_n =
\begin{pmatrix}
(V_n-\epsilon)^2+1 & -(V_n-\epsilon) \\
-(V_n-\epsilon) & 1
\end{pmatrix}.
</math>

The trace is
<math display="block">
\mathrm{Tr}(T_n^T T_n) = (V_n-\epsilon)^2 + 2,
</math>
and the determinant is
<math display="block">
\det(T_n^T T_n) = \det(T_n)^2 = 1.
</math>

Therefore the eigenvalues satisfy
<math display="block">
\lambda_+ \lambda_- = 1,
</math>
and the largest one obeys
<math display="block">
\lambda_+ \ge 1.
</math>

More precisely, as soon as <math>V_n-\epsilon \neq 0</math>, one has
<math display="block">
\lambda_+ > 1.
</math>

Thus each matrix expands at least one direction, except for the fine-tuned case <math>V_n=\epsilon</math>, which has zero probability for a continuous disorder distribution. The stretching condition is therefore satisfied almost surely.

=== (ii) Mixing of directions ===

We now verify that the angular dynamics is not confined to a finite set of directions.

Define the angle
<math display="block">
\theta_n = \arctan\!\left(\frac{\psi_{n-1}}{\psi_n}\right),
</math>
which represents the direction of the vector <math>(\psi_n,\psi_{n-1})</math>.

The transfer matrix induces the map
<math display="block">
\theta_{n+1} =
\arctan\!\left(\frac{1}{V_n-\epsilon-\tan\theta_n}\right).
</math>

For fixed <math>\theta_n</math>, this expression depends continuously on the random variable <math>V_n</math>. If the disorder has a continuous distribution (for instance Gaussian), the image of a given angle is not restricted to a finite set.

As a consequence, the sequence <math>\theta_n</math> is not confined to a finite set of directions: under iteration of the product
<math display="block">
T_n T_{n-1} \cdots T_1,
</math>
the angular dynamics can access a continuum of directions.

This exploration is not required to be uniform; it is sufficient that all directions are accessible. This is precisely the mixing condition required by Furstenberg's theorem.

=== Conclusion ===

The transfer matrices of the one-dimensional Anderson model satisfy both conditions:

# they expand at least one direction;
# they do not confine the angular dynamics to a finite set.

As a consequence,
<math display="block">
\lim_{n\to\infty} \frac{1}{n}\log \|\Pi_n\| = \gamma > 0,
</math>
so the Lyapunov exponent is positive for generic disorder.

---

== Localization length ==

The transfer-matrix recursion corresponds to fixing the wavefunction at one boundary and propagating it through the system.

Typical solutions grow exponentially

<math display="block">
|\psi_n|\sim e^{\gamma n}.
</math>

However a physical eigenstate must satisfy boundary conditions at both ends of the system. Matching two such solutions leads to exponentially localized eigenstates

<math display="block">
|\psi_n|\sim e^{-|n-n_0|/\xi_{\text{loc}}}.
</math>

The localization length is

<math display="block">
\xi_{\text{loc}}(\epsilon)=\frac{1}{\gamma(\epsilon)}.
</math>

Thus in one dimension arbitrarily weak disorder localizes all eigenstates.
This result is consistent with the scaling theory of localization discussed earlier, which predicts that for <math>d\le2</math> disorder inevitably drives the system toward the insulating regime.

---

== Fluctuations ==

Quantities such as

<math>
|\psi_n|,\quad \|\Pi_n\|,\quad G
</math>

show strong sample-to-sample fluctuations, while their logarithm is self-averaging.

For instance

<math display="block">
\ln|\psi_n|
\sim
\gamma n + O(\sqrt n)
</math>

so that the logarithm of the wavefunction performs a random walk with a positive drift.

L-8

2026-03-22T15:03:57Z

Rosso: /* Density of states */

'''Goal.''' We introduce the Anderson model and study the statistical properties of its eigenstates.
In one dimension disorder leads to localization of all eigenstates, which can be understood using products of random matrices.

= Anderson model (tight-binding model) =

We consider non-interacting particles hopping between nearest-neighbour sites of a lattice in the presence of disorder.

The Hamiltonian reads

<math display="block">
H =
- t \sum_{\langle i,j\rangle} (c_i^\dagger c_j + c_j^\dagger c_i)
+
\sum_i V_i c_i^\dagger c_i .
</math>

The random variables <math>V_i</math> represent on-site disorder.
For simplicity we set <math>t=1</math>.

We assume that the disorder variables are independent and identically distributed, uniformly in the interval

<math display="block">
V_i \in \left(-\frac{W}{2},\frac{W}{2}\right).
</math>

In one dimension, the single-particle Hamiltonian is represented by the tridiagonal matrix

<math display="block">
H =
\begin{pmatrix}
V_1 & -1 & 0 & 0 & \dots \\
-1 & V_2 & -1 & 0 & \dots \\
0 & -1 & V_3 & -1 & \dots \\
0 & 0 & -1 & \ddots & -1 \\
\dots & \dots & \dots & -1 & V_L
\end{pmatrix}.
</math>

We study the statistical properties of the eigenvalue problem

<math display="block">
H\psi = \epsilon \psi ,
\qquad
\sum_{n=1}^L |\psi_n|^2 = 1 .
</math>

In the one-particle sector, this becomes a discrete Schrödinger equation.

---

== Density of states ==

Without disorder the dispersion relation is

<math display="block">
\epsilon(k) = -2\cos k,
\qquad
k\in(-\pi,\pi).
</math>

The energy band is therefore

<math display="block">
-2 < \epsilon < 2 .
</math>

The density of states is

<math display="block">
\rho(\epsilon)
=
\int_{-\pi}^{\pi}
\frac{dk}{2\pi}
\delta(\epsilon-\epsilon(k))
=
\frac{1}{\pi\sqrt{4-\epsilon^2}}
\qquad
(\epsilon\in(-2,2)).
</math>

In the presence of disorder the density of states broadens and becomes sample dependent.

== Transfer matrices ==

The discrete Schrödinger equation reads

<math display="block">
-\psi_{n+1} - \psi_{n-1} + V_n \psi_n = \epsilon \psi_n .
</math>

It can be rewritten as

<math display="block">
\begin{pmatrix}
\psi_{n+1} \\
\psi_n
\end{pmatrix}
=
T_n
\begin{pmatrix}
\psi_n \\
\psi_{n-1}
\end{pmatrix}
</math>

with

<math display="block">
T_n =
\begin{pmatrix}
V_n-\epsilon & -1 \\
1 & 0
\end{pmatrix}.
</math>

Iterating gives

<math display="block">
\begin{pmatrix}
\psi_{n+1} \\
\psi_n
\end{pmatrix}
=
\Pi_n
\begin{pmatrix}
\psi_1 \\
\psi_0
\end{pmatrix},
\qquad
\Pi_n = T_n T_{n-1} \cdots T_1 .
</math>

Thus the wavefunction is controlled by a product of random matrices.

=== Remark ===

The transfer matrix formulation rewrites the problem as a recursion, starting from initial data and propagating the solution along the chain.

This corresponds to a Cauchy problem rather than a boundary value problem. Nevertheless, the growth properties of the solutions contain direct information about the nature of the eigenstates: exponential growth is associated with localization, while bounded solutions correspond to extended states.

== Furstenberg theorem (physical formulation) ==

The exponential growth of the product of random matrices
<math display="block">
\Pi_n = T_n T_{n-1} \cdots T_1
</math>
can be understood as a matrix analogue of the law of large numbers. A sufficient condition for this behavior is provided by a theorem of Harry Furstenberg.

To quantify this growth, we introduce the norm of a matrix <math>A</math> as
<math display="block">
\|A\|^2 = \frac{a_{11}^2 + a_{12}^2 + a_{21}^2 + a_{22}^2}{2}.
</math>

In physical terms, the theorem relies on two key ingredients:

=== (i) Stretching ===

Each matrix must be able to expand at least one direction. This means that for each <math>n</math> there exists a vector <math>v</math> such that
<math display="block">
\|T_n v\| = \sigma_{\max}(T_n)\, \|v\|,
\qquad \sigma_{\max}(T_n) > 1.
</math>

Equivalently, <math>\sigma_{\max}^2(T_n)</math> is the largest eigenvalue of the symmetric matrix
<math display="block">
T_n^T T_n.
</math>

=== (ii) Mixing of directions ===

The angular dynamics is not confined: under iteration of the product
<math display="block">
T_n T_{n-1} \cdots T_1,
</math>
the direction of a vector can explore the whole angular space. This exploration is not required to be uniform; it is sufficient that all directions are accessible.

=== Consequence ===

Under these conditions, the norm of the product grows exponentially with probability one:
<math display="block">
\lim_{n\to\infty} \frac{1}{n} \log \|\Pi_n\| = \gamma > 0,
</math>
where <math>\gamma</math> is the Lyapunov exponent.

This result is the matrix analogue of the fact that the logarithm of a product of independent random variables becomes self-averaging (see Exercise 15).

== Verification of Furstenberg's hypotheses for the Anderson model ==

We now check explicitly that the transfer matrices of the one-dimensional Anderson model satisfy the two conditions stated above.

The transfer matrices are
<math display="block">
T_n =
\begin{pmatrix}
V_n-\epsilon & -1 \\
1 & 0
\end{pmatrix}.
</math>

=== (i) Stretching ===

We first verify that each matrix expands at least one direction.

For a generic <math>2\times2</math> matrix, the maximal stretching factor is controlled by the largest eigenvalue of <math>T_n^T T_n</math>. Here
<math display="block">
T_n^T T_n =
\begin{pmatrix}
(V_n-\epsilon)^2+1 & -(V_n-\epsilon) \\
-(V_n-\epsilon) & 1
\end{pmatrix}.
</math>

The trace is
<math display="block">
\mathrm{Tr}(T_n^T T_n) = (V_n-\epsilon)^2 + 2,
</math>
and the determinant is
<math display="block">
\det(T_n^T T_n) = \det(T_n)^2 = 1.
</math>

Therefore the eigenvalues satisfy
<math display="block">
\lambda_+ \lambda_- = 1,
</math>
and the largest one obeys
<math display="block">
\lambda_+ \ge 1.
</math>

More precisely, as soon as <math>V_n-\epsilon \neq 0</math>, one has
<math display="block">
\lambda_+ > 1.
</math>

Thus each matrix expands at least one direction, except for the fine-tuned case <math>V_n=\epsilon</math>, which has zero probability for a continuous disorder distribution. The stretching condition is therefore satisfied almost surely.

=== (ii) Mixing of directions ===

We now verify that the angular dynamics is not confined to a finite set of directions.

Define the angle
<math display="block">
\theta_n = \arctan\!\left(\frac{\psi_{n-1}}{\psi_n}\right),
</math>
which represents the direction of the vector <math>(\psi_n,\psi_{n-1})</math>.

The transfer matrix induces the map
<math display="block">
\theta_{n+1} =
\arctan\!\left(\frac{1}{V_n-\epsilon-\tan\theta_n}\right).
</math>

For fixed <math>\theta_n</math>, this expression depends continuously on the random variable <math>V_n</math>. If the disorder has a continuous distribution (for instance Gaussian), the image of a given angle is not restricted to a finite set.

As a consequence, the sequence <math>\theta_n</math> is not confined to a finite set of directions: under iteration of the product
<math display="block">
T_n T_{n-1} \cdots T_1,
</math>
the angular dynamics can access a continuum of directions.

This exploration is not required to be uniform; it is sufficient that all directions are accessible. This is precisely the mixing condition required by Furstenberg's theorem.

=== Conclusion ===

The transfer matrices of the one-dimensional Anderson model satisfy both conditions:

# they expand at least one direction;
# they do not confine the angular dynamics to a finite set.

As a consequence,
<math display="block">
\lim_{n\to\infty} \frac{1}{n}\log \|\Pi_n\| = \gamma > 0,
</math>
so the Lyapunov exponent is positive for generic disorder.

---

== Localization length ==

The transfer-matrix recursion corresponds to fixing the wavefunction at one boundary and propagating it through the system.

Typical solutions grow exponentially

<math display="block">
|\psi_n|\sim e^{\gamma n}.
</math>

However a physical eigenstate must satisfy boundary conditions at both ends of the system. Matching two such solutions leads to exponentially localized eigenstates

<math display="block">
|\psi_n|\sim e^{-|n-n_0|/\xi_{\text{loc}}}.
</math>

The localization length is

<math display="block">
\xi_{\text{loc}}(\epsilon)=\frac{1}{\gamma(\epsilon)}.
</math>

Thus in one dimension arbitrarily weak disorder localizes all eigenstates.
This result is consistent with the scaling theory of localization discussed earlier, which predicts that for <math>d\le2</math> disorder inevitably drives the system toward the insulating regime.

---

== Fluctuations ==

Quantities such as

<math>
|\psi_n|,\quad \|\Pi_n\|,\quad G
</math>

show strong sample-to-sample fluctuations, while their logarithm is self-averaging.

For instance

<math display="block">
\ln|\psi_n|
\sim
\gamma n + O(\sqrt n)
</math>

so that the logarithm of the wavefunction performs a random walk with a positive drift.

L-8

2026-03-21T15:58:06Z

Rosso: /* Transfer matrices */

'''Goal.''' We introduce the Anderson model and study the statistical properties of its eigenstates.
In one dimension disorder leads to localization of all eigenstates, which can be understood using products of random matrices.

= Anderson model (tight-binding model) =

We consider non-interacting particles hopping between nearest-neighbour sites of a lattice in the presence of disorder.

The Hamiltonian reads

<math display="block">
H =
- t \sum_{\langle i,j\rangle} (c_i^\dagger c_j + c_j^\dagger c_i)
+
\sum_i V_i c_i^\dagger c_i .
</math>

The random variables <math>V_i</math> represent on-site disorder.
For simplicity we set <math>t=1</math>.

We assume that the disorder variables are independent and identically distributed, uniformly in the interval

<math display="block">
V_i \in \left(-\frac{W}{2},\frac{W}{2}\right).
</math>

In one dimension, the single-particle Hamiltonian is represented by the tridiagonal matrix

<math display="block">
H =
\begin{pmatrix}
V_1 & -1 & 0 & 0 & \dots \\
-1 & V_2 & -1 & 0 & \dots \\
0 & -1 & V_3 & -1 & \dots \\
0 & 0 & -1 & \ddots & -1 \\
\dots & \dots & \dots & -1 & V_L
\end{pmatrix}.
</math>

We study the statistical properties of the eigenvalue problem

<math display="block">
H\psi = \epsilon \psi ,
\qquad
\sum_{n=1}^L |\psi_n|^2 = 1 .
</math>

In the one-particle sector, this becomes a discrete Schrödinger equation.

---

== Density of states ==

Without disorder the dispersion relation is

<math display="block">
\epsilon(k) = -2\cos k,
\qquad
k\in(-\pi,\pi).
</math>

The energy band is therefore

<math display="block">
-2 < \epsilon < 2 .
</math>

The density of states is

<math display="block">
\rho(\epsilon)
=
\int_{-\pi}^{\pi}
\frac{dk}{2\pi}
\delta(\epsilon-\epsilon(k))
=
\frac{1}{\pi\sqrt{4-\epsilon^2}}
\qquad
(\epsilon\in(-2,2)).
</math>

In the presence of disorder the density of states broadens and becomes sample dependent.

---

== Transfer matrices ==

The discrete Schrödinger equation reads

<math display="block">
-\psi_{n+1} - \psi_{n-1} + V_n \psi_n = \epsilon \psi_n .
</math>

It can be rewritten as

<math display="block">
\begin{pmatrix}
\psi_{n+1} \\
\psi_n
\end{pmatrix}
=
T_n
\begin{pmatrix}
\psi_n \\
\psi_{n-1}
\end{pmatrix}
</math>

with

<math display="block">
T_n =
\begin{pmatrix}
V_n-\epsilon & -1 \\
1 & 0
\end{pmatrix}.
</math>

Iterating gives

<math display="block">
\begin{pmatrix}
\psi_{n+1} \\
\psi_n
\end{pmatrix}
=
\Pi_n
\begin{pmatrix}
\psi_1 \\
\psi_0
\end{pmatrix},
\qquad
\Pi_n = T_n T_{n-1} \cdots T_1 .
</math>

Thus the wavefunction is controlled by a product of random matrices.

=== Remark ===

The transfer matrix formulation rewrites the problem as a recursion, starting from initial data and propagating the solution along the chain.

This corresponds to a Cauchy problem rather than a boundary value problem. Nevertheless, the growth properties of the solutions contain direct information about the nature of the eigenstates: exponential growth is associated with localization, while bounded solutions correspond to extended states.

== Furstenberg theorem (physical formulation) ==

The exponential growth of the product of random matrices
<math display="block">
\Pi_n = T_n T_{n-1} \cdots T_1
</math>
can be understood as a matrix analogue of the law of large numbers. A sufficient condition for this behavior is provided by a theorem of Harry Furstenberg.

To quantify this growth, we introduce the norm of a matrix <math>A</math> as
<math display="block">
\|A\|^2 = \frac{a_{11}^2 + a_{12}^2 + a_{21}^2 + a_{22}^2}{2}.
</math>

In physical terms, the theorem relies on two key ingredients:

=== (i) Stretching ===

Each matrix must be able to expand at least one direction. This means that for each <math>n</math> there exists a vector <math>v</math> such that
<math display="block">
\|T_n v\| = \sigma_{\max}(T_n)\, \|v\|,
\qquad \sigma_{\max}(T_n) > 1.
</math>

Equivalently, <math>\sigma_{\max}^2(T_n)</math> is the largest eigenvalue of the symmetric matrix
<math display="block">
T_n^T T_n.
</math>

=== (ii) Mixing of directions ===

The angular dynamics is not confined: under iteration of the product
<math display="block">
T_n T_{n-1} \cdots T_1,
</math>
the direction of a vector can explore the whole angular space. This exploration is not required to be uniform; it is sufficient that all directions are accessible.

=== Consequence ===

Under these conditions, the norm of the product grows exponentially with probability one:
<math display="block">
\lim_{n\to\infty} \frac{1}{n} \log \|\Pi_n\| = \gamma > 0,
</math>
where <math>\gamma</math> is the Lyapunov exponent.

This result is the matrix analogue of the fact that the logarithm of a product of independent random variables becomes self-averaging (see Exercise 15).

== Verification of Furstenberg's hypotheses for the Anderson model ==

We now check explicitly that the transfer matrices of the one-dimensional Anderson model satisfy the two conditions stated above.

The transfer matrices are
<math display="block">
T_n =
\begin{pmatrix}
V_n-\epsilon & -1 \\
1 & 0
\end{pmatrix}.
</math>

=== (i) Stretching ===

We first verify that each matrix expands at least one direction.

For a generic <math>2\times2</math> matrix, the maximal stretching factor is controlled by the largest eigenvalue of <math>T_n^T T_n</math>. Here
<math display="block">
T_n^T T_n =
\begin{pmatrix}
(V_n-\epsilon)^2+1 & -(V_n-\epsilon) \\
-(V_n-\epsilon) & 1
\end{pmatrix}.
</math>

The trace is
<math display="block">
\mathrm{Tr}(T_n^T T_n) = (V_n-\epsilon)^2 + 2,
</math>
and the determinant is
<math display="block">
\det(T_n^T T_n) = \det(T_n)^2 = 1.
</math>

Therefore the eigenvalues satisfy
<math display="block">
\lambda_+ \lambda_- = 1,
</math>
and the largest one obeys
<math display="block">
\lambda_+ \ge 1.
</math>

More precisely, as soon as <math>V_n-\epsilon \neq 0</math>, one has
<math display="block">
\lambda_+ > 1.
</math>

Thus each matrix expands at least one direction, except for the fine-tuned case <math>V_n=\epsilon</math>, which has zero probability for a continuous disorder distribution. The stretching condition is therefore satisfied almost surely.

=== (ii) Mixing of directions ===

We now verify that the angular dynamics is not confined to a finite set of directions.

Define the angle
<math display="block">
\theta_n = \arctan\!\left(\frac{\psi_{n-1}}{\psi_n}\right),
</math>
which represents the direction of the vector <math>(\psi_n,\psi_{n-1})</math>.

The transfer matrix induces the map
<math display="block">
\theta_{n+1} =
\arctan\!\left(\frac{1}{V_n-\epsilon-\tan\theta_n}\right).
</math>

For fixed <math>\theta_n</math>, this expression depends continuously on the random variable <math>V_n</math>. If the disorder has a continuous distribution (for instance Gaussian), the image of a given angle is not restricted to a finite set.

As a consequence, the sequence <math>\theta_n</math> is not confined to a finite set of directions: under iteration of the product
<math display="block">
T_n T_{n-1} \cdots T_1,
</math>
the angular dynamics can access a continuum of directions.

This exploration is not required to be uniform; it is sufficient that all directions are accessible. This is precisely the mixing condition required by Furstenberg's theorem.

=== Conclusion ===

The transfer matrices of the one-dimensional Anderson model satisfy both conditions:

# they expand at least one direction;
# they do not confine the angular dynamics to a finite set.

As a consequence,
<math display="block">
\lim_{n\to\infty} \frac{1}{n}\log \|\Pi_n\| = \gamma > 0,
</math>
so the Lyapunov exponent is positive for generic disorder.

---

== Localization length ==

The transfer-matrix recursion corresponds to fixing the wavefunction at one boundary and propagating it through the system.

Typical solutions grow exponentially

<math display="block">
|\psi_n|\sim e^{\gamma n}.
</math>

However a physical eigenstate must satisfy boundary conditions at both ends of the system. Matching two such solutions leads to exponentially localized eigenstates

<math display="block">
|\psi_n|\sim e^{-|n-n_0|/\xi_{\text{loc}}}.
</math>

The localization length is

<math display="block">
\xi_{\text{loc}}(\epsilon)=\frac{1}{\gamma(\epsilon)}.
</math>

Thus in one dimension arbitrarily weak disorder localizes all eigenstates.
This result is consistent with the scaling theory of localization discussed earlier, which predicts that for <math>d\le2</math> disorder inevitably drives the system toward the insulating regime.

---

== Fluctuations ==

Quantities such as

<math>
|\psi_n|,\quad \|\Pi_n\|,\quad G
</math>

show strong sample-to-sample fluctuations, while their logarithm is self-averaging.

For instance

<math display="block">
\ln|\psi_n|
\sim
\gamma n + O(\sqrt n)
</math>

so that the logarithm of the wavefunction performs a random walk with a positive drift.

L-8

2026-03-21T15:56:05Z

Rosso: /* Anderson model (tight-binding model) */

'''Goal.''' We introduce the Anderson model and study the statistical properties of its eigenstates.
In one dimension disorder leads to localization of all eigenstates, which can be understood using products of random matrices.

= Anderson model (tight-binding model) =

We consider non-interacting particles hopping between nearest-neighbour sites of a lattice in the presence of disorder.

The Hamiltonian reads

<math display="block">
H =
- t \sum_{\langle i,j\rangle} (c_i^\dagger c_j + c_j^\dagger c_i)
+
\sum_i V_i c_i^\dagger c_i .
</math>

The random variables <math>V_i</math> represent on-site disorder.
For simplicity we set <math>t=1</math>.

We assume that the disorder variables are independent and identically distributed, uniformly in the interval

<math display="block">
V_i \in \left(-\frac{W}{2},\frac{W}{2}\right).
</math>

In one dimension, the single-particle Hamiltonian is represented by the tridiagonal matrix

<math display="block">
H =
\begin{pmatrix}
V_1 & -1 & 0 & 0 & \dots \\
-1 & V_2 & -1 & 0 & \dots \\
0 & -1 & V_3 & -1 & \dots \\
0 & 0 & -1 & \ddots & -1 \\
\dots & \dots & \dots & -1 & V_L
\end{pmatrix}.
</math>

We study the statistical properties of the eigenvalue problem

<math display="block">
H\psi = \epsilon \psi ,
\qquad
\sum_{n=1}^L |\psi_n|^2 = 1 .
</math>

In the one-particle sector, this becomes a discrete Schrödinger equation.

---

== Density of states ==

Without disorder the dispersion relation is

<math display="block">
\epsilon(k) = -2\cos k,
\qquad
k\in(-\pi,\pi).
</math>

The energy band is therefore

<math display="block">
-2 < \epsilon < 2 .
</math>

The density of states is

<math display="block">
\rho(\epsilon)
=
\int_{-\pi}^{\pi}
\frac{dk}{2\pi}
\delta(\epsilon-\epsilon(k))
=
\frac{1}{\pi\sqrt{4-\epsilon^2}}
\qquad
(\epsilon\in(-2,2)).
</math>

In the presence of disorder the density of states broadens and becomes sample dependent.

---

== Transfer matrices ==

After multiplying the eigenvalue equation by <math>-1</math> and redefining
<math>-V_n \to V_n</math> and <math>-\epsilon \to \epsilon</math>, the discrete Schrödinger equation reads

<math display="block">
\psi_{n+1} + \psi_{n-1} + V_n \psi_n = \epsilon \psi_n .
</math>

It can be rewritten as

<math display="block">
\begin{pmatrix}
\psi_{n+1} \\
\psi_n
\end{pmatrix}
=
T_n
\begin{pmatrix}
\psi_n \\
\psi_{n-1}
\end{pmatrix}
</math>

with

<math display="block">
T_n =
\begin{pmatrix}
V_n-\epsilon & -1 \\
1 & 0
\end{pmatrix}.
</math>

Iterating gives

<math display="block">
\begin{pmatrix}
\psi_{n+1} \\
\psi_n
\end{pmatrix}
=
\Pi_n
\begin{pmatrix}
\psi_1 \\
\psi_0
\end{pmatrix},
\qquad
\Pi_n = T_n T_{n-1} \cdots T_1 .
</math>

Thus the wavefunction is controlled by a product of random matrices.

=== Remark ===

The transfer matrix formulation rewrites the problem as a recursion, starting from initial data and propagating the solution along the chain.

This corresponds to a Cauchy problem rather than a boundary value problem. Nevertheless, the growth properties of the solutions contain direct information about the nature of the eigenstates: exponential growth is associated with localization, while bounded solutions correspond to extended states.

== Furstenberg theorem (physical formulation) ==

The exponential growth of the product of random matrices
<math display="block">
\Pi_n = T_n T_{n-1} \cdots T_1
</math>
can be understood as a matrix analogue of the law of large numbers. A sufficient condition for this behavior is provided by a theorem of Harry Furstenberg.

To quantify this growth, we introduce the norm of a matrix <math>A</math> as
<math display="block">
\|A\|^2 = \frac{a_{11}^2 + a_{12}^2 + a_{21}^2 + a_{22}^2}{2}.
</math>

In physical terms, the theorem relies on two key ingredients:

=== (i) Stretching ===

Each matrix must be able to expand at least one direction. This means that for each <math>n</math> there exists a vector <math>v</math> such that
<math display="block">
\|T_n v\| = \sigma_{\max}(T_n)\, \|v\|,
\qquad \sigma_{\max}(T_n) > 1.
</math>

Equivalently, <math>\sigma_{\max}^2(T_n)</math> is the largest eigenvalue of the symmetric matrix
<math display="block">
T_n^T T_n.
</math>

=== (ii) Mixing of directions ===

The angular dynamics is not confined: under iteration of the product
<math display="block">
T_n T_{n-1} \cdots T_1,
</math>
the direction of a vector can explore the whole angular space. This exploration is not required to be uniform; it is sufficient that all directions are accessible.

=== Consequence ===

Under these conditions, the norm of the product grows exponentially with probability one:
<math display="block">
\lim_{n\to\infty} \frac{1}{n} \log \|\Pi_n\| = \gamma > 0,
</math>
where <math>\gamma</math> is the Lyapunov exponent.

This result is the matrix analogue of the fact that the logarithm of a product of independent random variables becomes self-averaging (see Exercise 15).

== Verification of Furstenberg's hypotheses for the Anderson model ==

We now check explicitly that the transfer matrices of the one-dimensional Anderson model satisfy the two conditions stated above.

The transfer matrices are
<math display="block">
T_n =
\begin{pmatrix}
V_n-\epsilon & -1 \\
1 & 0
\end{pmatrix}.
</math>

=== (i) Stretching ===

We first verify that each matrix expands at least one direction.

For a generic <math>2\times2</math> matrix, the maximal stretching factor is controlled by the largest eigenvalue of <math>T_n^T T_n</math>. Here
<math display="block">
T_n^T T_n =
\begin{pmatrix}
(V_n-\epsilon)^2+1 & -(V_n-\epsilon) \\
-(V_n-\epsilon) & 1
\end{pmatrix}.
</math>

The trace is
<math display="block">
\mathrm{Tr}(T_n^T T_n) = (V_n-\epsilon)^2 + 2,
</math>
and the determinant is
<math display="block">
\det(T_n^T T_n) = \det(T_n)^2 = 1.
</math>

Therefore the eigenvalues satisfy
<math display="block">
\lambda_+ \lambda_- = 1,
</math>
and the largest one obeys
<math display="block">
\lambda_+ \ge 1.
</math>

More precisely, as soon as <math>V_n-\epsilon \neq 0</math>, one has
<math display="block">
\lambda_+ > 1.
</math>

Thus each matrix expands at least one direction, except for the fine-tuned case <math>V_n=\epsilon</math>, which has zero probability for a continuous disorder distribution. The stretching condition is therefore satisfied almost surely.

=== (ii) Mixing of directions ===

We now verify that the angular dynamics is not confined to a finite set of directions.

Define the angle
<math display="block">
\theta_n = \arctan\!\left(\frac{\psi_{n-1}}{\psi_n}\right),
</math>
which represents the direction of the vector <math>(\psi_n,\psi_{n-1})</math>.

The transfer matrix induces the map
<math display="block">
\theta_{n+1} =
\arctan\!\left(\frac{1}{V_n-\epsilon-\tan\theta_n}\right).
</math>

For fixed <math>\theta_n</math>, this expression depends continuously on the random variable <math>V_n</math>. If the disorder has a continuous distribution (for instance Gaussian), the image of a given angle is not restricted to a finite set.

As a consequence, the sequence <math>\theta_n</math> is not confined to a finite set of directions: under iteration of the product
<math display="block">
T_n T_{n-1} \cdots T_1,
</math>
the angular dynamics can access a continuum of directions.

This exploration is not required to be uniform; it is sufficient that all directions are accessible. This is precisely the mixing condition required by Furstenberg's theorem.

=== Conclusion ===

The transfer matrices of the one-dimensional Anderson model satisfy both conditions:

# they expand at least one direction;
# they do not confine the angular dynamics to a finite set.

As a consequence,
<math display="block">
\lim_{n\to\infty} \frac{1}{n}\log \|\Pi_n\| = \gamma > 0,
</math>
so the Lyapunov exponent is positive for generic disorder.

---

== Localization length ==

The transfer-matrix recursion corresponds to fixing the wavefunction at one boundary and propagating it through the system.

Typical solutions grow exponentially

<math display="block">
|\psi_n|\sim e^{\gamma n}.
</math>

However a physical eigenstate must satisfy boundary conditions at both ends of the system. Matching two such solutions leads to exponentially localized eigenstates

<math display="block">
|\psi_n|\sim e^{-|n-n_0|/\xi_{\text{loc}}}.
</math>

The localization length is

<math display="block">
\xi_{\text{loc}}(\epsilon)=\frac{1}{\gamma(\epsilon)}.
</math>

Thus in one dimension arbitrarily weak disorder localizes all eigenstates.
This result is consistent with the scaling theory of localization discussed earlier, which predicts that for <math>d\le2</math> disorder inevitably drives the system toward the insulating regime.

---

== Fluctuations ==

Quantities such as

<math>
|\psi_n|,\quad \|\Pi_n\|,\quad G
</math>

show strong sample-to-sample fluctuations, while their logarithm is self-averaging.

For instance

<math display="block">
\ln|\psi_n|
\sim
\gamma n + O(\sqrt n)
</math>

so that the logarithm of the wavefunction performs a random walk with a positive drift.

L-8

2026-03-20T22:37:42Z

Rosso: /* Lyapunov exponent */

'''Goal.''' We introduce the Anderson model and study the statistical properties of its eigenstates.
In one dimension disorder leads to localization of all eigenstates, which can be understood using products of random matrices.

= Anderson model (tight-binding model) =

We consider non-interacting particles hopping between nearest-neighbour sites of a lattice in the presence of disorder.

The Hamiltonian reads

<math display="block">
H =
- t \sum_{\langle i,j\rangle} (c_i^\dagger c_j + c_j^\dagger c_i)
+
\sum_i V_i c_i^\dagger c_i .
</math>

The random variables <math>V_i</math> represent on-site disorder.

For simplicity we set

<math>t=1</math>.

The disorder variables are independent random variables drawn from the box distribution

<math display="block">
V_i \in \left(-\frac{W}{2},\frac{W}{2}\right).
</math>

In one dimension the single-particle Hamiltonian corresponds to a tridiagonal matrix

<math display="block">
H =
\begin{pmatrix}
V_1 & -1 & 0 & 0 & \dots \\
-1 & V_2 & -1 & 0 & \dots \\
0 & -1 & V_3 & -1 & \dots \\
0 & 0 & -1 & \ddots & -1 \\
\dots & \dots & \dots & -1 & V_L
\end{pmatrix}.
</math>

We study the statistical properties of the eigenvalue problem

<math display="block">
H\psi = \epsilon \psi ,
\qquad
\sum_{n=1}^L |\psi_n|^2 = 1 .
</math>

---

== Density of states ==

Without disorder the dispersion relation is

<math display="block">
\epsilon(k) = -2\cos k,
\qquad
k\in(-\pi,\pi).
</math>

The energy band is therefore

<math display="block">
-2 < \epsilon < 2 .
</math>

The density of states is

<math display="block">
\rho(\epsilon)
=
\int_{-\pi}^{\pi}
\frac{dk}{2\pi}
\delta(\epsilon-\epsilon(k))
=
\frac{1}{\pi\sqrt{4-\epsilon^2}}
\qquad
(\epsilon\in(-2,2)).
</math>

In the presence of disorder the density of states broadens and becomes sample dependent.

---

== Transfer matrices ==

The discrete Schrödinger equation reads

<math display="block">
\psi_{n+1} + \psi_{n-1} + V_n \psi_n = \epsilon \psi_n .
</math>

It can be rewritten as

<math display="block">
\begin{pmatrix}
\psi_{n+1} \\
\psi_n
\end{pmatrix}
=
T_n
\begin{pmatrix}
\psi_n \\
\psi_{n-1}
\end{pmatrix}
</math>

with

<math display="block">
T_n =
\begin{pmatrix}
V_n-\epsilon & -1 \\
1 & 0
\end{pmatrix}.
</math>

Iterating gives

<math display="block">
\begin{pmatrix}
\psi_{n+1} \\
\psi_n
\end{pmatrix}
=
\Pi_n
\begin{pmatrix}
\psi_1 \\
\psi_0
\end{pmatrix},
\qquad
\Pi_n = T_n T_{n-1} \cdots T_1 .
</math>

Thus the wavefunction is controlled by a product of random matrices.

---

== Furstenberg theorem (physical formulation) ==

The exponential growth of the product of random matrices
<math display="block">
\Pi_n = T_n T_{n-1} \cdots T_1
</math>
can be understood as a matrix analogue of the law of large numbers. A sufficient condition for this behavior is provided by a theorem of Harry Furstenberg.

To quantify this growth, we introduce the norm of a matrix <math>A</math> as
<math display="block">
\|A\|^2 = \frac{a_{11}^2 + a_{12}^2 + a_{21}^2 + a_{22}^2}{2}.
</math>

In physical terms, the theorem relies on two key ingredients:

=== (i) Stretching ===

Each matrix must be able to expand at least one direction. This means that for each <math>n</math> there exists a vector <math>v</math> such that
<math display="block">
\|T_n v\| = \sigma_{\max}(T_n)\, \|v\|,
\qquad \sigma_{\max}(T_n) > 1.
</math>

Equivalently, <math>\sigma_{\max}^2(T_n)</math> is the largest eigenvalue of the symmetric matrix
<math display="block">
T_n^T T_n.
</math>

=== (ii) Mixing of directions ===

The angular dynamics is not confined: under iteration of the product
<math display="block">
T_n T_{n-1} \cdots T_1,
</math>
the direction of a vector can explore the whole angular space. This exploration is not required to be uniform; it is sufficient that all directions are accessible.

=== Consequence ===

Under these conditions, the norm of the product grows exponentially with probability one:
<math display="block">
\lim_{n\to\infty} \frac{1}{n} \log \|\Pi_n\| = \gamma > 0,
</math>
where <math>\gamma</math> is the Lyapunov exponent.

This result is the matrix analogue of the fact that the logarithm of a product of independent random variables becomes self-averaging (see Exercise 15).

== Verification of Furstenberg's hypotheses for the Anderson model ==

We now check explicitly that the transfer matrices of the one-dimensional Anderson model satisfy the two conditions stated above.

The transfer matrices are
<math display="block">
T_n =
\begin{pmatrix}
V_n-\epsilon & -1 \\
1 & 0
\end{pmatrix}.
</math>

=== (i) Stretching ===

We first verify that each matrix expands at least one direction.

For a generic <math>2\times2</math> matrix, the maximal stretching factor is controlled by the largest eigenvalue of <math>T_n^T T_n</math>. Here
<math display="block">
T_n^T T_n =
\begin{pmatrix}
(V_n-\epsilon)^2+1 & -(V_n-\epsilon) \\
-(V_n-\epsilon) & 1
\end{pmatrix}.
</math>

The trace is
<math display="block">
\mathrm{Tr}(T_n^T T_n) = (V_n-\epsilon)^2 + 2,
</math>
and the determinant is
<math display="block">
\det(T_n^T T_n) = \det(T_n)^2 = 1.
</math>

Therefore the eigenvalues satisfy
<math display="block">
\lambda_+ \lambda_- = 1,
</math>
and the largest one obeys
<math display="block">
\lambda_+ \ge 1.
</math>

More precisely, as soon as <math>V_n-\epsilon \neq 0</math>, one has
<math display="block">
\lambda_+ > 1.
</math>

Thus each matrix expands at least one direction, except for the fine-tuned case <math>V_n=\epsilon</math>, which has zero probability for a continuous disorder distribution. The stretching condition is therefore satisfied almost surely.

=== (ii) Mixing of directions ===

We now verify that the angular dynamics is not confined to a finite set of directions.

Define the angle
<math display="block">
\theta_n = \arctan\!\left(\frac{\psi_{n-1}}{\psi_n}\right),
</math>
which represents the direction of the vector <math>(\psi_n,\psi_{n-1})</math>.

The transfer matrix induces the map
<math display="block">
\theta_{n+1} =
\arctan\!\left(\frac{1}{V_n-\epsilon-\tan\theta_n}\right).
</math>

For fixed <math>\theta_n</math>, this expression depends continuously on the random variable <math>V_n</math>. If the disorder has a continuous distribution (for instance Gaussian), the image of a given angle is not restricted to a finite set.

As a consequence, the sequence <math>\theta_n</math> is not confined to a finite set of directions: under iteration of the product
<math display="block">
T_n T_{n-1} \cdots T_1,
</math>
the angular dynamics can access a continuum of directions.

This exploration is not required to be uniform; it is sufficient that all directions are accessible. This is precisely the mixing condition required by Furstenberg's theorem.

=== Conclusion ===

The transfer matrices of the one-dimensional Anderson model satisfy both conditions:

# they expand at least one direction;
# they do not confine the angular dynamics to a finite set.

As a consequence,
<math display="block">
\lim_{n\to\infty} \frac{1}{n}\log \|\Pi_n\| = \gamma > 0,
</math>
so the Lyapunov exponent is positive for generic disorder.

---

== Localization length ==

The transfer-matrix recursion corresponds to fixing the wavefunction at one boundary and propagating it through the system.

Typical solutions grow exponentially

<math display="block">
|\psi_n|\sim e^{\gamma n}.
</math>

However a physical eigenstate must satisfy boundary conditions at both ends of the system. Matching two such solutions leads to exponentially localized eigenstates

<math display="block">
|\psi_n|\sim e^{-|n-n_0|/\xi_{\text{loc}}}.
</math>

The localization length is

<math display="block">
\xi_{\text{loc}}(\epsilon)=\frac{1}{\gamma(\epsilon)}.
</math>

Thus in one dimension arbitrarily weak disorder localizes all eigenstates.
This result is consistent with the scaling theory of localization discussed earlier, which predicts that for <math>d\le2</math> disorder inevitably drives the system toward the insulating regime.

---

== Fluctuations ==

Quantities such as

<math>
|\psi_n|,\quad \|\Pi_n\|,\quad G
</math>

show strong sample-to-sample fluctuations, while their logarithm is self-averaging.

For instance

<math display="block">
\ln|\psi_n|
\sim
\gamma n + O(\sqrt n)
</math>

so that the logarithm of the wavefunction performs a random walk with a positive drift.

File:Exercises 13 15.pdf

2026-03-20T16:56:05Z

Rosso: Rosso uploaded a new version of File:Exercises 13 15.pdf

L-9

2026-03-17T14:20:22Z

Rosso: /* Delocalized eigenstates */

= Eigenstates =

Without disorder, the eigenstates are delocalized plane waves.

In the presence of disorder, three scenarios can arise:

* '''Delocalized eigenstates''', where the wavefunction remains extended over the whole system.
* '''Localized eigenstates''', where the wavefunction is exponentially confined to a finite region.
* '''Multifractal eigenstates''', occurring at the '''mobility edge''' of the Anderson transition, where the wavefunction exhibits a complex, scale–dependent structure.

In three dimensions, increasing the disorder strength leads to a transition between a metallic phase with delocalized eigenstates and an insulating phase with localized eigenstates. The critical energy separating these two regimes is called the mobility edge. Eigenstates at the mobility edge display multifractal statistics.

To characterize these different regimes it is useful to introduce the inverse participation ratio (IPR)

<math display="block">
\mathrm{IPR}(q)=\sum_n |\psi_n|^{2 q}.
</math>

For a system of linear size <math>L</math>, one typically observes a scaling

<math display="block">
\mathrm{IPR}(q) \sim L^{-\tau_q}.
</math>

The exponent <math>\tau_q</math> characterizes the spatial structure of the eigenstate.

== Delocalized eigenstates ==

In a delocalized state the probability is spread uniformly over the system:

<math display="block">
|\psi_n|^{2} \approx L^{-d}.
</math>

Therefore

<math display="block">
\mathrm{IPR}(q)
=
\sum_n |\psi_n|^{2q}
\sim L^d (L^{-d})^q
=
L^{d(1-q)}.
</math>

Thus

<math display="block">
\tau_q=d(q-1).
</math>

== Localized eigenstates ==

In a localized state the wavefunction is concentrated within a region of size <math>\xi_{\text{loc}}</math>.

Roughly

<math display="block">
|\psi_n|^2 \sim \xi_{\text{loc}}^{-d}
</math>

on <math>\xi_{\text{loc}}^d</math> sites and negligible elsewhere. Hence

<math display="block">
\mathrm{IPR}(q)\sim \text{const},
\qquad
\tau_q=0.
</math>

Thus localized states do not scale with system size.

== Multifractal eigenstates ==

At the mobility edge the eigenstates are neither extended nor localized. Instead, the amplitudes fluctuate strongly across the system.

The scaling of the IPR is characterized by a nonlinear function

<math display="block">
\tau_q.
</math>

The corresponding multifractal dimensions are defined as

<math display="block">
D_q=\frac{\tau_q}{q-1}.
</math>

It is useful to contrast fractal and multifractal scaling.

* In a '''fractal object''', a single exponent describes the scaling of the measure. If the wavefunction amplitudes scale with a single exponent <math>D</math>, then

<math display="block">
\mathrm{IPR}(q)\sim L^{-D(q-1)} .
</math>

All moments are controlled by the same dimension <math>D</math>.

* In a '''multifractal object''', different regions of the system scale with different exponents. In this case there is no single fractal dimension: different moments probe different effective dimensions of the wavefunction.

== Multifractal spectrum ==

To describe this structure it is useful to introduce the '''multifractal spectrum''' <math>f(\alpha)</math>.

We assume that the wavefunction amplitudes scale as

<math display="block">
|\psi_n|^2 \sim L^{-\alpha}
</math>

on approximately

<math display="block">
L^{f(\alpha)}
</math>

sites.

The function <math>f(\alpha)</math> therefore describes the fractal dimension of the set of sites where the wavefunction has exponent <math>\alpha</math>.

Using this representation,

<math display="block">
\mathrm{IPR}(q)
=
\sum_n |\psi_n|^{2q}
\sim
\int d\alpha \,
L^{-\alpha q} L^{f(\alpha)} .
</math>

For large <math>L</math>, the integral is dominated by the saddle point, giving

<math display="block">
\tau(q)=\min_{\alpha}\left(\alpha q-f(\alpha)\right).
</math>

Thus <math>\tau(q)</math> and <math>f(\alpha)</math> are related by a Legendre transform.

If <math>\alpha^*(q)</math> satisfies

<math display="block">
f'(\alpha^*(q))=q,
</math>

then

<math display="block">
\tau(q)=\alpha^*(q)q-f(\alpha^*(q)).
</math>

Multifractal wavefunctions exhibit a smooth spectrum with maximum

<math display="block">
f(\alpha_0)=d,
</math>

which corresponds to the most typical amplitude of the wavefunction.

It is useful to contrast this situation with the case of a simple fractal.

In a fractal object, the measure has a single scaling exponent. In our notation this means that the wavefunction amplitudes scale with a single value <math>\alpha_0</math> on a set of fractal dimension <math>D</math>:

<math display="block">
|\psi_n|^2 \sim L^{-\alpha_0}
</math>

on approximately

<math display="block">
L^{D}
</math>

sites, and are essentially negligible elsewhere.

In terms of the multifractal spectrum this corresponds to a trivial spectrum

<math display="block">
f(\alpha_0)=D,
\qquad
f(\alpha\neq\alpha_0)=-\infty.
</math>

By contrast, in a multifractal state the amplitudes are broadly distributed and many values of <math>\alpha</math> contribute. The function <math>f(\alpha)</math> then becomes a smooth curve describing the distribution of scaling exponents of the wavefunction.

= Larkin model =

In your homework you solved a toy model for the interface

<math display="block">
\partial_t h(r,t) = \nabla^2 h(r,t) + F(r).
</math>

For simplicity, we assume Gaussian disorder

<math display="block">
\overline{F(r)}=0,
\qquad
\overline{F(r)F(r')}=\sigma^2 \delta^d(r-r').
</math>

You proved that:

* the roughness exponent of this model is <math>\zeta_L=\frac{4-d}{2}</math> below dimension 4
* the force per unit length acting on the center of the interface is <math>f= \sigma/\sqrt{L^d}</math>
* at long times the interface shape is

<math display="block">
\overline{h(q)h(-q)} = \frac{\sigma^2}{q^{d+2\zeta_L}}.
</math>

In the real depinning model the disorder is, however, a non-linear function of <math>h</math>. The idea of Larkin is that this linearization is valid only up to a scale <math>r_f</math>, the correlation length of the disorder along the <math>h</math> direction. This defines the '''Larkin length'''.

Indeed, starting from

<math display="block">
\overline{(h(r)-h(0))^2}
= \int d^d q \,\overline{h(q)h(-q)}\,(1-\cos(q r))
\sim \sigma^2 r^{2\zeta_L},
</math>

we obtain

<math display="block">
\overline{(h(\ell_L)-h(0))^2}= r_f^2,
\qquad
\ell_L=\left(\frac{r_f}{\sigma}\right)^{1/\zeta_L}.
</math>

Above this scale the roughness changes and pinning sets in with a critical force

<math display="block">
f_c= \frac{\sigma}{\ell_L^{d/(2 \zeta_L)}}.
</math>

In <math>d=1</math> we therefore obtain

<math display="block">
\ell_L=\left(\frac{r_f}{\sigma}\right)^{2/3}.
</math>

L-9

2026-03-15T16:33:04Z

Rosso: /* Larkin model */

= Eigenstates =

Without disorder, the eigenstates are delocalized plane waves.

In the presence of disorder, three scenarios can arise:

* '''Delocalized eigenstates''', where the wavefunction remains extended over the whole system.
* '''Localized eigenstates''', where the wavefunction is exponentially confined to a finite region.
* '''Multifractal eigenstates''', occurring at the '''mobility edge''' of the Anderson transition, where the wavefunction exhibits a complex, scale–dependent structure.

In three dimensions, increasing the disorder strength leads to a transition between a metallic phase with delocalized eigenstates and an insulating phase with localized eigenstates. The critical energy separating these two regimes is called the mobility edge. Eigenstates at the mobility edge display multifractal statistics.

To characterize these different regimes it is useful to introduce the inverse participation ratio (IPR)

<math display="block">
\mathrm{IPR}(q)=\sum_n |\psi_n|^{2 q}.
</math>

For a system of linear size <math>L</math>, one typically observes a scaling

<math display="block">
\mathrm{IPR}(q) \sim L^{-\tau_q}.
</math>

The exponent <math>\tau_q</math> characterizes the spatial structure of the eigenstate.

== Delocalized eigenstates ==

In a delocalized state the probability is spread uniformly over the system:

<math display="block">
|\psi_n|^{2} \approx L^{-d}.
</math>

Therefore

<math display="block">
\mathrm{IPR}(q)
=
\sum_n |\psi_n|^{2q}
\sim L^d (L^{-d})^q
=
L^{d(1-q)}.
</math>

Thus

<math display="block">
\tau_q=d(1-q).
</math>

== Localized eigenstates ==

In a localized state the wavefunction is concentrated within a region of size <math>\xi_{\text{loc}}</math>.

Roughly

<math display="block">
|\psi_n|^2 \sim \xi_{\text{loc}}^{-d}
</math>

on <math>\xi_{\text{loc}}^d</math> sites and negligible elsewhere. Hence

<math display="block">
\mathrm{IPR}(q)\sim \text{const},
\qquad
\tau_q=0.
</math>

Thus localized states do not scale with system size.

== Multifractal eigenstates ==

At the mobility edge the eigenstates are neither extended nor localized. Instead, the amplitudes fluctuate strongly across the system.

The scaling of the IPR is characterized by a nonlinear function

<math display="block">
\tau_q.
</math>

The corresponding multifractal dimensions are defined as

<math display="block">
D_q=\frac{\tau_q}{q-1}.
</math>

It is useful to contrast fractal and multifractal scaling.

* In a '''fractal object''', a single exponent describes the scaling of the measure. If the wavefunction amplitudes scale with a single exponent <math>D</math>, then

<math display="block">
\mathrm{IPR}(q)\sim L^{-D(q-1)} .
</math>

All moments are controlled by the same dimension <math>D</math>.

* In a '''multifractal object''', different regions of the system scale with different exponents. In this case there is no single fractal dimension: different moments probe different effective dimensions of the wavefunction.

== Multifractal spectrum ==

To describe this structure it is useful to introduce the '''multifractal spectrum''' <math>f(\alpha)</math>.

We assume that the wavefunction amplitudes scale as

<math display="block">
|\psi_n|^2 \sim L^{-\alpha}
</math>

on approximately

<math display="block">
L^{f(\alpha)}
</math>

sites.

The function <math>f(\alpha)</math> therefore describes the fractal dimension of the set of sites where the wavefunction has exponent <math>\alpha</math>.

Using this representation,

<math display="block">
\mathrm{IPR}(q)
=
\sum_n |\psi_n|^{2q}
\sim
\int d\alpha \,
L^{-\alpha q} L^{f(\alpha)} .
</math>

For large <math>L</math>, the integral is dominated by the saddle point, giving

<math display="block">
\tau(q)=\min_{\alpha}\left(\alpha q-f(\alpha)\right).
</math>

Thus <math>\tau(q)</math> and <math>f(\alpha)</math> are related by a Legendre transform.

If <math>\alpha^*(q)</math> satisfies

<math display="block">
f'(\alpha^*(q))=q,
</math>

then

<math display="block">
\tau(q)=\alpha^*(q)q-f(\alpha^*(q)).
</math>

Multifractal wavefunctions exhibit a smooth spectrum with maximum

<math display="block">
f(\alpha_0)=d,
</math>

which corresponds to the most typical amplitude of the wavefunction.

It is useful to contrast this situation with the case of a simple fractal.

In a fractal object, the measure has a single scaling exponent. In our notation this means that the wavefunction amplitudes scale with a single value <math>\alpha_0</math> on a set of fractal dimension <math>D</math>:

<math display="block">
|\psi_n|^2 \sim L^{-\alpha_0}
</math>

on approximately

<math display="block">
L^{D}
</math>

sites, and are essentially negligible elsewhere.

In terms of the multifractal spectrum this corresponds to a trivial spectrum

<math display="block">
f(\alpha_0)=D,
\qquad
f(\alpha\neq\alpha_0)=-\infty.
</math>

By contrast, in a multifractal state the amplitudes are broadly distributed and many values of <math>\alpha</math> contribute. The function <math>f(\alpha)</math> then becomes a smooth curve describing the distribution of scaling exponents of the wavefunction.

= Larkin model =

In your homework you solved a toy model for the interface

<math display="block">
\partial_t h(r,t) = \nabla^2 h(r,t) + F(r).
</math>

For simplicity, we assume Gaussian disorder

<math display="block">
\overline{F(r)}=0,
\qquad
\overline{F(r)F(r')}=\sigma^2 \delta^d(r-r').
</math>

You proved that:

* the roughness exponent of this model is <math>\zeta_L=\frac{4-d}{2}</math> below dimension 4
* the force per unit length acting on the center of the interface is <math>f= \sigma/\sqrt{L^d}</math>
* at long times the interface shape is

<math display="block">
\overline{h(q)h(-q)} = \frac{\sigma^2}{q^{d+2\zeta_L}}.
</math>

In the real depinning model the disorder is, however, a non-linear function of <math>h</math>. The idea of Larkin is that this linearization is valid only up to a scale <math>r_f</math>, the correlation length of the disorder along the <math>h</math> direction. This defines the '''Larkin length'''.

Indeed, starting from

<math display="block">
\overline{(h(r)-h(0))^2}
= \int d^d q \,\overline{h(q)h(-q)}\,(1-\cos(q r))
\sim \sigma^2 r^{2\zeta_L},
</math>

we obtain

<math display="block">
\overline{(h(\ell_L)-h(0))^2}= r_f^2,
\qquad
\ell_L=\left(\frac{r_f}{\sigma}\right)^{1/\zeta_L}.
</math>

Above this scale the roughness changes and pinning sets in with a critical force

<math display="block">
f_c= \frac{\sigma}{\ell_L^{d/(2 \zeta_L)}}.
</math>

In <math>d=1</math> we therefore obtain

<math display="block">
\ell_L=\left(\frac{r_f}{\sigma}\right)^{2/3}.
</math>

L-9

2026-03-15T16:31:35Z

Rosso: /* Eigenstates */

= Eigenstates =

Without disorder, the eigenstates are delocalized plane waves.

In the presence of disorder, three scenarios can arise:

* '''Delocalized eigenstates''', where the wavefunction remains extended over the whole system.
* '''Localized eigenstates''', where the wavefunction is exponentially confined to a finite region.
* '''Multifractal eigenstates''', occurring at the '''mobility edge''' of the Anderson transition, where the wavefunction exhibits a complex, scale–dependent structure.

In three dimensions, increasing the disorder strength leads to a transition between a metallic phase with delocalized eigenstates and an insulating phase with localized eigenstates. The critical energy separating these two regimes is called the mobility edge. Eigenstates at the mobility edge display multifractal statistics.

To characterize these different regimes it is useful to introduce the inverse participation ratio (IPR)

<math display="block">
\mathrm{IPR}(q)=\sum_n |\psi_n|^{2 q}.
</math>

For a system of linear size <math>L</math>, one typically observes a scaling

<math display="block">
\mathrm{IPR}(q) \sim L^{-\tau_q}.
</math>

The exponent <math>\tau_q</math> characterizes the spatial structure of the eigenstate.

== Delocalized eigenstates ==

In a delocalized state the probability is spread uniformly over the system:

<math display="block">
|\psi_n|^{2} \approx L^{-d}.
</math>

Therefore

<math display="block">
\mathrm{IPR}(q)
=
\sum_n |\psi_n|^{2q}
\sim L^d (L^{-d})^q
=
L^{d(1-q)}.
</math>

Thus

<math display="block">
\tau_q=d(1-q).
</math>

== Localized eigenstates ==

In a localized state the wavefunction is concentrated within a region of size <math>\xi_{\text{loc}}</math>.

Roughly

<math display="block">
|\psi_n|^2 \sim \xi_{\text{loc}}^{-d}
</math>

on <math>\xi_{\text{loc}}^d</math> sites and negligible elsewhere. Hence

<math display="block">
\mathrm{IPR}(q)\sim \text{const},
\qquad
\tau_q=0.
</math>

Thus localized states do not scale with system size.

== Multifractal eigenstates ==

At the mobility edge the eigenstates are neither extended nor localized. Instead, the amplitudes fluctuate strongly across the system.

The scaling of the IPR is characterized by a nonlinear function

<math display="block">
\tau_q.
</math>

The corresponding multifractal dimensions are defined as

<math display="block">
D_q=\frac{\tau_q}{q-1}.
</math>

It is useful to contrast fractal and multifractal scaling.

* In a '''fractal object''', a single exponent describes the scaling of the measure. If the wavefunction amplitudes scale with a single exponent <math>D</math>, then

<math display="block">
\mathrm{IPR}(q)\sim L^{-D(q-1)} .
</math>

All moments are controlled by the same dimension <math>D</math>.

* In a '''multifractal object''', different regions of the system scale with different exponents. In this case there is no single fractal dimension: different moments probe different effective dimensions of the wavefunction.

== Multifractal spectrum ==

To describe this structure it is useful to introduce the '''multifractal spectrum''' <math>f(\alpha)</math>.

We assume that the wavefunction amplitudes scale as

<math display="block">
|\psi_n|^2 \sim L^{-\alpha}
</math>

on approximately

<math display="block">
L^{f(\alpha)}
</math>

sites.

The function <math>f(\alpha)</math> therefore describes the fractal dimension of the set of sites where the wavefunction has exponent <math>\alpha</math>.

Using this representation,

<math display="block">
\mathrm{IPR}(q)
=
\sum_n |\psi_n|^{2q}
\sim
\int d\alpha \,
L^{-\alpha q} L^{f(\alpha)} .
</math>

For large <math>L</math>, the integral is dominated by the saddle point, giving

<math display="block">
\tau(q)=\min_{\alpha}\left(\alpha q-f(\alpha)\right).
</math>

Thus <math>\tau(q)</math> and <math>f(\alpha)</math> are related by a Legendre transform.

If <math>\alpha^*(q)</math> satisfies

<math display="block">
f'(\alpha^*(q))=q,
</math>

then

<math display="block">
\tau(q)=\alpha^*(q)q-f(\alpha^*(q)).
</math>

Multifractal wavefunctions exhibit a smooth spectrum with maximum

<math display="block">
f(\alpha_0)=d,
</math>

which corresponds to the most typical amplitude of the wavefunction.

It is useful to contrast this situation with the case of a simple fractal.

In a fractal object, the measure has a single scaling exponent. In our notation this means that the wavefunction amplitudes scale with a single value <math>\alpha_0</math> on a set of fractal dimension <math>D</math>:

<math display="block">
|\psi_n|^2 \sim L^{-\alpha_0}
</math>

on approximately

<math display="block">
L^{D}
</math>

sites, and are essentially negligible elsewhere.

In terms of the multifractal spectrum this corresponds to a trivial spectrum

<math display="block">
f(\alpha_0)=D,
\qquad
f(\alpha\neq\alpha_0)=-\infty.
</math>

By contrast, in a multifractal state the amplitudes are broadly distributed and many values of <math>\alpha</math> contribute. The function <math>f(\alpha)</math> then becomes a smooth curve describing the distribution of scaling exponents of the wavefunction.

= Larkin model =

In your homework you solved a toy model for the interface:
<math display="block">
\partial_t h(r,t) = \nabla^2 h(r,t) + F(r).
</math>
For simplicity, we assume Gaussian disorder <math>\overline{F(r)}=0</math>, <math>\overline{F(r)F(r')}=\sigma^2 \delta^d(r-r')</math>.

You proved that:

* the roughness exponent of this model is <math>\zeta_L=\frac{4-d}{2}</math> below dimension 4
* the force per unit length acting on the center of the interface is <math>f= \sigma/\sqrt{L^d}</math>
* at long times the interface shape is
<math display="block">
\overline{h(q)h(-q)} = \frac{\sigma^2}{q^{d+2\zeta_L}}.
</math>

In the real depinning model the disorder is, however, a non-linear function of <math>h</math>. The idea of Larkin is that this linearization is correct up to <math>r_f</math>, the correlation length of the disorder along the <math>h</math> direction. This defines a Larkin length.

Indeed, from
<math display="block">
\overline{(h(r)-h(0))^2}
= \int d^d q \,\overline{h(q)h(-q)}\,(1-\cos(q r))
\sim \sigma^2 r^{2\zeta_L},
</math>
you get
<math display="block">
\overline{(h(\ell_L)-h(0))^2}= r_f^2,
\qquad
\ell_L=\left(\frac{r_f}{\sigma}\right)^{1/\zeta_L}.
</math>

Above this scale, roughness changes and pinning starts with a critical force
<math display="block">
f_c= \frac{\sigma}{\ell_L^{d/(2 \zeta_L)}}.
</math>

In <math>d=1</math> we have <math>\ell_L=\left(\frac{r_f}{\sigma}\right)^{2/3}</math>.

L-9

2026-03-15T16:28:52Z

Rosso: /* Eigenstates */

= Eigenstates =

Without disorder, the eigenstates are delocalized plane waves.

In the presence of disorder, three scenarios can arise:

* '''Delocalized eigenstates''', where the wavefunction remains extended over the whole system.
* '''Localized eigenstates''', where the wavefunction is exponentially confined to a finite region.
* '''Multifractal eigenstates''', occurring at the '''mobility edge''' of the Anderson transition, where the wavefunction exhibits a complex, scale–dependent structure.

In three dimensions, increasing the disorder strength leads to a transition between a metallic phase with delocalized eigenstates and an insulating phase with localized eigenstates. The critical energy separating these two regimes is called the '''mobility edge'''. Eigenstates at the mobility edge display multifractal statistics.

To characterize these different regimes it is useful to introduce the '''inverse participation ratio''' (IPR)

<math display="block">
\mathrm{IPR}(q)=\sum_n |\psi_n|^{2 q}.
</math>

For a system of linear size <math>L</math>, one typically observes a scaling

<math display="block">
\mathrm{IPR}(q) \sim L^{-\tau_q}.
</math>

The exponent <math>\tau_q</math> characterizes the spatial structure of the eigenstate.

== Delocalized eigenstates ==

In a delocalized state the probability is spread uniformly over the system:

<math display="block">
|\psi_n|^{2} \approx L^{-d}.
</math>

Therefore

<math display="block">
\mathrm{IPR}(q)
=
\sum_n |\psi_n|^{2q}
\sim L^d (L^{-d})^q
=
L^{d(1-q)}.
</math>

Thus

<math display="block">
\tau_q=d(1-q).
</math>

== Localized eigenstates ==

In a localized state the wavefunction is concentrated within a region of size <math>\xi_{\text{loc}}</math>.

Roughly

<math display="block">
|\psi_n|^2 \sim \xi_{\text{loc}}^{-d}
</math>

on <math>\xi_{\text{loc}}^d</math> sites and negligible elsewhere. Hence

<math display="block">
\mathrm{IPR}(q)\sim \text{const},
\qquad
\tau_q=0.
</math>

Thus localized states do not scale with system size.

== Multifractal eigenstates ==

At the mobility edge the eigenstates are neither extended nor localized. Instead, the amplitudes fluctuate strongly across the system.

The scaling of the IPR is characterized by a nonlinear function

<math display="block">
\tau_q.
</math>

The corresponding '''multifractal dimensions''' are defined as

<math display="block">
D_q=\frac{\tau_q}{q-1}.
</math>

It is useful to contrast '''fractal''' and '''multifractal''' scaling.

* In a '''fractal object''', a single exponent describes the scaling of the measure. If the wavefunction amplitudes scale with a single exponent <math>D</math>, then

<math display="block">
\mathrm{IPR}(q)\sim L^{-D(q-1)} .
</math>

All moments are controlled by the same dimension <math>D</math>.

* In a '''multifractal object''', different regions of the system scale with different exponents. In this case there is no single fractal dimension: different moments probe different effective dimensions of the wavefunction.

== Multifractal spectrum ==

To describe this structure it is useful to introduce the '''multifractal spectrum''' <math>f(\alpha)</math>.

We assume that the wavefunction amplitudes scale as

<math display="block">
|\psi_n|^2 \sim L^{-\alpha}
</math>

on approximately

<math display="block">
L^{f(\alpha)}
</math>

sites.

The function <math>f(\alpha)</math> therefore describes the fractal dimension of the set of sites where the wavefunction has exponent <math>\alpha</math>.

Using this representation,

<math display="block">
\mathrm{IPR}(q)
=
\sum_n |\psi_n|^{2q}
\sim
\int d\alpha \,
L^{-\alpha q} L^{f(\alpha)} .
</math>

For large <math>L</math>, the integral is dominated by the saddle point, giving

<math display="block">
\tau(q)=\min_{\alpha}\left(\alpha q-f(\alpha)\right).
</math>

Thus <math>\tau(q)</math> and <math>f(\alpha)</math> are related by a '''Legendre transform'''.

If <math>\alpha^*(q)</math> satisfies

<math display="block">
f'(\alpha^*(q))=q,
</math>

then

<math display="block">
\tau(q)=\alpha^*(q)q-f(\alpha^*(q)).
</math>

Multifractal wavefunctions exhibit a smooth spectrum with maximum

<math display="block">
f(\alpha_0)=d,
</math>

which corresponds to the most typical amplitude of the wavefunction.

It is useful to contrast this situation with the case of a simple fractal.

In a '''fractal object''', the measure has a single scaling exponent. In our notation this means that the wavefunction amplitudes scale with a single value <math>\alpha_0</math> on a set of fractal dimension <math>D</math>:

<math display="block">
|\psi_n|^2 \sim L^{-\alpha_0}
</math>

on approximately

<math display="block">
L^{D}
</math>

sites, and are essentially negligible elsewhere.

In terms of the multifractal spectrum this corresponds to a trivial spectrum

<math display="block">
f(\alpha_0)=D,
\qquad
f(\alpha\neq\alpha_0)=-\infty.
</math>

By contrast, in a '''multifractal state''' the amplitudes are broadly distributed and many values of <math>\alpha</math> contribute. The function <math>f(\alpha)</math> then becomes a smooth curve describing the distribution of scaling exponents of the wavefunction.

= Larkin model =

In your homework you solved a toy model for the interface:
<math display="block">
\partial_t h(r,t) = \nabla^2 h(r,t) + F(r).
</math>
For simplicity, we assume Gaussian disorder <math>\overline{F(r)}=0</math>, <math>\overline{F(r)F(r')}=\sigma^2 \delta^d(r-r')</math>.

You proved that:

* the roughness exponent of this model is <math>\zeta_L=\frac{4-d}{2}</math> below dimension 4
* the force per unit length acting on the center of the interface is <math>f= \sigma/\sqrt{L^d}</math>
* at long times the interface shape is
<math display="block">
\overline{h(q)h(-q)} = \frac{\sigma^2}{q^{d+2\zeta_L}}.
</math>

In the real depinning model the disorder is, however, a non-linear function of <math>h</math>. The idea of Larkin is that this linearization is correct up to <math>r_f</math>, the correlation length of the disorder along the <math>h</math> direction. This defines a Larkin length.

Indeed, from
<math display="block">
\overline{(h(r)-h(0))^2}
= \int d^d q \,\overline{h(q)h(-q)}\,(1-\cos(q r))
\sim \sigma^2 r^{2\zeta_L},
</math>
you get
<math display="block">
\overline{(h(\ell_L)-h(0))^2}= r_f^2,
\qquad
\ell_L=\left(\frac{r_f}{\sigma}\right)^{1/\zeta_L}.
</math>

Above this scale, roughness changes and pinning starts with a critical force
<math display="block">
f_c= \frac{\sigma}{\ell_L^{d/(2 \zeta_L)}}.
</math>

In <math>d=1</math> we have <math>\ell_L=\left(\frac{r_f}{\sigma}\right)^{2/3}</math>.

L-9

2026-03-15T16:25:49Z

Rosso: /* Multifractal spectrum */

= Eigenstates =

Without disorder, the eigenstates are delocalized plane waves.

In the presence of disorder, three scenarios can arise:

* '''Delocalized eigenstates''', where the wavefunction remains extended over the whole system.
* '''Localized eigenstates''', where the wavefunction is exponentially confined to a finite region.
* '''Multifractal eigenstates''', occurring at the '''mobility edge''' of the Anderson transition, where the wavefunction exhibits a complex, scale–dependent structure.

In three dimensions, increasing the disorder strength leads to a transition between a metallic phase with delocalized eigenstates and an insulating phase with localized eigenstates. The critical energy separating these two regimes is called the '''mobility edge'''. Eigenstates at the mobility edge display multifractal statistics.

To characterize these different regimes it is useful to introduce the '''inverse participation ratio''' (IPR)

<math display="block">
\mathrm{IPR}(q)=\sum_n |\psi_n|^{2 q}.
</math>

For a system of linear size <math>L</math>, one typically observes a scaling

<math display="block">
\mathrm{IPR}(q) \sim L^{-\tau_q}.
</math>

The exponent <math>\tau_q</math> characterizes the spatial structure of the eigenstate.

== Delocalized eigenstates ==

In a delocalized state the probability is spread uniformly over the system:

<math display="block">
|\psi_n|^{2} \approx L^{-d}.
</math>

Therefore

<math display="block">
\mathrm{IPR}(q)
=
\sum_n |\psi_n|^{2q}
\sim L^d (L^{-d})^q
=
L^{d(1-q)}.
</math>

Thus

<math display="block">
\tau_q=d(1-q).
</math>

== Localized eigenstates ==

In a localized state the wavefunction is concentrated within a region of size <math>\xi_{\text{loc}}</math>.

Roughly

<math display="block">
|\psi_n|^2 \sim \xi_{\text{loc}}^{-d}
</math>

on <math>\xi_{\text{loc}}^d</math> sites and negligible elsewhere. Hence

<math display="block">
\mathrm{IPR}(q)\sim \text{const},
\qquad
\tau_q=0.
</math>

Thus localized states do not scale with system size.

== Multifractal eigenstates ==

At the mobility edge the eigenstates are neither extended nor localized. Instead, the amplitudes fluctuate strongly across the system.

The scaling of the IPR is characterized by a nonlinear function

<math display="block">
\tau_q.
</math>

The corresponding '''multifractal dimensions''' are defined as

<math display="block">
D_q=\frac{\tau_q}{q-1}.
</math>

It is useful to contrast '''fractal''' and '''multifractal''' scaling.

* In a '''fractal object''', a single exponent describes the scaling of the measure. For instance, if a probability measure is uniformly distributed on a fractal set of dimension <math>D</math>, one has
<math display="block">
\mathrm{IPR}(q)\sim L^{-D(q-1)} .
</math>
All moments are controlled by the same dimension <math>D</math>.

* In a '''multifractal object''', different regions of the system scale with different exponents.
In this case there is no single fractal dimension: different moments probe different effective dimensions of the wavefunction.

== Multifractal spectrum ==

To describe this structure it is useful to introduce the '''multifractal spectrum''' <math>f(\alpha)</math>.

We assume that the wavefunction amplitudes scale as

<math display="block">
|\psi_n|^2 \sim L^{-\alpha}
</math>

on approximately

<math display="block">
L^{f(\alpha)}
</math>

sites.

The function <math>f(\alpha)</math> describes the fractal dimension of the set of sites where the wavefunction has exponent <math>\alpha</math>.

Using this representation,

<math display="block">
\mathrm{IPR}(q)
=
\sum_n |\psi_n|^{2q}
\sim
\int d\alpha \,
L^{-\alpha q} L^{f(\alpha)} .
</math>

For large <math>L</math>, the integral is dominated by the saddle point, giving

<math display="block">
\tau(q)=\min_{\alpha}\left(\alpha q-f(\alpha)\right).
</math>

Thus <math>\tau(q)</math> and <math>f(\alpha)</math> are related by a '''Legendre transform'''.

If <math>\alpha^*(q)</math> satisfies

<math display="block">
f'(\alpha^*(q))=q,
</math>

then

<math display="block">
\tau(q)=\alpha^*(q)q-f(\alpha^*(q)).
</math>

Multifractal wavefunctions exhibit a smooth spectrum with maximum

<math display="block">
f(\alpha_0)=d,
</math>

It is useful to contrast this situation with the case of a simple fractal.

In a '''fractal object''', the measure has a single scaling exponent. In our notation this means that the wavefunction amplitudes scale with a single value <math>\alpha_0</math> on a set of fractal dimension <math>D</math>:

<math display="block">
|\psi_n|^2 \sim L^{-\alpha_0}
</math>

on approximately

<math display="block">
L^{D}
</math>

sites, and are essentially negligible elsewhere.

In terms of the multifractal spectrum this corresponds to a trivial spectrum

<math display="block">
f(\alpha_0)=D,
\qquad
f(\alpha\neq\alpha_0)=-\infty.
</math>

By contrast, in a '''multifractal state''' the amplitudes are broadly distributed and many values of <math>\alpha</math> contribute. The function <math>f(\alpha)</math> then becomes a smooth curve.
indicating that the wavefunction explores the entire system but with strong spatial fluctuations.

= Larkin model =

In your homework you solved a toy model for the interface:
<math display="block">
\partial_t h(r,t) = \nabla^2 h(r,t) + F(r).
</math>
For simplicity, we assume Gaussian disorder <math>\overline{F(r)}=0</math>, <math>\overline{F(r)F(r')}=\sigma^2 \delta^d(r-r')</math>.

You proved that:

* the roughness exponent of this model is <math>\zeta_L=\frac{4-d}{2}</math> below dimension 4
* the force per unit length acting on the center of the interface is <math>f= \sigma/\sqrt{L^d}</math>
* at long times the interface shape is
<math display="block">
\overline{h(q)h(-q)} = \frac{\sigma^2}{q^{d+2\zeta_L}}.
</math>

In the real depinning model the disorder is, however, a non-linear function of <math>h</math>. The idea of Larkin is that this linearization is correct up to <math>r_f</math>, the correlation length of the disorder along the <math>h</math> direction. This defines a Larkin length.

Indeed, from
<math display="block">
\overline{(h(r)-h(0))^2}
= \int d^d q \,\overline{h(q)h(-q)}\,(1-\cos(q r))
\sim \sigma^2 r^{2\zeta_L},
</math>
you get
<math display="block">
\overline{(h(\ell_L)-h(0))^2}= r_f^2,
\qquad
\ell_L=\left(\frac{r_f}{\sigma}\right)^{1/\zeta_L}.
</math>

Above this scale, roughness changes and pinning starts with a critical force
<math display="block">
f_c= \frac{\sigma}{\ell_L^{d/(2 \zeta_L)}}.
</math>

In <math>d=1</math> we have <math>\ell_L=\left(\frac{r_f}{\sigma}\right)^{2/3}</math>.

L-9

2026-03-15T16:19:01Z

Rosso: /* Multifractal eigenstates */

= Eigenstates =

Without disorder, the eigenstates are delocalized plane waves.

In the presence of disorder, three scenarios can arise:

* '''Delocalized eigenstates''', where the wavefunction remains extended over the whole system.
* '''Localized eigenstates''', where the wavefunction is exponentially confined to a finite region.
* '''Multifractal eigenstates''', occurring at the '''mobility edge''' of the Anderson transition, where the wavefunction exhibits a complex, scale–dependent structure.

In three dimensions, increasing the disorder strength leads to a transition between a metallic phase with delocalized eigenstates and an insulating phase with localized eigenstates. The critical energy separating these two regimes is called the '''mobility edge'''. Eigenstates at the mobility edge display multifractal statistics.

To characterize these different regimes it is useful to introduce the '''inverse participation ratio''' (IPR)

<math display="block">
\mathrm{IPR}(q)=\sum_n |\psi_n|^{2 q}.
</math>

For a system of linear size <math>L</math>, one typically observes a scaling

<math display="block">
\mathrm{IPR}(q) \sim L^{-\tau_q}.
</math>

The exponent <math>\tau_q</math> characterizes the spatial structure of the eigenstate.

== Delocalized eigenstates ==

In a delocalized state the probability is spread uniformly over the system:

<math display="block">
|\psi_n|^{2} \approx L^{-d}.
</math>

Therefore

<math display="block">
\mathrm{IPR}(q)
=
\sum_n |\psi_n|^{2q}
\sim L^d (L^{-d})^q
=
L^{d(1-q)}.
</math>

Thus

<math display="block">
\tau_q=d(1-q).
</math>

== Localized eigenstates ==

In a localized state the wavefunction is concentrated within a region of size <math>\xi_{\text{loc}}</math>.

Roughly

<math display="block">
|\psi_n|^2 \sim \xi_{\text{loc}}^{-d}
</math>

on <math>\xi_{\text{loc}}^d</math> sites and negligible elsewhere. Hence

<math display="block">
\mathrm{IPR}(q)\sim \text{const},
\qquad
\tau_q=0.
</math>

Thus localized states do not scale with system size.

== Multifractal eigenstates ==

At the mobility edge the eigenstates are neither extended nor localized. Instead, the amplitudes fluctuate strongly across the system.

The scaling of the IPR is characterized by a nonlinear function

<math display="block">
\tau_q.
</math>

The corresponding '''multifractal dimensions''' are defined as

<math display="block">
D_q=\frac{\tau_q}{q-1}.
</math>

It is useful to contrast '''fractal''' and '''multifractal''' scaling.

* In a '''fractal object''', a single exponent describes the scaling of the measure. For instance, if a probability measure is uniformly distributed on a fractal set of dimension <math>D</math>, one has
<math display="block">
\mathrm{IPR}(q)\sim L^{-D(q-1)} .
</math>
All moments are controlled by the same dimension <math>D</math>.

* In a '''multifractal object''', different regions of the system scale with different exponents.
In this case there is no single fractal dimension: different moments probe different effective dimensions of the wavefunction.

== Multifractal spectrum ==

To describe this structure it is useful to introduce the '''multifractal spectrum''' <math>f(\alpha)</math>.

We assume that the wavefunction amplitudes scale as

<math display="block">
|\psi_n|^2 \sim L^{-\alpha}
</math>

on approximately

<math display="block">
L^{f(\alpha)}
</math>

sites.

The function <math>f(\alpha)</math> describes the fractal dimension of the set of sites where the wavefunction has exponent <math>\alpha</math>.

Using this representation,

<math display="block">
\mathrm{IPR}(q)
=
\sum_n |\psi_n|^{2q}
\sim
\int d\alpha \,
L^{-\alpha q} L^{f(\alpha)} .
</math>

For large <math>L</math>, the integral is dominated by the saddle point, giving

<math display="block">
\tau(q)=\min_{\alpha}\left(\alpha q-f(\alpha)\right).
</math>

Thus <math>\tau(q)</math> and <math>f(\alpha)</math> are related by a '''Legendre transform'''.

If <math>\alpha^*(q)</math> satisfies

<math display="block">
f'(\alpha^*(q))=q,
</math>

then

<math display="block">
\tau(q)=\alpha^*(q)q-f(\alpha^*(q)).
</math>

Multifractal wavefunctions exhibit a smooth spectrum with maximum

<math display="block">
f(\alpha_0)=d,
</math>

indicating that the wavefunction explores the entire system but with strong spatial fluctuations.

= Larkin model =

In your homework you solved a toy model for the interface:
<math display="block">
\partial_t h(r,t) = \nabla^2 h(r,t) + F(r).
</math>
For simplicity, we assume Gaussian disorder <math>\overline{F(r)}=0</math>, <math>\overline{F(r)F(r')}=\sigma^2 \delta^d(r-r')</math>.

You proved that:

* the roughness exponent of this model is <math>\zeta_L=\frac{4-d}{2}</math> below dimension 4
* the force per unit length acting on the center of the interface is <math>f= \sigma/\sqrt{L^d}</math>
* at long times the interface shape is
<math display="block">
\overline{h(q)h(-q)} = \frac{\sigma^2}{q^{d+2\zeta_L}}.
</math>

In the real depinning model the disorder is, however, a non-linear function of <math>h</math>. The idea of Larkin is that this linearization is correct up to <math>r_f</math>, the correlation length of the disorder along the <math>h</math> direction. This defines a Larkin length.

Indeed, from
<math display="block">
\overline{(h(r)-h(0))^2}
= \int d^d q \,\overline{h(q)h(-q)}\,(1-\cos(q r))
\sim \sigma^2 r^{2\zeta_L},
</math>
you get
<math display="block">
\overline{(h(\ell_L)-h(0))^2}= r_f^2,
\qquad
\ell_L=\left(\frac{r_f}{\sigma}\right)^{1/\zeta_L}.
</math>

Above this scale, roughness changes and pinning starts with a critical force
<math display="block">
f_c= \frac{\sigma}{\ell_L^{d/(2 \zeta_L)}}.
</math>

In <math>d=1</math> we have <math>\ell_L=\left(\frac{r_f}{\sigma}\right)^{2/3}</math>.

L-9

2026-03-15T16:08:22Z

Rosso: /* Eigenstates */

= Eigenstates =

Without disorder, the eigenstates are delocalized plane waves.

In the presence of disorder, three scenarios can arise:

* '''Delocalized eigenstates''', where the wavefunction remains extended over the whole system.
* '''Localized eigenstates''', where the wavefunction is exponentially confined to a finite region.
* '''Multifractal eigenstates''', occurring at the '''mobility edge''' of the Anderson transition, where the wavefunction exhibits a complex, scale–dependent structure.

In three dimensions, increasing the disorder strength leads to a transition between a metallic phase with delocalized eigenstates and an insulating phase with localized eigenstates. The critical energy separating these two regimes is called the '''mobility edge'''. Eigenstates at the mobility edge display multifractal statistics.

To characterize these different regimes it is useful to introduce the '''inverse participation ratio''' (IPR)

<math display="block">
\mathrm{IPR}(q)=\sum_n |\psi_n|^{2 q}.
</math>

For a system of linear size <math>L</math>, one typically observes a scaling

<math display="block">
\mathrm{IPR}(q) \sim L^{-\tau_q}.
</math>

The exponent <math>\tau_q</math> characterizes the spatial structure of the eigenstate.

== Delocalized eigenstates ==

In a delocalized state the probability is spread uniformly over the system:

<math display="block">
|\psi_n|^{2} \approx L^{-d}.
</math>

Therefore

<math display="block">
\mathrm{IPR}(q)
=
\sum_n |\psi_n|^{2q}
\sim L^d (L^{-d})^q
=
L^{d(1-q)}.
</math>

Thus

<math display="block">
\tau_q=d(1-q).
</math>

== Localized eigenstates ==

In a localized state the wavefunction is concentrated within a region of size <math>\xi_{\text{loc}}</math>.

Roughly

<math display="block">
|\psi_n|^2 \sim \xi_{\text{loc}}^{-d}
</math>

on <math>\xi_{\text{loc}}^d</math> sites and negligible elsewhere. Hence

<math display="block">
\mathrm{IPR}(q)\sim \text{const},
\qquad
\tau_q=0.
</math>

Thus localized states do not scale with system size.

== Multifractal eigenstates ==

At the mobility edge the eigenstates are neither extended nor localized. Instead, the amplitudes fluctuate strongly across the system.

The scaling of the IPR is characterized by a '''nonlinear function'''

<math display="block">
\tau_q.
</math>

The corresponding '''multifractal dimensions''' are defined as

<math display="block">
D_q=\frac{\tau_q}{q-1}.
</math>

Multifractality means that different moments of the wavefunction probe different effective dimensions.

== Multifractal spectrum ==

To describe this structure it is useful to introduce the '''multifractal spectrum''' <math>f(\alpha)</math>.

We assume that the wavefunction amplitudes scale as

<math display="block">
|\psi_n|^2 \sim L^{-\alpha}
</math>

on approximately

<math display="block">
L^{f(\alpha)}
</math>

sites.

The function <math>f(\alpha)</math> describes the fractal dimension of the set of sites where the wavefunction has exponent <math>\alpha</math>.

Using this representation,

<math display="block">
\mathrm{IPR}(q)
=
\sum_n |\psi_n|^{2q}
\sim
\int d\alpha \,
L^{-\alpha q} L^{f(\alpha)} .
</math>

For large <math>L</math>, the integral is dominated by the saddle point, giving

<math display="block">
\tau(q)=\min_{\alpha}\left(\alpha q-f(\alpha)\right).
</math>

Thus <math>\tau(q)</math> and <math>f(\alpha)</math> are related by a '''Legendre transform'''.

If <math>\alpha^*(q)</math> satisfies

<math display="block">
f'(\alpha^*(q))=q,
</math>

then

<math display="block">
\tau(q)=\alpha^*(q)q-f(\alpha^*(q)).
</math>

Multifractal wavefunctions exhibit a smooth spectrum with maximum

<math display="block">
f(\alpha_0)=d,
</math>

indicating that the wavefunction explores the entire system but with strong spatial fluctuations.

= Larkin model =

In your homework you solved a toy model for the interface:
<math display="block">
\partial_t h(r,t) = \nabla^2 h(r,t) + F(r).
</math>
For simplicity, we assume Gaussian disorder <math>\overline{F(r)}=0</math>, <math>\overline{F(r)F(r')}=\sigma^2 \delta^d(r-r')</math>.

You proved that:

* the roughness exponent of this model is <math>\zeta_L=\frac{4-d}{2}</math> below dimension 4
* the force per unit length acting on the center of the interface is <math>f= \sigma/\sqrt{L^d}</math>
* at long times the interface shape is
<math display="block">
\overline{h(q)h(-q)} = \frac{\sigma^2}{q^{d+2\zeta_L}}.
</math>

In the real depinning model the disorder is, however, a non-linear function of <math>h</math>. The idea of Larkin is that this linearization is correct up to <math>r_f</math>, the correlation length of the disorder along the <math>h</math> direction. This defines a Larkin length.

Indeed, from
<math display="block">
\overline{(h(r)-h(0))^2}
= \int d^d q \,\overline{h(q)h(-q)}\,(1-\cos(q r))
\sim \sigma^2 r^{2\zeta_L},
</math>
you get
<math display="block">
\overline{(h(\ell_L)-h(0))^2}= r_f^2,
\qquad
\ell_L=\left(\frac{r_f}{\sigma}\right)^{1/\zeta_L}.
</math>

Above this scale, roughness changes and pinning starts with a critical force
<math display="block">
f_c= \frac{\sigma}{\ell_L^{d/(2 \zeta_L)}}.
</math>

In <math>d=1</math> we have <math>\ell_L=\left(\frac{r_f}{\sigma}\right)^{2/3}</math>.

L-8

2026-03-12T20:39:21Z

Rosso:

File:DISSYTS.pdf

2026-03-12T09:52:55Z

Rosso:

Main Page

2026-03-09T15:54:32Z

Rosso: /* Exercises */

Welcome to the WIKI page of the M2 ICFP course on the Statistical Physics of Disordered Systems, second semester 2026.

= Where and When =

* Each Monday at 2pm - 6 pm, from January 19th to March 23rd. No lecture on 23/02/26.
* Room 202 in Jussieu campus, Tours 54-55 until 16th February
* Room 209 in Jussieu campus, Tours 56-66 209 from 2nd March '''Attention: ROOM CHANGE!'''
* Each session is a mixture of lectures and exercises.

= The Team =

* [https://vale1925.wixsite.com/vros Valentina Ros] - vale1925@gmail.com
* [http://lptms.u-psud.fr/alberto_rosso/ Alberto Rosso] - alberto.rosso74@gmail.com

= Course description =

This course deals with systems in which the presence of impurities or amorphous structures (in other words, of disorder) influences radically the physics, generating novel phenomena. These phenomena involve the properties of the system at equilibrium (freezing and glass transitions), as well as their dynamical evolution out-of-equilibrium (pinning, avalanches), giving rise to ergodicity breaking both in absence and in presence of quantum fluctuations (classical metastability, quantum localization). We discuss the main statistical physics models that are able to capture the phenomenology of these systems, as well as the powerful theoretical tools (replica theory, large deviations, random matrix theory, scaling arguments, strong-disorder expansions) that have been developed to characterize quantitatively their physics. These theoretical tools nowadays have a huge impact in a variety of fields that go well-beyond statistical physics (computer science, probability, condensed matter, theoretical biology). Below is a list of topics discussed during the course.

'''Finite-dimensional disordered systems:'''

* Introduction to disordered systems and to the spin glass transition.
* Interface growth. Directed polymers in random media. Scenarios for the glass transition: the glass transition in KPZ in d>2.
* Depinning and avalanches. Bienaymé-Galton-Watson processes.
* Anderson localization: introduction. Localization in 1D: transfer matrix and Lyapunov.

'''Mean-field disordered systems:'''

* The simplest spin-glass: solution of the Random Energy Model.
* The replica method: the solution of the spherical p-spin model. Sketch of the solution of Sherrington Kirkpatrick model (full RSB).
* Towards glassy dynamics: rugged landscapes. Slow dynamics and aging: the trap model.
* The Anderson model on the Bethe lattice: the mobility edge.

=Lectures and tutorials=

'''If the layout of the formulas is bad, it might be because you are using Safari. Try opening the wiki with Firefox or Chrome.'''

{| class="wikitable" border="1"
|-
! width="100"|Date
! width="500"| 14h00-15h45
! width="500"| 16h00-17h45

|-valign="top"

| Week 1 (19/01)
|
* [[L1_ICTS| Spin Glass Transition (Alberto)]]

|
* [[T-I| A dictionary. The REM: energy landscape (Valentina)]]  [[Media:2025 P1 solutions.pdf| Sol Prob.1 ]]
|-valign=“top"

|-valign=“top"
| Week 2 (26/01)
|
* [[L2_ICFP| Stochastic Interfaces and growth (Alberto)]]
|
* [[T-2|The REM: freezing, condensation, glassiness (Valentina)]]   [[Media:2025 P2 solutions.pdf| Sol Prob.2 ]]
|-valign=“top"

|-valign=“top"
| Week 3 (02/02)
|
* [[L-3|Directed polymer in random media (Alberto)]]
|
* [[L-4| KPZ and glassiness in finite dimension (Alberto)]] [[https://colab.research.google.com/drive/1PTya42ZS2kU87A-BxQFFIUDTs_k47men?usp=sharing| notebook]]
|-valign=“top"
| Week 4 (9/02) and Week 5 (16/02)
|
* [[T-3| Spin glasses, equilibrium: replicas, the steps (Valentina)]]  [[Media:2025 P3 solutions.pdf| Sol Prob.3 ]]
|
* [[T-4| Spin glasses, equilibrium: replicas, the interpretation (Valentina)]]   [[Media:2025 P4 solutions.pdf| Sol Probs. 4 ]] 
[[Media:2025_Parisi_scheme.pdf| Notes: Probing states with replicas]]
|-valign=“top"
| Week 6 (02/03)
|
* [[LBan-IV| Driven Disordered Materials (Alberto)]] [[Media:DISSYTS.pdf| Slides ]]
|
* [[LBan-V| Avalanches in Disordered Materials (Alberto)]]
|-valign=“top"

| Week 7 (9/03)
|
* [[L-7| Anderson localization: introduction (Alberto)]]
|
* [[T-5| Rugged landscapes: counting metastable states (Valentina)]]  
|-valign=“top"
| Week 8 (16/03)
|
* [[T-6| Rugged landscapes: stability of metastable states (Valentina)]] 
|
* [[T-7| Trap model and aging dynamics (Valentina)]] 
|-valign=“top"
| Week 9 (23/03)
|
* [[L-8| Localization in 1D, transfer matrix and Lyapunov exponent (Alberto)]] [[https://colab.research.google.com/drive/1ZJ0yvMrtflWNNmPfaRQ8KTteoWfm0bqk?usp=sharing| notebook]]
|
* [[L-9|Multifractality, tails (Alberto)]]
|}




= Exercises =

<li> Week 1: [[Media:Exercises_1-3.pdf| Exercises 1-3 on extreme value statistics]]
</li>

<li> Week 2:
[[Media:Tutorial_and_Exercise_4.pdf| Tutorial and Exercise 4 on random matrices]] 
[[Media:Exercises_5-6.pdf| Exercises 5-6 on the random energy model]]
</li>

<li> Week 3:
[[Media:Exercises 7&8.pdf| Exercises 7-8 on interfaces]]
</li>

<li> Week 4:
[[Media:Exercises 9-10.pdf| Exercises 9-10 on glassiness]]
</li>

<li> Week 5 and 6:
[[Media:11-12_Exercises.pdf| Exercises 11-12 on dynamics]]
</li>

<li> Week 7:
[[Media:Exercises_13_15.pdf| Exercises 13-15 on branching and localization]]
</li>

<li> Week 8:
[[XX| Exercises 16-17 on trap model and localization]]
</li>

<li>
[[Media:DISSYTS.pdf| Slides ]]
</li>

= Evaluation and exam =

The exam will be on '''Monday, March 30th 2026'''. It will be written, 3h long. It consists of two parts:

Part 1: theory questions, see here for an example.

Part 2: you will be asked to solve pieces of the 17 exercises given to you in advance.

'''You are not allowed to bring any material (printed notes, handwritten notes) nor to use any device during the exam.'''

All relevant formulas will be provided in the text of the exam. There will be one printed version of the WIKI pages available to you to consult.

Main Page

2026-03-09T15:50:30Z

Rosso: /* Lectures and tutorials */

Welcome to the WIKI page of the M2 ICFP course on the Statistical Physics of Disordered Systems, second semester 2026.

= Where and When =

* Each Monday at 2pm - 6 pm, from January 19th to March 23rd. No lecture on 23/02/26.
* Room 202 in Jussieu campus, Tours 54-55 until 16th February
* Room 209 in Jussieu campus, Tours 56-66 209 from 2nd March '''Attention: ROOM CHANGE!'''
* Each session is a mixture of lectures and exercises.

= The Team =

* [https://vale1925.wixsite.com/vros Valentina Ros] - vale1925@gmail.com
* [http://lptms.u-psud.fr/alberto_rosso/ Alberto Rosso] - alberto.rosso74@gmail.com

= Course description =

This course deals with systems in which the presence of impurities or amorphous structures (in other words, of disorder) influences radically the physics, generating novel phenomena. These phenomena involve the properties of the system at equilibrium (freezing and glass transitions), as well as their dynamical evolution out-of-equilibrium (pinning, avalanches), giving rise to ergodicity breaking both in absence and in presence of quantum fluctuations (classical metastability, quantum localization). We discuss the main statistical physics models that are able to capture the phenomenology of these systems, as well as the powerful theoretical tools (replica theory, large deviations, random matrix theory, scaling arguments, strong-disorder expansions) that have been developed to characterize quantitatively their physics. These theoretical tools nowadays have a huge impact in a variety of fields that go well-beyond statistical physics (computer science, probability, condensed matter, theoretical biology). Below is a list of topics discussed during the course.

'''Finite-dimensional disordered systems:'''

* Introduction to disordered systems and to the spin glass transition.
* Interface growth. Directed polymers in random media. Scenarios for the glass transition: the glass transition in KPZ in d>2.
* Depinning and avalanches. Bienaymé-Galton-Watson processes.
* Anderson localization: introduction. Localization in 1D: transfer matrix and Lyapunov.

'''Mean-field disordered systems:'''

* The simplest spin-glass: solution of the Random Energy Model.
* The replica method: the solution of the spherical p-spin model. Sketch of the solution of Sherrington Kirkpatrick model (full RSB).
* Towards glassy dynamics: rugged landscapes. Slow dynamics and aging: the trap model.
* The Anderson model on the Bethe lattice: the mobility edge.

=Lectures and tutorials=

'''If the layout of the formulas is bad, it might be because you are using Safari. Try opening the wiki with Firefox or Chrome.'''

{| class="wikitable" border="1"
|-
! width="100"|Date
! width="500"| 14h00-15h45
! width="500"| 16h00-17h45

|-valign="top"

| Week 1 (19/01)
|
* [[L1_ICTS| Spin Glass Transition (Alberto)]]

|
* [[T-I| A dictionary. The REM: energy landscape (Valentina)]]  [[Media:2025 P1 solutions.pdf| Sol Prob.1 ]]
|-valign=“top"

|-valign=“top"
| Week 2 (26/01)
|
* [[L2_ICFP| Stochastic Interfaces and growth (Alberto)]]
|
* [[T-2|The REM: freezing, condensation, glassiness (Valentina)]]   [[Media:2025 P2 solutions.pdf| Sol Prob.2 ]]
|-valign=“top"

|-valign=“top"
| Week 3 (02/02)
|
* [[L-3|Directed polymer in random media (Alberto)]]
|
* [[L-4| KPZ and glassiness in finite dimension (Alberto)]] [[https://colab.research.google.com/drive/1PTya42ZS2kU87A-BxQFFIUDTs_k47men?usp=sharing| notebook]]
|-valign=“top"
| Week 4 (9/02) and Week 5 (16/02)
|
* [[T-3| Spin glasses, equilibrium: replicas, the steps (Valentina)]]  [[Media:2025 P3 solutions.pdf| Sol Prob.3 ]]
|
* [[T-4| Spin glasses, equilibrium: replicas, the interpretation (Valentina)]]   [[Media:2025 P4 solutions.pdf| Sol Probs. 4 ]] 
[[Media:2025_Parisi_scheme.pdf| Notes: Probing states with replicas]]
|-valign=“top"
| Week 6 (02/03)
|
* [[LBan-IV| Driven Disordered Materials (Alberto)]] [[Media:DISSYTS.pdf| Slides ]]
|
* [[LBan-V| Avalanches in Disordered Materials (Alberto)]]
|-valign=“top"

| Week 7 (9/03)
|
* [[L-7| Anderson localization: introduction (Alberto)]]
|
* [[T-5| Rugged landscapes: counting metastable states (Valentina)]]  
|-valign=“top"
| Week 8 (16/03)
|
* [[T-6| Rugged landscapes: stability of metastable states (Valentina)]] 
|
* [[T-7| Trap model and aging dynamics (Valentina)]] 
|-valign=“top"
| Week 9 (23/03)
|
* [[L-8| Localization in 1D, transfer matrix and Lyapunov exponent (Alberto)]] [[https://colab.research.google.com/drive/1ZJ0yvMrtflWNNmPfaRQ8KTteoWfm0bqk?usp=sharing| notebook]]
|
* [[L-9|Multifractality, tails (Alberto)]]
|}




= Exercises =

<li> Week 1: [[Media:Exercises_1-3.pdf| Exercises 1-3 on extreme value statistics]]
</li>

<li> Week 2:
[[Media:Tutorial_and_Exercise_4.pdf| Tutorial and Exercise 4 on random matrices]] 
[[Media:Exercises_5-6.pdf| Exercises 5-6 on the random energy model]]
</li>

<li> Week 3:
[[Media:Exercises 7&8.pdf| Exercises 7-8 on interfaces]]
</li>

<li> Week 4:
[[Media:Exercises 9-10.pdf| Exercises 9-10 on glassiness]]
</li>

<li> Week 5 and 6:
[[Media:11-12_Exercises.pdf| Exercises 11-12 on dynamics]]
</li>

<li> Week 7:
[[Media:Exercises_13_15.pdf| Exercises 13-15 on branching and localization]]
</li>

<li> Week 8:
[[XX| Exercises 16-17 on trap model and localization]]
</li>

= Evaluation and exam =

The exam will be on '''Monday, March 30th 2026'''. It will be written, 3h long. It consists of two parts:

Part 1: theory questions, see here for an example.

Part 2: you will be asked to solve pieces of the 17 exercises given to you in advance.

'''You are not allowed to bring any material (printed notes, handwritten notes) nor to use any device during the exam.'''

All relevant formulas will be provided in the text of the exam. There will be one printed version of the WIKI pages available to you to consult.

Main Page

2026-03-09T15:49:30Z

Rosso: /* Lectures and tutorials */

Welcome to the WIKI page of the M2 ICFP course on the Statistical Physics of Disordered Systems, second semester 2026.

= Where and When =

* Each Monday at 2pm - 6 pm, from January 19th to March 23rd. No lecture on 23/02/26.
* Room 202 in Jussieu campus, Tours 54-55 until 16th February
* Room 209 in Jussieu campus, Tours 56-66 209 from 2nd March '''Attention: ROOM CHANGE!'''
* Each session is a mixture of lectures and exercises.

= The Team =

* [https://vale1925.wixsite.com/vros Valentina Ros] - vale1925@gmail.com
* [http://lptms.u-psud.fr/alberto_rosso/ Alberto Rosso] - alberto.rosso74@gmail.com

= Course description =

This course deals with systems in which the presence of impurities or amorphous structures (in other words, of disorder) influences radically the physics, generating novel phenomena. These phenomena involve the properties of the system at equilibrium (freezing and glass transitions), as well as their dynamical evolution out-of-equilibrium (pinning, avalanches), giving rise to ergodicity breaking both in absence and in presence of quantum fluctuations (classical metastability, quantum localization). We discuss the main statistical physics models that are able to capture the phenomenology of these systems, as well as the powerful theoretical tools (replica theory, large deviations, random matrix theory, scaling arguments, strong-disorder expansions) that have been developed to characterize quantitatively their physics. These theoretical tools nowadays have a huge impact in a variety of fields that go well-beyond statistical physics (computer science, probability, condensed matter, theoretical biology). Below is a list of topics discussed during the course.

'''Finite-dimensional disordered systems:'''

* Introduction to disordered systems and to the spin glass transition.
* Interface growth. Directed polymers in random media. Scenarios for the glass transition: the glass transition in KPZ in d>2.
* Depinning and avalanches. Bienaymé-Galton-Watson processes.
* Anderson localization: introduction. Localization in 1D: transfer matrix and Lyapunov.

'''Mean-field disordered systems:'''

* The simplest spin-glass: solution of the Random Energy Model.
* The replica method: the solution of the spherical p-spin model. Sketch of the solution of Sherrington Kirkpatrick model (full RSB).
* Towards glassy dynamics: rugged landscapes. Slow dynamics and aging: the trap model.
* The Anderson model on the Bethe lattice: the mobility edge.

=Lectures and tutorials=

'''If the layout of the formulas is bad, it might be because you are using Safari. Try opening the wiki with Firefox or Chrome.'''

{| class="wikitable" border="1"
|-
! width="100"|Date
! width="500"| 14h00-15h45
! width="500"| 16h00-17h45

|-valign="top"

| Week 1 (19/01)
|
* [[L1_ICTS| Spin Glass Transition (Alberto)]]

|
* [[T-I| A dictionary. The REM: energy landscape (Valentina)]]  [[Media:2025 P1 solutions.pdf| Sol Prob.1 ]]
|-valign=“top"

|-valign=“top"
| Week 2 (26/01)
|
* [[L2_ICFP| Stochastic Interfaces and growth (Alberto)]]
|
* [[T-2|The REM: freezing, condensation, glassiness (Valentina)]]   [[Media:2025 P2 solutions.pdf| Sol Prob.2 ]]
|-valign=“top"

|-valign=“top"
| Week 3 (02/02)
|
* [[L-3|Directed polymer in random media (Alberto)]]
|
* [[L-4| KPZ and glassiness in finite dimension (Alberto)]] [[https://colab.research.google.com/drive/1PTya42ZS2kU87A-BxQFFIUDTs_k47men?usp=sharing| notebook]]
|-valign=“top"
| Week 4 (9/02) and Week 5 (16/02)
|
* [[T-3| Spin glasses, equilibrium: replicas, the steps (Valentina)]]  [[Media:2025 P3 solutions.pdf| Sol Prob.3 ]]
|
* [[T-4| Spin glasses, equilibrium: replicas, the interpretation (Valentina)]]   [[Media:2025 P4 solutions.pdf| Sol Probs. 4 ]] 
[[Media:2025_Parisi_scheme.pdf| Notes: Probing states with replicas]]
|-valign=“top"
| Week 6 (02/03)
|
* [[LBan-IV| Driven Disordered Materials <--[[Media:DISSYTS.pdf| Slides ]]--> (Alberto)]]
|
* [[LBan-V| Avalanches in Disordered Materials (Alberto)]]
|-valign=“top"

| Week 7 (9/03)
|
* [[L-7| Anderson localization: introduction (Alberto)]]
|
* [[T-5| Rugged landscapes: counting metastable states (Valentina)]]  
|-valign=“top"
| Week 8 (16/03)
|
* [[T-6| Rugged landscapes: stability of metastable states (Valentina)]] 
|
* [[T-7| Trap model and aging dynamics (Valentina)]] 
|-valign=“top"
| Week 9 (23/03)
|
* [[L-8| Localization in 1D, transfer matrix and Lyapunov exponent (Alberto)]] [[https://colab.research.google.com/drive/1ZJ0yvMrtflWNNmPfaRQ8KTteoWfm0bqk?usp=sharing| notebook]]
|
* [[L-9|Multifractality, tails (Alberto)]]
|}




= Exercises =

<li> Week 1: [[Media:Exercises_1-3.pdf| Exercises 1-3 on extreme value statistics]]
</li>

<li> Week 2:
[[Media:Tutorial_and_Exercise_4.pdf| Tutorial and Exercise 4 on random matrices]] 
[[Media:Exercises_5-6.pdf| Exercises 5-6 on the random energy model]]
</li>

<li> Week 3:
[[Media:Exercises 7&8.pdf| Exercises 7-8 on interfaces]]
</li>

<li> Week 4:
[[Media:Exercises 9-10.pdf| Exercises 9-10 on glassiness]]
</li>

<li> Week 5 and 6:
[[Media:11-12_Exercises.pdf| Exercises 11-12 on dynamics]]
</li>

<li> Week 7:
[[Media:Exercises_13_15.pdf| Exercises 13-15 on branching and localization]]
</li>

<li> Week 8:
[[XX| Exercises 16-17 on trap model and localization]]
</li>

= Evaluation and exam =

The exam will be on '''Monday, March 30th 2026'''. It will be written, 3h long. It consists of two parts:

Part 1: theory questions, see here for an example.

Part 2: you will be asked to solve pieces of the 17 exercises given to you in advance.

'''You are not allowed to bring any material (printed notes, handwritten notes) nor to use any device during the exam.'''

All relevant formulas will be provided in the text of the exam. There will be one printed version of the WIKI pages available to you to consult.

Main Page

2026-03-09T15:48:54Z

Rosso: /* Lectures and tutorials */

Welcome to the WIKI page of the M2 ICFP course on the Statistical Physics of Disordered Systems, second semester 2026.

= Where and When =

* Each Monday at 2pm - 6 pm, from January 19th to March 23rd. No lecture on 23/02/26.
* Room 202 in Jussieu campus, Tours 54-55 until 16th February
* Room 209 in Jussieu campus, Tours 56-66 209 from 2nd March '''Attention: ROOM CHANGE!'''
* Each session is a mixture of lectures and exercises.

= The Team =

* [https://vale1925.wixsite.com/vros Valentina Ros] - vale1925@gmail.com
* [http://lptms.u-psud.fr/alberto_rosso/ Alberto Rosso] - alberto.rosso74@gmail.com

= Course description =

This course deals with systems in which the presence of impurities or amorphous structures (in other words, of disorder) influences radically the physics, generating novel phenomena. These phenomena involve the properties of the system at equilibrium (freezing and glass transitions), as well as their dynamical evolution out-of-equilibrium (pinning, avalanches), giving rise to ergodicity breaking both in absence and in presence of quantum fluctuations (classical metastability, quantum localization). We discuss the main statistical physics models that are able to capture the phenomenology of these systems, as well as the powerful theoretical tools (replica theory, large deviations, random matrix theory, scaling arguments, strong-disorder expansions) that have been developed to characterize quantitatively their physics. These theoretical tools nowadays have a huge impact in a variety of fields that go well-beyond statistical physics (computer science, probability, condensed matter, theoretical biology). Below is a list of topics discussed during the course.

'''Finite-dimensional disordered systems:'''

* Introduction to disordered systems and to the spin glass transition.
* Interface growth. Directed polymers in random media. Scenarios for the glass transition: the glass transition in KPZ in d>2.
* Depinning and avalanches. Bienaymé-Galton-Watson processes.
* Anderson localization: introduction. Localization in 1D: transfer matrix and Lyapunov.

'''Mean-field disordered systems:'''

* The simplest spin-glass: solution of the Random Energy Model.
* The replica method: the solution of the spherical p-spin model. Sketch of the solution of Sherrington Kirkpatrick model (full RSB).
* Towards glassy dynamics: rugged landscapes. Slow dynamics and aging: the trap model.
* The Anderson model on the Bethe lattice: the mobility edge.

=Lectures and tutorials=

'''If the layout of the formulas is bad, it might be because you are using Safari. Try opening the wiki with Firefox or Chrome.'''

{| class="wikitable" border="1"
|-
! width="100"|Date
! width="500"| 14h00-15h45
! width="500"| 16h00-17h45

|-valign="top"

| Week 1 (19/01)
|
* [[L1_ICTS| Spin Glass Transition (Alberto)]]

|
* [[T-I| A dictionary. The REM: energy landscape (Valentina)]]  [[Media:2025 P1 solutions.pdf| Sol Prob.1 ]]
|-valign=“top"

|-valign=“top"
| Week 2 (26/01)
|
* [[L2_ICFP| Stochastic Interfaces and growth (Alberto)]]
|
* [[T-2|The REM: freezing, condensation, glassiness (Valentina)]]   [[Media:2025 P2 solutions.pdf| Sol Prob.2 ]]
|-valign=“top"

|-valign=“top"
| Week 3 (02/02)
|
* [[L-3|Directed polymer in random media (Alberto)]]
|
* [[L-4| KPZ and glassiness in finite dimension (Alberto)]] [[https://colab.research.google.com/drive/1PTya42ZS2kU87A-BxQFFIUDTs_k47men?usp=sharing| notebook]]
|-valign=“top"
| Week 4 (9/02) and Week 5 (16/02)
|
* [[T-3| Spin glasses, equilibrium: replicas, the steps (Valentina)]]  [[Media:2025 P3 solutions.pdf| Sol Prob.3 ]]
|
* [[T-4| Spin glasses, equilibrium: replicas, the interpretation (Valentina)]]   [[Media:2025 P4 solutions.pdf| Sol Probs. 4 ]] 
[[Media:2025_Parisi_scheme.pdf| Notes: Probing states with replicas]]
|-valign=“top"
| Week 6 (02/03)
|
* [[LBan-IV| Driven Disordered Materials  (Alberto)]]
|
* [[LBan-V| Avalanches in Disordered Materials (Alberto)]]
|-valign=“top"

| Week 7 (9/03)
|
* [[L-7| Anderson localization: introduction (Alberto)]]
|
* [[T-5| Rugged landscapes: counting metastable states (Valentina)]]  
|-valign=“top"
| Week 8 (16/03)
|
* [[T-6| Rugged landscapes: stability of metastable states (Valentina)]] 
|
* [[T-7| Trap model and aging dynamics (Valentina)]] 
|-valign=“top"
| Week 9 (23/03)
|
* [[L-8| Localization in 1D, transfer matrix and Lyapunov exponent (Alberto)]] [[https://colab.research.google.com/drive/1ZJ0yvMrtflWNNmPfaRQ8KTteoWfm0bqk?usp=sharing| notebook]]
|
* [[L-9|Multifractality, tails (Alberto)]]
|}




= Exercises =

<li> Week 1: [[Media:Exercises_1-3.pdf| Exercises 1-3 on extreme value statistics]]
</li>

<li> Week 2:
[[Media:Tutorial_and_Exercise_4.pdf| Tutorial and Exercise 4 on random matrices]] 
[[Media:Exercises_5-6.pdf| Exercises 5-6 on the random energy model]]
</li>

<li> Week 3:
[[Media:Exercises 7&8.pdf| Exercises 7-8 on interfaces]]
</li>

<li> Week 4:
[[Media:Exercises 9-10.pdf| Exercises 9-10 on glassiness]]
</li>

<li> Week 5 and 6:
[[Media:11-12_Exercises.pdf| Exercises 11-12 on dynamics]]
</li>

<li> Week 7:
[[Media:Exercises_13_15.pdf| Exercises 13-15 on branching and localization]]
</li>

<li> Week 8:
[[XX| Exercises 16-17 on trap model and localization]]
</li>

= Evaluation and exam =

The exam will be on '''Monday, March 30th 2026'''. It will be written, 3h long. It consists of two parts:

Part 1: theory questions, see here for an example.

Part 2: you will be asked to solve pieces of the 17 exercises given to you in advance.

'''You are not allowed to bring any material (printed notes, handwritten notes) nor to use any device during the exam.'''

All relevant formulas will be provided in the text of the exam. There will be one printed version of the WIKI pages available to you to consult.

File:DISSYST.pdf

2026-03-09T15:46:24Z

Rosso:

L-7

2026-03-09T09:50:37Z

Rosso: /* Evolution of a Gaussian wave packet */

'''Goal.''' This lecture introduces the phenomenon of localization. Localization is a '''wave phenomenon induced by disorder''' that suppresses transport in a system.

== Short recap: wavefunctions and eigenstates ==

Before discussing localization we briefly recall a few basic notions of quantum mechanics.

A quantum particle in one dimension is described by a '''wavefunction''' <math>\psi(x,t)</math>.
The quantity <math>|\psi(x,t)|^2</math> is the probability density of finding the particle at position <math>x</math> at time <math>t</math>. The wavefunction therefore satisfies the normalization condition

<math display="block">
\int_{-\infty}^{\infty} dx\, |\psi(x,t)|^2 = 1 .
</math>

The time evolution of the wavefunction is governed by the Schrödinger equation

<math display="block">
i\hbar \partial_t \psi(x,t) = H\psi(x,t),
</math>

where <math>H</math> is the Hamiltonian of the system. For a particle moving in one dimension in a potential <math>V(x)</math>, the Hamiltonian reads

<math display="block">
H = -\frac{\hbar^2}{2m}\partial_x^2 + V(x).
</math>

=== Eigenstates ===

A particularly important class of solutions are the '''eigenstates''' of the Hamiltonian

<math display="block">
H\psi_n(x) = E_n \psi_n(x).
</math>

If the particle is in an eigenstate the full solution reads

<math display="block">
\psi_n(x,t)=\psi_n(x)e^{-iE_n t/\hbar}.
</math>

The probability density <math>|\psi_n(x,t)|^2</math> is therefore independent of time: eigenstates are stationary states.
Two qualitatively different situations may occur depending on the form of the potential.

=== Discrete and continuous spectra ===

* '''Discrete spectrum (bound states)'''

The energies take isolated values <math>E_n</math>.
This typically happens when the particle is confined in a finite region
(for instance in a potential well).

The corresponding eigenfunctions are normalizable,

<math display="block">
\int dx\, |\psi_n(x)|^2 = 1,
</math>

and the particle remains localized in space. These states are called
bound states.

* '''Continuous spectrum (continuum states)'''

The energy can take any value in a continuous interval.
This occurs for instance for a free particle or for a particle with
energy above the confining potential.

A simple example is provided by plane waves

<math display="block">
\psi_k(x)=\frac{1}{\sqrt{2\pi}} e^{ikx},
\qquad
E=\frac{\hbar^2 k^2}{2m}.
</math>
These states are not normalizable in the usual sense. They are instead
normalized using Dirac delta functions,

<math display="block">
\int dx\, \psi_k^*(x)\psi_{k'}(x)=\delta(k-k').
</math>

They form a continuous basis that can be used to construct physical
wave packets.

=== Superposition principle ===

The Schrödinger equation is linear. As a consequence, any linear combination
of solutions is again a solution. This property is known as the
superposition principle.

If the spectrum is discrete, an arbitrary wavefunction can be expanded
in the basis of eigenstates

<math display="block">
\psi(x,t)=\sum_n c_n e^{-iE_n t/\hbar}\psi_n(x).
</math>

If the spectrum is continuous the expansion becomes an integral

<math display="block">
\psi(x,t)=\int dk\, c(k)\psi_k(x,t),
\qquad
\int dk\, |c(k)|^2=1.
</math>

By choosing the coefficients <math>c(k)</math> appropriately one can construct
a localized '''wave packet''' describing a particle initially confined in space.

In many physical situations the Hamiltonian has a mixed spectrum, containing both discrete bound states and continuum states. In this case a general wavefunction can be expressed as a superposition involving contributions from both sets of eigenstates.
A simple example is a finite potential well: it supports a finite number of bound states with discrete energies, while particles with higher energies belong to the continuum and correspond to scattering states.

=== Probability current ===

Besides the probability density one can define a '''probability current'''

<math display="block">
J(x,t)=\frac{\hbar}{2mi}
\left(
\psi^* \frac{d\psi}{dx}
-
\psi \frac{d\psi^*}{dx}
\right).
</math>

This quantity measures the '''flow of probability''' across a point in space.

Its expression follows from combining the Schrödinger equation with the
conservation of probability. Indeed the probability density

<math>
\rho(x,t)=|\psi(x,t)|^2
</math>

satisfies the continuity equation

<math display="block">
\partial_t \rho(x,t) + \partial_x J(x,t)=0,
</math>

which has the same structure as a conservation law in hydrodynamics.

For a plane wave

<math display="block">
\psi(x)=e^{ikx}
</math>

one finds

<math display="block">
J=\frac{\hbar k}{m}.
</math>

Thus plane waves describe particles propagating through space and carrying
a non–zero probability current.

If the wavefunction is real the probability current vanishes, since the two
terms in the expression of <math>J</math> cancel. This is the case for bound
states in one dimension: for a real potential the eigenfunctions can be
chosen real, and bound states in 1D are non–degenerate. Physically this
corresponds to a '''standing wave''' rather than a propagating wave.

=== Scattering states ===

Transport problems often involve a localized potential (for instance a sample
or a potential barrier) surrounded by free space. Outside this region the
solutions of the Schrödinger equation are plane waves.

When a particle interacts with the sample, the wavefunction generally contains
three contributions:

* an incoming wave,
* a reflected wave,
* a transmitted wave.

The corresponding solutions are called '''scattering states'''.

For example, a particle incoming from the left is described asymptotically by

<math display="block">
\psi_{k,L}(x)=
\begin{cases}
e^{ikx}+r\, e^{-ikx} & x\to -\infty \\
t\, e^{ikx} & x\to +\infty
\end{cases}
</math>

The first term represents the incoming wave, the second the reflected wave,
while the transmitted wave propagates to the right with amplitude <math>t</math>.

Using the expression of the probability current, one finds that a plane wave
<math>e^{ikx}</math> carries a current

<math display="block">
J=\frac{\hbar k}{m}.
</math>

Thus the incoming, reflected and transmitted waves correspond to incoming,
reflected and transmitted probability currents.

Since probability is conserved, the current must be the same on both sides
of the sample. This implies the relation

<math display="block">
R + T = 1,
</math>

where

<math display="block">
R = |r|^2, \qquad T = |t|^2
</math>

are the reflection and transmission probabilities.

== Free particles and ballistic behaviour ==

We now illustrate how a localized quantum particle evolves in the absence of disorder.

For a free particle the Hamiltonian reads

<math display="block">
H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}.
</math>

As discussed above, the eigenstates of this Hamiltonian are plane waves with a continuous spectrum

<math display="block">
\psi_k(x,t)=\frac{1}{\sqrt{2\pi}}e^{ikx}e^{-iE_k t/\hbar},
\qquad
E_k=\frac{\hbar^2k^2}{2m}.
</math>

Since plane waves are completely delocalized, a physical particle must be described by a superposition of eigenstates, forming a '''wave packet'''

<math display="block">
\psi(x,t)=\int_{-\infty}^{\infty}dk\,c(k)\psi_k(x,t),
\qquad
\int_{-\infty}^{\infty}dk\,|c(k)|^2=1.
</math>

=== Evolution of a Gaussian wave packet ===

* '''Initial state'''

At time <math>t=0</math> consider a Gaussian wave packet

<math display="block">
\psi(x,0)=\frac{e^{-x^2/(4a^2)}}{(2\pi a^2)^{1/4}},
\qquad
|\psi(x,0)|^2=\frac{e^{-x^2/(2a^2)}}{\sqrt{2\pi a^2}}.
</math>

Show that the coefficients of the plane-wave decomposition are

<math display="block">
c(k)=\left(\frac{2a^2}{\pi}\right)^{1/4}e^{-a^2k^2}.
</math>

* '''Time evolution'''

Define the spreading velocity

<math display="block">
v_s=\frac{\hbar}{2ma}.
</math>

Show that the time evolution of the packet is

<math display="block">
\psi(x,t)=
\frac{e^{-x^2/(4a^2(1+i (v_s/a) t))}}
{[2\pi a^2(1+i (v_s/a) t)]^{1/4}}.
</math>

* '''Ballistic spreading'''

The probability density becomes

<math display="block">
|\psi(x,t)|^2=
\frac{e^{-x^2/(2a^2(1+(v_s/a)^2 t^2))}}
{\sqrt{2\pi a^2(1+(v_s/a)^2 t^2)}}.
</math>

Hence

<math display="block">
\sqrt{\langle x^2\rangle}
=
\left(\int dx\,x^2|\psi(x,t)|^2\right)^{1/2}
=
a\sqrt{1+(v_s/a)^2 t^2}.
</math>

At long times

<math display="block">
\sqrt{\langle x^2\rangle}\sim v_s t.
</math>

This linear growth is called '''ballistic spreading'''.

It should be contrasted with two other possible transport regimes:

* '''Diffusive motion'''
<math>\sqrt{\langle x^2\rangle}\sim \sqrt{t}</math>

* '''Localized regime'''
<math>\sqrt{\langle x^2\rangle}</math> saturates at long times.

Understanding how disorder modifies ballistic motion and eventually suppresses transport is the main goal of the following sections.

== Localization of the packet: general idea and experiment ==

[[File:Localization1DB.png|thumb|left|x240px|BEC condensate expanding in a 1D disordered potential. Billy et al., Nature (2008).]]

In 1958, P. Anderson proposed that disorder can suppress transport in quantum systems. This phenomenon, known as '''Anderson localization''', has since been observed both numerically and experimentally.

In the experiment shown here, a Gaussian packet of a Bose–Einstein condensate is prepared in a harmonic trap. When the trap is removed the cloud initially expands but quickly stops spreading and remains localized.

To understand this behaviour we must solve the eigenstate problem of the Hamiltonian in the presence of disorder.

[[File:Localization1dB.png|thumb|left|x140px|Semilog plot of the particle density. Billy et al., Nature (2008).]]

In a disordered potential an eigenstate of energy <math>E_k</math> has the form

<math display="block">
\psi_k(x,t)=\psi_k(x)e^{-iE_k t/\hbar}.
</math>

The spatial part of the wavefunction is localized around some position <math>\bar{x}</math> and decays exponentially

<math display="block">
\psi_k(x)\sim
e^{-|x-\bar{x}|/\xi_{\text{loc}}}.
</math>

Here <math>\xi_{\text{loc}}</math> is the localization length.

Because the eigenstates are localized, the wave packet is built from states localized near its initial position. Eigenstates far from the packet contribute exponentially little, and states composing the packet decay exponentially away from their center.

As a consequence transport far from the initial position of the particle is '''exponentially suppressed'''.
== Diffusive transport ==
In most materials weak disorder does not lead to localization but to diffusive transport. A simple microscopic description is provided by the Drude model.

Electrons in a metal do not start from rest. Even in the absence of an electric field they move with a typical velocity of order the Fermi velocity

<math display="block">
v_F=\frac{\hbar k_F}{m}.
</math>

Their motion is interrupted by scattering events with impurities, defects, or phonons. These collisions occur on average after a characteristic time <math>\tau</math>, called the '''mean free time'''.

Between two collisions electrons move essentially freely. The typical distance traveled during this time is the mean free path:

<math display="block">
\ell = v_F \tau .
</math>

The motion therefore has two distinct regimes:

* On length scales smaller than <math>\ell</math>, electrons propagate ballistically.
* On length scales much larger than <math>\ell</math>, repeated random scattering events lead to diffusion.

We now introduce an external electric field <math>E</math>. An electron of charge <math>-e</math> obeys the equation of motion

<math display="block">
m\frac{dv}{dt}=-eE.
</math>

Between two collisions the electron accelerates under the electric field. After each collision its velocity is randomized again. Averaging over many such scattering events leads to a small systematic drift velocity

<math display="block">
v_d=-\frac{eE\tau}{m}.
</math>

Note that this drift velocity is extremely small compared with the typical electron velocity, <math>
v_d \ll v_F </math>. Thus electrons move very rapidly with random directions, while the electric field only produces a tiny bias in this motion. If the electron density is <math>n</math>, the electric current density is

<math display="block">
j=-ne\,v_d.
</math>

Substituting the drift velocity gives

<math display="block">
j=\frac{ne^2\tau}{m}E.
</math>

This provides the microscopic origin of Ohm's law

<math display="block">
j=\sigma E,
\qquad
\sigma=\frac{ne^2\tau}{m}.
</math>

It is important to stress that the Drude model is purely '''classical'''. It ignores the wave nature of electrons and quantum interference effects. A fully quantum derivation of Ohm's law is much more subtle and typically relies on Green functions and diagrammatic methods.

In the following we will focus on the regime where quantum interference becomes important and can eventually suppress diffusion altogether, leading to localization.

== Conductance ==

The conductance <math>G</math> is the inverse of the resistance. Understanding how it depends on the system size will be the starting point for the scaling theory of localization.

=== Diffusive transport ===

Consider a wire of length <math>L</math> and cross section <math>S</math>. The total current is

<math display="block">
I=jS,
</math>

while the voltage drop is

<math display="block">
V=EL.
</math>

Using Ohm's law <math>j=\sigma E</math> we obtain

<math display="block">
I=\sigma \frac{S}{L} V.
</math>

Therefore

<math display="block">
G=\frac{I}{V}=\sigma \frac{S}{L}.
</math>

For a sample of linear size <math>L</math> in spatial dimension <math>d</math>, the cross section scales as

<math display="block">
S\sim L^{d-1}.
</math>

Hence

<math display="block">
G(L)\sim \sigma L^{d-2}.
</math>

This power-law scaling is the characteristic signature of diffusive transport.

=== Localized regime ===

When disorder is strong diffusion is suppressed and the system becomes insulating.

In Exercise 15 we studied transport using the Landauer picture. In this approach the conductance is proportional to the transmission probability through the sample:

<math display="block">
G=\frac{e^2}{\hbar}|t(E_F)|^2.
</math>

The factor <math>e^2/\hbar</math> sets the natural quantum scale of conductance for a single transport channel.

In a localized system the transmission probability typically decays exponentially with the system size

<math display="block">
|t(E_F)|^2 \sim e^{-2L/\xi_{\text{loc}}},
</math>

where <math>\xi_{\text{loc}}</math> is the localization length.

This leads to

<math display="block">
G(L)\sim e^{-2L/\xi_{\text{loc}}}.
</math>

Thus in the localized phase the conductance decreases exponentially with the system size.

== The “Gang of Four” scaling theory ==

The two behaviors discussed above raise a natural question: how does the conductance of a disordered system evolve when the system size is increased?

In a famous PRL (1979), Abrahams, Anderson, Licciardello and Ramakrishnan proposed a scaling theory of localization that addresses this question.

The key quantity is the '''dimensionless conductance'''

<math display="block">
g=\frac{G\hbar}{e^2}.
</math>

The factor <math>e^2/\hbar</math> is the natural quantum unit of conductance (see the Landauer formula). The quantity <math>g</math> therefore measures the effective number of conducting channels.

Instead of computing the conductance microscopically, the scaling theory studies how <math>g</math> evolves when the system size <math>L</math> is increased.

This evolution is described by the scaling equation

<math display="block">
\frac{d\ln g}{d\ln L}=\beta(g).
</math>

The function <math>\beta(g)</math> depends only on <math>g</math> and on the spatial dimension.

Two limiting regimes are known.

* '''Metallic regime''' (<math>g\gg1</math>)

Transport is diffusive. From the Drude result

<math display="block">
G(L)\sim L^{d-2},
</math>

we obtain

<math display="block">
\beta(g)\to d-2.
</math>

* '''Localized regime''' (<math>g\ll1</math>)

The conductance decreases exponentially

<math display="block">
g(L)\sim e^{-L/\xi_{\text{loc}}},
</math>

which implies

<math display="block">
\beta(g)\sim \ln g.
</math>

The simplest scenario is that the beta function is monotonic.

This leads to a striking dimensional prediction:

* for <math>d>2</math> the beta function changes sign and a '''metal–insulator transition''' exists
* for <math>d\le2</math> the beta function remains negative and '''all states are localized'''.

The scaling theory therefore predicts that in low dimension disorder inevitably destroys diffusion.

To understand how localization emerges microscopically, we must now study the spectrum and eigenstates of quantum particles moving in a random potential.

L-7

2026-03-09T08:25:48Z

Rosso: /* Diffusive transport */

'''Goal.''' This lecture introduces the phenomenon of localization. Localization is a '''wave phenomenon induced by disorder''' that suppresses transport in a system.

== Short recap: wavefunctions and eigenstates ==

Before discussing localization we briefly recall a few basic notions of quantum mechanics.

A quantum particle in one dimension is described by a '''wavefunction''' <math>\psi(x,t)</math>.
The quantity <math>|\psi(x,t)|^2</math> is the probability density of finding the particle at position <math>x</math> at time <math>t</math>. The wavefunction therefore satisfies the normalization condition

<math display="block">
\int_{-\infty}^{\infty} dx\, |\psi(x,t)|^2 = 1 .
</math>

The time evolution of the wavefunction is governed by the Schrödinger equation

<math display="block">
i\hbar \partial_t \psi(x,t) = H\psi(x,t),
</math>

where <math>H</math> is the Hamiltonian of the system. For a particle moving in one dimension in a potential <math>V(x)</math>, the Hamiltonian reads

<math display="block">
H = -\frac{\hbar^2}{2m}\partial_x^2 + V(x).
</math>

=== Eigenstates ===

A particularly important class of solutions are the '''eigenstates''' of the Hamiltonian

<math display="block">
H\psi_n(x) = E_n \psi_n(x).
</math>

If the particle is in an eigenstate the full solution reads

<math display="block">
\psi_n(x,t)=\psi_n(x)e^{-iE_n t/\hbar}.
</math>

The probability density <math>|\psi_n(x,t)|^2</math> is therefore independent of time: eigenstates are stationary states.
Two qualitatively different situations may occur depending on the form of the potential.

=== Discrete and continuous spectra ===

* '''Discrete spectrum (bound states)'''

The energies take isolated values <math>E_n</math>.
This typically happens when the particle is confined in a finite region
(for instance in a potential well).

The corresponding eigenfunctions are normalizable,

<math display="block">
\int dx\, |\psi_n(x)|^2 = 1,
</math>

and the particle remains localized in space. These states are called
bound states.

* '''Continuous spectrum (continuum states)'''

The energy can take any value in a continuous interval.
This occurs for instance for a free particle or for a particle with
energy above the confining potential.

A simple example is provided by plane waves

<math display="block">
\psi_k(x)=\frac{1}{\sqrt{2\pi}} e^{ikx},
\qquad
E=\frac{\hbar^2 k^2}{2m}.
</math>
These states are not normalizable in the usual sense. They are instead
normalized using Dirac delta functions,

<math display="block">
\int dx\, \psi_k^*(x)\psi_{k'}(x)=\delta(k-k').
</math>

They form a continuous basis that can be used to construct physical
wave packets.

=== Superposition principle ===

The Schrödinger equation is linear. As a consequence, any linear combination
of solutions is again a solution. This property is known as the
superposition principle.

If the spectrum is discrete, an arbitrary wavefunction can be expanded
in the basis of eigenstates

<math display="block">
\psi(x,t)=\sum_n c_n e^{-iE_n t/\hbar}\psi_n(x).
</math>

If the spectrum is continuous the expansion becomes an integral

<math display="block">
\psi(x,t)=\int dk\, c(k)\psi_k(x,t),
\qquad
\int dk\, |c(k)|^2=1.
</math>

By choosing the coefficients <math>c(k)</math> appropriately one can construct
a localized '''wave packet''' describing a particle initially confined in space.

In many physical situations the Hamiltonian has a mixed spectrum, containing both discrete bound states and continuum states. In this case a general wavefunction can be expressed as a superposition involving contributions from both sets of eigenstates.
A simple example is a finite potential well: it supports a finite number of bound states with discrete energies, while particles with higher energies belong to the continuum and correspond to scattering states.

=== Probability current ===

Besides the probability density one can define a '''probability current'''

<math display="block">
J(x,t)=\frac{\hbar}{2mi}
\left(
\psi^* \frac{d\psi}{dx}
-
\psi \frac{d\psi^*}{dx}
\right).
</math>

This quantity measures the '''flow of probability''' across a point in space.

Its expression follows from combining the Schrödinger equation with the
conservation of probability. Indeed the probability density

<math>
\rho(x,t)=|\psi(x,t)|^2
</math>

satisfies the continuity equation

<math display="block">
\partial_t \rho(x,t) + \partial_x J(x,t)=0,
</math>

which has the same structure as a conservation law in hydrodynamics.

For a plane wave

<math display="block">
\psi(x)=e^{ikx}
</math>

one finds

<math display="block">
J=\frac{\hbar k}{m}.
</math>

Thus plane waves describe particles propagating through space and carrying
a non–zero probability current.

If the wavefunction is real the probability current vanishes, since the two
terms in the expression of <math>J</math> cancel. This is the case for bound
states in one dimension: for a real potential the eigenfunctions can be
chosen real, and bound states in 1D are non–degenerate. Physically this
corresponds to a '''standing wave''' rather than a propagating wave.

=== Scattering states ===

Transport problems often involve a localized potential (for instance a sample
or a potential barrier) surrounded by free space. Outside this region the
solutions of the Schrödinger equation are plane waves.

When a particle interacts with the sample, the wavefunction generally contains
three contributions:

* an incoming wave,
* a reflected wave,
* a transmitted wave.

The corresponding solutions are called '''scattering states'''.

For example, a particle incoming from the left is described asymptotically by

<math display="block">
\psi_{k,L}(x)=
\begin{cases}
e^{ikx}+r\, e^{-ikx} & x\to -\infty \\
t\, e^{ikx} & x\to +\infty
\end{cases}
</math>

The first term represents the incoming wave, the second the reflected wave,
while the transmitted wave propagates to the right with amplitude <math>t</math>.

Using the expression of the probability current, one finds that a plane wave
<math>e^{ikx}</math> carries a current

<math display="block">
J=\frac{\hbar k}{m}.
</math>

Thus the incoming, reflected and transmitted waves correspond to incoming,
reflected and transmitted probability currents.

Since probability is conserved, the current must be the same on both sides
of the sample. This implies the relation

<math display="block">
R + T = 1,
</math>

where

<math display="block">
R = |r|^2, \qquad T = |t|^2
</math>

are the reflection and transmission probabilities.

== Free particles and ballistic behaviour ==

We now illustrate how a localized quantum particle evolves in the absence of disorder.

For a free particle the Hamiltonian reads

<math display="block">
H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}.
</math>

As discussed above, the eigenstates of this Hamiltonian are plane waves with a continuous spectrum

<math display="block">
\psi_k(x,t)=\frac{1}{\sqrt{2\pi}}e^{ikx}e^{-iE_k t/\hbar},
\qquad
E_k=\frac{\hbar^2k^2}{2m}.
</math>

Since plane waves are completely delocalized, a physical particle must be described by a superposition of eigenstates, forming a '''wave packet'''

<math display="block">
\psi(x,t)=\int_{-\infty}^{\infty}dk\,c(k)\psi_k(x,t),
\qquad
\int_{-\infty}^{\infty}dk\,|c(k)|^2=1.
</math>

=== Evolution of a Gaussian wave packet ===

* '''Initial state'''

At time <math>t=0</math> consider a Gaussian wave packet

<math display="block">
\psi(x,0)=\frac{e^{-x^2/(4a^2)}}{(2\pi a^2)^{1/4}},
\qquad
|\psi(x,0)|^2=\frac{e^{-x^2/(2a^2)}}{\sqrt{2\pi a^2}}.
</math>

Show that the coefficients of the plane-wave decomposition are

<math display="block">
c(k)=\left(\frac{2a^2}{\pi}\right)^{1/4}e^{-a^2k^2}.
</math>

* '''Time evolution'''

Define the spreading velocity

<math display="block">
v_s=\frac{\hbar}{2ma^2}.
</math>

Show that the time evolution of the packet is

<math display="block">
\psi(x,t)=
\frac{e^{-x^2/(4a^2(1+i v_s t))}}
{[2\pi a^2(1+i v_s t)]^{1/4}}.
</math>

* '''Ballistic spreading'''

The probability density becomes

<math display="block">
|\psi(x,t)|^2=
\frac{e^{-x^2/(2a^2(1+v_s^2 t^2))}}
{\sqrt{2\pi a^2(1+v_s^2 t^2)}}.
</math>

Hence

<math display="block">
\sqrt{\langle x^2\rangle}
=
\left(\int dx\,x^2|\psi(x,t)|^2\right)^{1/2}
=
a\sqrt{1+v_s^2 t^2}.
</math>

At long times

<math display="block">
\sqrt{\langle x^2\rangle}\sim v_s t.
</math>

This linear growth is called '''ballistic spreading'''.

It should be contrasted with two other possible transport regimes:

* '''Diffusive motion'''
<math>\sqrt{\langle x^2\rangle}\sim \sqrt{t}</math>

* '''Localized regime'''
<math>\sqrt{\langle x^2\rangle}</math> saturates at long times.

Understanding how disorder modifies ballistic motion and eventually suppresses transport is the main goal of the following sections.

== Localization of the packet: general idea and experiment ==

[[File:Localization1DB.png|thumb|left|x240px|BEC condensate expanding in a 1D disordered potential. Billy et al., Nature (2008).]]

In 1958, P. Anderson proposed that disorder can suppress transport in quantum systems. This phenomenon, known as '''Anderson localization''', has since been observed both numerically and experimentally.

In the experiment shown here, a Gaussian packet of a Bose–Einstein condensate is prepared in a harmonic trap. When the trap is removed the cloud initially expands but quickly stops spreading and remains localized.

To understand this behaviour we must solve the eigenstate problem of the Hamiltonian in the presence of disorder.

[[File:Localization1dB.png|thumb|left|x140px|Semilog plot of the particle density. Billy et al., Nature (2008).]]

In a disordered potential an eigenstate of energy <math>E_k</math> has the form

<math display="block">
\psi_k(x,t)=\psi_k(x)e^{-iE_k t/\hbar}.
</math>

The spatial part of the wavefunction is localized around some position <math>\bar{x}</math> and decays exponentially

<math display="block">
\psi_k(x)\sim
e^{-|x-\bar{x}|/\xi_{\text{loc}}}.
</math>

Here <math>\xi_{\text{loc}}</math> is the localization length.

Because the eigenstates are localized, the wave packet is built from states localized near its initial position. Eigenstates far from the packet contribute exponentially little, and states composing the packet decay exponentially away from their center.

As a consequence transport far from the initial position of the particle is '''exponentially suppressed'''.
== Diffusive transport ==
In most materials weak disorder does not lead to localization but to diffusive transport. A simple microscopic description is provided by the Drude model.

Electrons in a metal do not start from rest. Even in the absence of an electric field they move with a typical velocity of order the Fermi velocity

<math display="block">
v_F=\frac{\hbar k_F}{m}.
</math>

Their motion is interrupted by scattering events with impurities, defects, or phonons. These collisions occur on average after a characteristic time <math>\tau</math>, called the '''mean free time'''.

Between two collisions electrons move essentially freely. The typical distance traveled during this time is the mean free path:

<math display="block">
\ell = v_F \tau .
</math>

The motion therefore has two distinct regimes:

* On length scales smaller than <math>\ell</math>, electrons propagate ballistically.
* On length scales much larger than <math>\ell</math>, repeated random scattering events lead to diffusion.

We now introduce an external electric field <math>E</math>. An electron of charge <math>-e</math> obeys the equation of motion

<math display="block">
m\frac{dv}{dt}=-eE.
</math>

Between two collisions the electron accelerates under the electric field. After each collision its velocity is randomized again. Averaging over many such scattering events leads to a small systematic drift velocity

<math display="block">
v_d=-\frac{eE\tau}{m}.
</math>

Note that this drift velocity is extremely small compared with the typical electron velocity, <math>
v_d \ll v_F </math>. Thus electrons move very rapidly with random directions, while the electric field only produces a tiny bias in this motion. If the electron density is <math>n</math>, the electric current density is

<math display="block">
j=-ne\,v_d.
</math>

Substituting the drift velocity gives

<math display="block">
j=\frac{ne^2\tau}{m}E.
</math>

This provides the microscopic origin of Ohm's law

<math display="block">
j=\sigma E,
\qquad
\sigma=\frac{ne^2\tau}{m}.
</math>

It is important to stress that the Drude model is purely '''classical'''. It ignores the wave nature of electrons and quantum interference effects. A fully quantum derivation of Ohm's law is much more subtle and typically relies on Green functions and diagrammatic methods.

In the following we will focus on the regime where quantum interference becomes important and can eventually suppress diffusion altogether, leading to localization.

== Conductance ==

The conductance <math>G</math> is the inverse of the resistance. Understanding how it depends on the system size will be the starting point for the scaling theory of localization.

=== Diffusive transport ===

Consider a wire of length <math>L</math> and cross section <math>S</math>. The total current is

<math display="block">
I=jS,
</math>

while the voltage drop is

<math display="block">
V=EL.
</math>

Using Ohm's law <math>j=\sigma E</math> we obtain

<math display="block">
I=\sigma \frac{S}{L} V.
</math>

Therefore

<math display="block">
G=\frac{I}{V}=\sigma \frac{S}{L}.
</math>

For a sample of linear size <math>L</math> in spatial dimension <math>d</math>, the cross section scales as

<math display="block">
S\sim L^{d-1}.
</math>

Hence

<math display="block">
G(L)\sim \sigma L^{d-2}.
</math>

This power-law scaling is the characteristic signature of diffusive transport.

=== Localized regime ===

When disorder is strong diffusion is suppressed and the system becomes insulating.

In Exercise 15 we studied transport using the Landauer picture. In this approach the conductance is proportional to the transmission probability through the sample:

<math display="block">
G=\frac{e^2}{\hbar}|t(E_F)|^2.
</math>

The factor <math>e^2/\hbar</math> sets the natural quantum scale of conductance for a single transport channel.

In a localized system the transmission probability typically decays exponentially with the system size

<math display="block">
|t(E_F)|^2 \sim e^{-2L/\xi_{\text{loc}}},
</math>

where <math>\xi_{\text{loc}}</math> is the localization length.

This leads to

<math display="block">
G(L)\sim e^{-2L/\xi_{\text{loc}}}.
</math>

Thus in the localized phase the conductance decreases exponentially with the system size.

== The “Gang of Four” scaling theory ==

The two behaviors discussed above raise a natural question: how does the conductance of a disordered system evolve when the system size is increased?

In a famous PRL (1979), Abrahams, Anderson, Licciardello and Ramakrishnan proposed a scaling theory of localization that addresses this question.

The key quantity is the '''dimensionless conductance'''

<math display="block">
g=\frac{G\hbar}{e^2}.
</math>

The factor <math>e^2/\hbar</math> is the natural quantum unit of conductance (see the Landauer formula). The quantity <math>g</math> therefore measures the effective number of conducting channels.

Instead of computing the conductance microscopically, the scaling theory studies how <math>g</math> evolves when the system size <math>L</math> is increased.

This evolution is described by the scaling equation

<math display="block">
\frac{d\ln g}{d\ln L}=\beta(g).
</math>

The function <math>\beta(g)</math> depends only on <math>g</math> and on the spatial dimension.

Two limiting regimes are known.

* '''Metallic regime''' (<math>g\gg1</math>)

Transport is diffusive. From the Drude result

<math display="block">
G(L)\sim L^{d-2},
</math>

we obtain

<math display="block">
\beta(g)\to d-2.
</math>

* '''Localized regime''' (<math>g\ll1</math>)

The conductance decreases exponentially

<math display="block">
g(L)\sim e^{-L/\xi_{\text{loc}}},
</math>

which implies

<math display="block">
\beta(g)\sim \ln g.
</math>

The simplest scenario is that the beta function is monotonic.

This leads to a striking dimensional prediction:

* for <math>d>2</math> the beta function changes sign and a '''metal–insulator transition''' exists
* for <math>d\le2</math> the beta function remains negative and '''all states are localized'''.

The scaling theory therefore predicts that in low dimension disorder inevitably destroys diffusion.

To understand how localization emerges microscopically, we must now study the spectrum and eigenstates of quantum particles moving in a random potential.

L-7

2026-03-09T08:10:48Z

Rosso: /* Diffusive transport */

'''Goal.''' This lecture introduces the phenomenon of localization. Localization is a '''wave phenomenon induced by disorder''' that suppresses transport in a system.

== Short recap: wavefunctions and eigenstates ==

Before discussing localization we briefly recall a few basic notions of quantum mechanics.

A quantum particle in one dimension is described by a '''wavefunction''' <math>\psi(x,t)</math>.
The quantity <math>|\psi(x,t)|^2</math> is the probability density of finding the particle at position <math>x</math> at time <math>t</math>. The wavefunction therefore satisfies the normalization condition

<math display="block">
\int_{-\infty}^{\infty} dx\, |\psi(x,t)|^2 = 1 .
</math>

The time evolution of the wavefunction is governed by the Schrödinger equation

<math display="block">
i\hbar \partial_t \psi(x,t) = H\psi(x,t),
</math>

where <math>H</math> is the Hamiltonian of the system. For a particle moving in one dimension in a potential <math>V(x)</math>, the Hamiltonian reads

<math display="block">
H = -\frac{\hbar^2}{2m}\partial_x^2 + V(x).
</math>

=== Eigenstates ===

A particularly important class of solutions are the '''eigenstates''' of the Hamiltonian

<math display="block">
H\psi_n(x) = E_n \psi_n(x).
</math>

If the particle is in an eigenstate the full solution reads

<math display="block">
\psi_n(x,t)=\psi_n(x)e^{-iE_n t/\hbar}.
</math>

The probability density <math>|\psi_n(x,t)|^2</math> is therefore independent of time: eigenstates are stationary states.
Two qualitatively different situations may occur depending on the form of the potential.

=== Discrete and continuous spectra ===

* '''Discrete spectrum (bound states)'''

The energies take isolated values <math>E_n</math>.
This typically happens when the particle is confined in a finite region
(for instance in a potential well).

The corresponding eigenfunctions are normalizable,

<math display="block">
\int dx\, |\psi_n(x)|^2 = 1,
</math>

and the particle remains localized in space. These states are called
bound states.

* '''Continuous spectrum (continuum states)'''

The energy can take any value in a continuous interval.
This occurs for instance for a free particle or for a particle with
energy above the confining potential.

A simple example is provided by plane waves

<math display="block">
\psi_k(x)=\frac{1}{\sqrt{2\pi}} e^{ikx},
\qquad
E=\frac{\hbar^2 k^2}{2m}.
</math>
These states are not normalizable in the usual sense. They are instead
normalized using Dirac delta functions,

<math display="block">
\int dx\, \psi_k^*(x)\psi_{k'}(x)=\delta(k-k').
</math>

They form a continuous basis that can be used to construct physical
wave packets.

=== Superposition principle ===

The Schrödinger equation is linear. As a consequence, any linear combination
of solutions is again a solution. This property is known as the
superposition principle.

If the spectrum is discrete, an arbitrary wavefunction can be expanded
in the basis of eigenstates

<math display="block">
\psi(x,t)=\sum_n c_n e^{-iE_n t/\hbar}\psi_n(x).
</math>

If the spectrum is continuous the expansion becomes an integral

<math display="block">
\psi(x,t)=\int dk\, c(k)\psi_k(x,t),
\qquad
\int dk\, |c(k)|^2=1.
</math>

By choosing the coefficients <math>c(k)</math> appropriately one can construct
a localized '''wave packet''' describing a particle initially confined in space.

In many physical situations the Hamiltonian has a mixed spectrum, containing both discrete bound states and continuum states. In this case a general wavefunction can be expressed as a superposition involving contributions from both sets of eigenstates.
A simple example is a finite potential well: it supports a finite number of bound states with discrete energies, while particles with higher energies belong to the continuum and correspond to scattering states.

=== Probability current ===

Besides the probability density one can define a '''probability current'''

<math display="block">
J(x,t)=\frac{\hbar}{2mi}
\left(
\psi^* \frac{d\psi}{dx}
-
\psi \frac{d\psi^*}{dx}
\right).
</math>

This quantity measures the '''flow of probability''' across a point in space.

Its expression follows from combining the Schrödinger equation with the
conservation of probability. Indeed the probability density

<math>
\rho(x,t)=|\psi(x,t)|^2
</math>

satisfies the continuity equation

<math display="block">
\partial_t \rho(x,t) + \partial_x J(x,t)=0,
</math>

which has the same structure as a conservation law in hydrodynamics.

For a plane wave

<math display="block">
\psi(x)=e^{ikx}
</math>

one finds

<math display="block">
J=\frac{\hbar k}{m}.
</math>

Thus plane waves describe particles propagating through space and carrying
a non–zero probability current.

If the wavefunction is real the probability current vanishes, since the two
terms in the expression of <math>J</math> cancel. This is the case for bound
states in one dimension: for a real potential the eigenfunctions can be
chosen real, and bound states in 1D are non–degenerate. Physically this
corresponds to a '''standing wave''' rather than a propagating wave.

=== Scattering states ===

Transport problems often involve a localized potential (for instance a sample
or a potential barrier) surrounded by free space. Outside this region the
solutions of the Schrödinger equation are plane waves.

When a particle interacts with the sample, the wavefunction generally contains
three contributions:

* an incoming wave,
* a reflected wave,
* a transmitted wave.

The corresponding solutions are called '''scattering states'''.

For example, a particle incoming from the left is described asymptotically by

<math display="block">
\psi_{k,L}(x)=
\begin{cases}
e^{ikx}+r\, e^{-ikx} & x\to -\infty \\
t\, e^{ikx} & x\to +\infty
\end{cases}
</math>

The first term represents the incoming wave, the second the reflected wave,
while the transmitted wave propagates to the right with amplitude <math>t</math>.

Using the expression of the probability current, one finds that a plane wave
<math>e^{ikx}</math> carries a current

<math display="block">
J=\frac{\hbar k}{m}.
</math>

Thus the incoming, reflected and transmitted waves correspond to incoming,
reflected and transmitted probability currents.

Since probability is conserved, the current must be the same on both sides
of the sample. This implies the relation

<math display="block">
R + T = 1,
</math>

where

<math display="block">
R = |r|^2, \qquad T = |t|^2
</math>

are the reflection and transmission probabilities.

== Free particles and ballistic behaviour ==

We now illustrate how a localized quantum particle evolves in the absence of disorder.

For a free particle the Hamiltonian reads

<math display="block">
H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}.
</math>

As discussed above, the eigenstates of this Hamiltonian are plane waves with a continuous spectrum

<math display="block">
\psi_k(x,t)=\frac{1}{\sqrt{2\pi}}e^{ikx}e^{-iE_k t/\hbar},
\qquad
E_k=\frac{\hbar^2k^2}{2m}.
</math>

Since plane waves are completely delocalized, a physical particle must be described by a superposition of eigenstates, forming a '''wave packet'''

<math display="block">
\psi(x,t)=\int_{-\infty}^{\infty}dk\,c(k)\psi_k(x,t),
\qquad
\int_{-\infty}^{\infty}dk\,|c(k)|^2=1.
</math>

=== Evolution of a Gaussian wave packet ===

* '''Initial state'''

At time <math>t=0</math> consider a Gaussian wave packet

<math display="block">
\psi(x,0)=\frac{e^{-x^2/(4a^2)}}{(2\pi a^2)^{1/4}},
\qquad
|\psi(x,0)|^2=\frac{e^{-x^2/(2a^2)}}{\sqrt{2\pi a^2}}.
</math>

Show that the coefficients of the plane-wave decomposition are

<math display="block">
c(k)=\left(\frac{2a^2}{\pi}\right)^{1/4}e^{-a^2k^2}.
</math>

* '''Time evolution'''

Define the spreading velocity

<math display="block">
v_s=\frac{\hbar}{2ma^2}.
</math>

Show that the time evolution of the packet is

<math display="block">
\psi(x,t)=
\frac{e^{-x^2/(4a^2(1+i v_s t))}}
{[2\pi a^2(1+i v_s t)]^{1/4}}.
</math>

* '''Ballistic spreading'''

The probability density becomes

<math display="block">
|\psi(x,t)|^2=
\frac{e^{-x^2/(2a^2(1+v_s^2 t^2))}}
{\sqrt{2\pi a^2(1+v_s^2 t^2)}}.
</math>

Hence

<math display="block">
\sqrt{\langle x^2\rangle}
=
\left(\int dx\,x^2|\psi(x,t)|^2\right)^{1/2}
=
a\sqrt{1+v_s^2 t^2}.
</math>

At long times

<math display="block">
\sqrt{\langle x^2\rangle}\sim v_s t.
</math>

This linear growth is called '''ballistic spreading'''.

It should be contrasted with two other possible transport regimes:

* '''Diffusive motion'''
<math>\sqrt{\langle x^2\rangle}\sim \sqrt{t}</math>

* '''Localized regime'''
<math>\sqrt{\langle x^2\rangle}</math> saturates at long times.

Understanding how disorder modifies ballistic motion and eventually suppresses transport is the main goal of the following sections.

== Localization of the packet: general idea and experiment ==

[[File:Localization1DB.png|thumb|left|x240px|BEC condensate expanding in a 1D disordered potential. Billy et al., Nature (2008).]]

In 1958, P. Anderson proposed that disorder can suppress transport in quantum systems. This phenomenon, known as '''Anderson localization''', has since been observed both numerically and experimentally.

In the experiment shown here, a Gaussian packet of a Bose–Einstein condensate is prepared in a harmonic trap. When the trap is removed the cloud initially expands but quickly stops spreading and remains localized.

To understand this behaviour we must solve the eigenstate problem of the Hamiltonian in the presence of disorder.

[[File:Localization1dB.png|thumb|left|x140px|Semilog plot of the particle density. Billy et al., Nature (2008).]]

In a disordered potential an eigenstate of energy <math>E_k</math> has the form

<math display="block">
\psi_k(x,t)=\psi_k(x)e^{-iE_k t/\hbar}.
</math>

The spatial part of the wavefunction is localized around some position <math>\bar{x}</math> and decays exponentially

<math display="block">
\psi_k(x)\sim
e^{-|x-\bar{x}|/\xi_{\text{loc}}}.
</math>

Here <math>\xi_{\text{loc}}</math> is the localization length.

Because the eigenstates are localized, the wave packet is built from states localized near its initial position. Eigenstates far from the packet contribute exponentially little, and states composing the packet decay exponentially away from their center.

As a consequence transport far from the initial position of the particle is '''exponentially suppressed'''.
== Diffusive transport ==
In most materials weak disorder does not lead to localization but to diffusive transport. A simple microscopic description is provided by the Drude model.

Electrons in a metal do not start from rest. Even in the absence of an electric field they move with a typical velocity of order the '''Fermi velocity'''

<math display="block">
v_F=\frac{\hbar k_F}{m}.
</math>

Their motion is interrupted by scattering events with impurities, defects, or phonons. These collisions occur on average after a characteristic time <math>\tau</math>, called the '''mean free time'''.

Between two collisions electrons move essentially freely. The typical distance traveled during this time is the mean free path:

<math display="block">
\ell = v_F \tau .
</math>

The motion therefore has two distinct regimes:

* On length scales smaller than <math>\ell</math>, electrons propagate ballistically.
* On length scales much larger than <math>\ell</math>, repeated random scattering events lead to diffusion.

We now introduce an external electric field <math>E</math>. An electron of charge <math>-e</math> obeys the equation of motion

<math display="block">
m\frac{dv}{dt}=-eE.
</math>

Between two collisions the electron accelerates under the electric field. After each collision its velocity is randomized again. Averaging over many such scattering events leads to a small systematic drift velocity

<math display="block">
v_d=-\frac{eE\tau}{m}.
</math>

Note that this drift velocity is extremely small compared with the typical electron velocity, <math>
v_d \ll v_F </math>. Thus electrons move very rapidly with random directions, while the electric field only produces a tiny bias in this motion. If the electron density is <math>n</math>, the electric current density is

<math display="block">
j=-ne\,v_d.
</math>

Substituting the drift velocity gives

<math display="block">
j=\frac{ne^2\tau}{m}E.
</math>

This provides the microscopic origin of Ohm's law

<math display="block">
j=\sigma E,
\qquad
\sigma=\frac{ne^2\tau}{m}.
</math>

It is important to stress that the Drude model is purely '''classical'''. It ignores the wave nature of electrons and quantum interference effects. A fully quantum derivation of Ohm's law is much more subtle and typically relies on Green functions and diagrammatic methods.

In the following we will focus on the regime where quantum interference becomes important and can eventually suppress diffusion altogether, leading to localization.

== Conductance ==

The conductance <math>G</math> is the inverse of the resistance. Understanding how it depends on the system size will be the starting point for the scaling theory of localization.

=== Diffusive transport ===

Consider a wire of length <math>L</math> and cross section <math>S</math>. The total current is

<math display="block">
I=jS,
</math>

while the voltage drop is

<math display="block">
V=EL.
</math>

Using Ohm's law <math>j=\sigma E</math> we obtain

<math display="block">
I=\sigma \frac{S}{L} V.
</math>

Therefore

<math display="block">
G=\frac{I}{V}=\sigma \frac{S}{L}.
</math>

For a sample of linear size <math>L</math> in spatial dimension <math>d</math>, the cross section scales as

<math display="block">
S\sim L^{d-1}.
</math>

Hence

<math display="block">
G(L)\sim \sigma L^{d-2}.
</math>

This power-law scaling is the characteristic signature of diffusive transport.

=== Localized regime ===

When disorder is strong diffusion is suppressed and the system becomes insulating.

In Exercise 15 we studied transport using the Landauer picture. In this approach the conductance is proportional to the transmission probability through the sample:

<math display="block">
G=\frac{e^2}{\hbar}|t(E_F)|^2.
</math>

The factor <math>e^2/\hbar</math> sets the natural quantum scale of conductance for a single transport channel.

In a localized system the transmission probability typically decays exponentially with the system size

<math display="block">
|t(E_F)|^2 \sim e^{-2L/\xi_{\text{loc}}},
</math>

where <math>\xi_{\text{loc}}</math> is the localization length.

This leads to

<math display="block">
G(L)\sim e^{-2L/\xi_{\text{loc}}}.
</math>

Thus in the localized phase the conductance decreases exponentially with the system size.

== The “Gang of Four” scaling theory ==

The two behaviors discussed above raise a natural question: how does the conductance of a disordered system evolve when the system size is increased?

In a famous PRL (1979), Abrahams, Anderson, Licciardello and Ramakrishnan proposed a scaling theory of localization that addresses this question.

The key quantity is the '''dimensionless conductance'''

<math display="block">
g=\frac{G\hbar}{e^2}.
</math>

The factor <math>e^2/\hbar</math> is the natural quantum unit of conductance (see the Landauer formula). The quantity <math>g</math> therefore measures the effective number of conducting channels.

Instead of computing the conductance microscopically, the scaling theory studies how <math>g</math> evolves when the system size <math>L</math> is increased.

This evolution is described by the scaling equation

<math display="block">
\frac{d\ln g}{d\ln L}=\beta(g).
</math>

The function <math>\beta(g)</math> depends only on <math>g</math> and on the spatial dimension.

Two limiting regimes are known.

* '''Metallic regime''' (<math>g\gg1</math>)

Transport is diffusive. From the Drude result

<math display="block">
G(L)\sim L^{d-2},
</math>

we obtain

<math display="block">
\beta(g)\to d-2.
</math>

* '''Localized regime''' (<math>g\ll1</math>)

The conductance decreases exponentially

<math display="block">
g(L)\sim e^{-L/\xi_{\text{loc}}},
</math>

which implies

<math display="block">
\beta(g)\sim \ln g.
</math>

The simplest scenario is that the beta function is monotonic.

This leads to a striking dimensional prediction:

* for <math>d>2</math> the beta function changes sign and a '''metal–insulator transition''' exists
* for <math>d\le2</math> the beta function remains negative and '''all states are localized'''.

The scaling theory therefore predicts that in low dimension disorder inevitably destroys diffusion.

To understand how localization emerges microscopically, we must now study the spectrum and eigenstates of quantum particles moving in a random potential.

L-7

2026-03-08T15:29:03Z

Rosso: /* Superposition principle */

'''Goal.''' This lecture introduces the phenomenon of localization. Localization is a '''wave phenomenon induced by disorder''' that suppresses transport in a system.

== Short recap: wavefunctions and eigenstates ==

Before discussing localization we briefly recall a few basic notions of quantum mechanics.

A quantum particle in one dimension is described by a '''wavefunction''' <math>\psi(x,t)</math>.
The quantity <math>|\psi(x,t)|^2</math> is the probability density of finding the particle at position <math>x</math> at time <math>t</math>. The wavefunction therefore satisfies the normalization condition

<math display="block">
\int_{-\infty}^{\infty} dx\, |\psi(x,t)|^2 = 1 .
</math>

The time evolution of the wavefunction is governed by the Schrödinger equation

<math display="block">
i\hbar \partial_t \psi(x,t) = H\psi(x,t),
</math>

where <math>H</math> is the Hamiltonian of the system. For a particle moving in one dimension in a potential <math>V(x)</math>, the Hamiltonian reads

<math display="block">
H = -\frac{\hbar^2}{2m}\partial_x^2 + V(x).
</math>

=== Eigenstates ===

A particularly important class of solutions are the '''eigenstates''' of the Hamiltonian

<math display="block">
H\psi_n(x) = E_n \psi_n(x).
</math>

If the particle is in an eigenstate the full solution reads

<math display="block">
\psi_n(x,t)=\psi_n(x)e^{-iE_n t/\hbar}.
</math>

The probability density <math>|\psi_n(x,t)|^2</math> is therefore independent of time: eigenstates are stationary states.
Two qualitatively different situations may occur depending on the form of the potential.

=== Discrete and continuous spectra ===

* '''Discrete spectrum (bound states)'''

The energies take isolated values <math>E_n</math>.
This typically happens when the particle is confined in a finite region
(for instance in a potential well).

The corresponding eigenfunctions are normalizable,

<math display="block">
\int dx\, |\psi_n(x)|^2 = 1,
</math>

and the particle remains localized in space. These states are called
bound states.

* '''Continuous spectrum (continuum states)'''

The energy can take any value in a continuous interval.
This occurs for instance for a free particle or for a particle with
energy above the confining potential.

A simple example is provided by plane waves

<math display="block">
\psi_k(x)=\frac{1}{\sqrt{2\pi}} e^{ikx},
\qquad
E=\frac{\hbar^2 k^2}{2m}.
</math>
These states are not normalizable in the usual sense. They are instead
normalized using Dirac delta functions,

<math display="block">
\int dx\, \psi_k^*(x)\psi_{k'}(x)=\delta(k-k').
</math>

They form a continuous basis that can be used to construct physical
wave packets.

=== Superposition principle ===

The Schrödinger equation is linear. As a consequence, any linear combination
of solutions is again a solution. This property is known as the
superposition principle.

If the spectrum is discrete, an arbitrary wavefunction can be expanded
in the basis of eigenstates

<math display="block">
\psi(x,t)=\sum_n c_n e^{-iE_n t/\hbar}\psi_n(x).
</math>

If the spectrum is continuous the expansion becomes an integral

<math display="block">
\psi(x,t)=\int dk\, c(k)\psi_k(x,t),
\qquad
\int dk\, |c(k)|^2=1.
</math>

By choosing the coefficients <math>c(k)</math> appropriately one can construct
a localized '''wave packet''' describing a particle initially confined in space.

In many physical situations the Hamiltonian has a mixed spectrum, containing both discrete bound states and continuum states. In this case a general wavefunction can be expressed as a superposition involving contributions from both sets of eigenstates.
A simple example is a finite potential well: it supports a finite number of bound states with discrete energies, while particles with higher energies belong to the continuum and correspond to scattering states.

=== Probability current ===

Besides the probability density one can define a '''probability current'''

<math display="block">
J(x,t)=\frac{\hbar}{2mi}
\left(
\psi^* \frac{d\psi}{dx}
-
\psi \frac{d\psi^*}{dx}
\right).
</math>

This quantity measures the '''flow of probability''' across a point in space.

Its expression follows from combining the Schrödinger equation with the
conservation of probability. Indeed the probability density

<math>
\rho(x,t)=|\psi(x,t)|^2
</math>

satisfies the continuity equation

<math display="block">
\partial_t \rho(x,t) + \partial_x J(x,t)=0,
</math>

which has the same structure as a conservation law in hydrodynamics.

For a plane wave

<math display="block">
\psi(x)=e^{ikx}
</math>

one finds

<math display="block">
J=\frac{\hbar k}{m}.
</math>

Thus plane waves describe particles propagating through space and carrying
a non–zero probability current.

If the wavefunction is real the probability current vanishes, since the two
terms in the expression of <math>J</math> cancel. This is the case for bound
states in one dimension: for a real potential the eigenfunctions can be
chosen real, and bound states in 1D are non–degenerate. Physically this
corresponds to a '''standing wave''' rather than a propagating wave.

=== Scattering states ===

Transport problems often involve a localized potential (for instance a sample
or a potential barrier) surrounded by free space. Outside this region the
solutions of the Schrödinger equation are plane waves.

When a particle interacts with the sample, the wavefunction generally contains
three contributions:

* an incoming wave,
* a reflected wave,
* a transmitted wave.

The corresponding solutions are called '''scattering states'''.

For example, a particle incoming from the left is described asymptotically by

<math display="block">
\psi_{k,L}(x)=
\begin{cases}
e^{ikx}+r\, e^{-ikx} & x\to -\infty \\
t\, e^{ikx} & x\to +\infty
\end{cases}
</math>

The first term represents the incoming wave, the second the reflected wave,
while the transmitted wave propagates to the right with amplitude <math>t</math>.

Using the expression of the probability current, one finds that a plane wave
<math>e^{ikx}</math> carries a current

<math display="block">
J=\frac{\hbar k}{m}.
</math>

Thus the incoming, reflected and transmitted waves correspond to incoming,
reflected and transmitted probability currents.

Since probability is conserved, the current must be the same on both sides
of the sample. This implies the relation

<math display="block">
R + T = 1,
</math>

where

<math display="block">
R = |r|^2, \qquad T = |t|^2
</math>

are the reflection and transmission probabilities.

== Free particles and ballistic behaviour ==

We now illustrate how a localized quantum particle evolves in the absence of disorder.

For a free particle the Hamiltonian reads

<math display="block">
H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}.
</math>

As discussed above, the eigenstates of this Hamiltonian are plane waves with a continuous spectrum

<math display="block">
\psi_k(x,t)=\frac{1}{\sqrt{2\pi}}e^{ikx}e^{-iE_k t/\hbar},
\qquad
E_k=\frac{\hbar^2k^2}{2m}.
</math>

Since plane waves are completely delocalized, a physical particle must be described by a superposition of eigenstates, forming a '''wave packet'''

<math display="block">
\psi(x,t)=\int_{-\infty}^{\infty}dk\,c(k)\psi_k(x,t),
\qquad
\int_{-\infty}^{\infty}dk\,|c(k)|^2=1.
</math>

=== Evolution of a Gaussian wave packet ===

* '''Initial state'''

At time <math>t=0</math> consider a Gaussian wave packet

<math display="block">
\psi(x,0)=\frac{e^{-x^2/(4a^2)}}{(2\pi a^2)^{1/4}},
\qquad
|\psi(x,0)|^2=\frac{e^{-x^2/(2a^2)}}{\sqrt{2\pi a^2}}.
</math>

Show that the coefficients of the plane-wave decomposition are

<math display="block">
c(k)=\left(\frac{2a^2}{\pi}\right)^{1/4}e^{-a^2k^2}.
</math>

* '''Time evolution'''

Define the spreading velocity

<math display="block">
v_s=\frac{\hbar}{2ma^2}.
</math>

Show that the time evolution of the packet is

<math display="block">
\psi(x,t)=
\frac{e^{-x^2/(4a^2(1+i v_s t))}}
{[2\pi a^2(1+i v_s t)]^{1/4}}.
</math>

* '''Ballistic spreading'''

The probability density becomes

<math display="block">
|\psi(x,t)|^2=
\frac{e^{-x^2/(2a^2(1+v_s^2 t^2))}}
{\sqrt{2\pi a^2(1+v_s^2 t^2)}}.
</math>

Hence

<math display="block">
\sqrt{\langle x^2\rangle}
=
\left(\int dx\,x^2|\psi(x,t)|^2\right)^{1/2}
=
a\sqrt{1+v_s^2 t^2}.
</math>

At long times

<math display="block">
\sqrt{\langle x^2\rangle}\sim v_s t.
</math>

This linear growth is called '''ballistic spreading'''.

It should be contrasted with two other possible transport regimes:

* '''Diffusive motion'''
<math>\sqrt{\langle x^2\rangle}\sim \sqrt{t}</math>

* '''Localized regime'''
<math>\sqrt{\langle x^2\rangle}</math> saturates at long times.

Understanding how disorder modifies ballistic motion and eventually suppresses transport is the main goal of the following sections.

== Localization of the packet: general idea and experiment ==

[[File:Localization1DB.png|thumb|left|x240px|BEC condensate expanding in a 1D disordered potential. Billy et al., Nature (2008).]]

In 1958, P. Anderson proposed that disorder can suppress transport in quantum systems. This phenomenon, known as '''Anderson localization''', has since been observed both numerically and experimentally.

In the experiment shown here, a Gaussian packet of a Bose–Einstein condensate is prepared in a harmonic trap. When the trap is removed the cloud initially expands but quickly stops spreading and remains localized.

To understand this behaviour we must solve the eigenstate problem of the Hamiltonian in the presence of disorder.

[[File:Localization1dB.png|thumb|left|x140px|Semilog plot of the particle density. Billy et al., Nature (2008).]]

In a disordered potential an eigenstate of energy <math>E_k</math> has the form

<math display="block">
\psi_k(x,t)=\psi_k(x)e^{-iE_k t/\hbar}.
</math>

The spatial part of the wavefunction is localized around some position <math>\bar{x}</math> and decays exponentially

<math display="block">
\psi_k(x)\sim
e^{-|x-\bar{x}|/\xi_{\text{loc}}}.
</math>

Here <math>\xi_{\text{loc}}</math> is the localization length.

Because the eigenstates are localized, the wave packet is built from states localized near its initial position. Eigenstates far from the packet contribute exponentially little, and states composing the packet decay exponentially away from their center.

As a consequence transport far from the initial position of the particle is '''exponentially suppressed'''.
== Diffusive transport ==

In most materials weak disorder does not lead to localization but to diffusive transport. A simple microscopic description is provided by the Drude model. In this classical picture electrons move freely between collisions with impurities or lattice defects. These scattering events occur on average after a characteristic time <math>\tau</math>, called the mean free time.

Under the action of an electric field <math>E</math>, an electron of charge <math>-e</math> obeys

<math display="block">
m\frac{dv}{dt}=-eE.
</math>

Between two collisions the electron accelerates, but collisions randomize its velocity. Averaging over many such events leads to a stationary drift velocity

<math display="block">
v_d=-\frac{eE\tau}{m}.
</math>

If the electron density is <math>n</math>, the electric current density is

<math display="block">
j=-ne\,v_d.
</math>

Substituting the drift velocity gives

<math display="block">
j=\frac{ne^2\tau}{m}E.
</math>

This is the microscopic origin of Ohm's law

<math display="block">
j=\sigma E,
\qquad
\sigma=\frac{ne^2\tau}{m}.
</math>

It is important to stress that the Drude model is purely '''classical'''. It ignores the wave nature of electrons and quantum interference effects. A fully quantum derivation of Ohm's law is much more subtle and typically relies on Green functions and diagrammatic methods. Here we will instead focus on the regime where quantum interference becomes crucial and can eventually suppress diffusion altogether.
== Conductance ==

The conductance <math>G</math> is the inverse of the resistance. Understanding how it depends on the system size will be the starting point for the scaling theory of localization.

=== Diffusive transport ===

Consider a wire of length <math>L</math> and cross section <math>S</math>. The total current is

<math display="block">
I=jS,
</math>

while the voltage drop is

<math display="block">
V=EL.
</math>

Using Ohm's law <math>j=\sigma E</math> we obtain

<math display="block">
I=\sigma \frac{S}{L} V.
</math>

Therefore

<math display="block">
G=\frac{I}{V}=\sigma \frac{S}{L}.
</math>

For a sample of linear size <math>L</math> in spatial dimension <math>d</math>, the cross section scales as

<math display="block">
S\sim L^{d-1}.
</math>

Hence

<math display="block">
G(L)\sim \sigma L^{d-2}.
</math>

This power-law scaling is the characteristic signature of diffusive transport.

=== Localized regime ===

When disorder is strong diffusion is suppressed and the system becomes insulating.

In Exercise 15 we studied transport using the Landauer picture. In this approach the conductance is proportional to the transmission probability through the sample:

<math display="block">
G=\frac{e^2}{\hbar}|t(E_F)|^2.
</math>

The factor <math>e^2/\hbar</math> sets the natural quantum scale of conductance for a single transport channel.

In a localized system the transmission probability typically decays exponentially with the system size

<math display="block">
|t(E_F)|^2 \sim e^{-2L/\xi_{\text{loc}}},
</math>

where <math>\xi_{\text{loc}}</math> is the localization length.

This leads to

<math display="block">
G(L)\sim e^{-2L/\xi_{\text{loc}}}.
</math>

Thus in the localized phase the conductance decreases exponentially with the system size.

== The “Gang of Four” scaling theory ==

The two behaviors discussed above raise a natural question: how does the conductance of a disordered system evolve when the system size is increased?

In a famous PRL (1979), Abrahams, Anderson, Licciardello and Ramakrishnan proposed a scaling theory of localization that addresses this question.

The key quantity is the '''dimensionless conductance'''

<math display="block">
g=\frac{G\hbar}{e^2}.
</math>

The factor <math>e^2/\hbar</math> is the natural quantum unit of conductance (see the Landauer formula). The quantity <math>g</math> therefore measures the effective number of conducting channels.

Instead of computing the conductance microscopically, the scaling theory studies how <math>g</math> evolves when the system size <math>L</math> is increased.

This evolution is described by the scaling equation

<math display="block">
\frac{d\ln g}{d\ln L}=\beta(g).
</math>

The function <math>\beta(g)</math> depends only on <math>g</math> and on the spatial dimension.

Two limiting regimes are known.

* '''Metallic regime''' (<math>g\gg1</math>)

Transport is diffusive. From the Drude result

<math display="block">
G(L)\sim L^{d-2},
</math>

we obtain

<math display="block">
\beta(g)\to d-2.
</math>

* '''Localized regime''' (<math>g\ll1</math>)

The conductance decreases exponentially

<math display="block">
g(L)\sim e^{-L/\xi_{\text{loc}}},
</math>

which implies

<math display="block">
\beta(g)\sim \ln g.
</math>

The simplest scenario is that the beta function is monotonic.

This leads to a striking dimensional prediction:

* for <math>d>2</math> the beta function changes sign and a '''metal–insulator transition''' exists
* for <math>d\le2</math> the beta function remains negative and '''all states are localized'''.

The scaling theory therefore predicts that in low dimension disorder inevitably destroys diffusion.

To understand how localization emerges microscopically, we must now study the spectrum and eigenstates of quantum particles moving in a random potential.

L-7

2026-03-08T15:28:36Z

Rosso: /* Eigenstates */

'''Goal.''' This lecture introduces the phenomenon of localization. Localization is a '''wave phenomenon induced by disorder''' that suppresses transport in a system.

== Short recap: wavefunctions and eigenstates ==

Before discussing localization we briefly recall a few basic notions of quantum mechanics.

A quantum particle in one dimension is described by a '''wavefunction''' <math>\psi(x,t)</math>.
The quantity <math>|\psi(x,t)|^2</math> is the probability density of finding the particle at position <math>x</math> at time <math>t</math>. The wavefunction therefore satisfies the normalization condition

<math display="block">
\int_{-\infty}^{\infty} dx\, |\psi(x,t)|^2 = 1 .
</math>

The time evolution of the wavefunction is governed by the Schrödinger equation

<math display="block">
i\hbar \partial_t \psi(x,t) = H\psi(x,t),
</math>

where <math>H</math> is the Hamiltonian of the system. For a particle moving in one dimension in a potential <math>V(x)</math>, the Hamiltonian reads

<math display="block">
H = -\frac{\hbar^2}{2m}\partial_x^2 + V(x).
</math>

=== Eigenstates ===

A particularly important class of solutions are the '''eigenstates''' of the Hamiltonian

<math display="block">
H\psi_n(x) = E_n \psi_n(x).
</math>

If the particle is in an eigenstate the full solution reads

<math display="block">
\psi_n(x,t)=\psi_n(x)e^{-iE_n t/\hbar}.
</math>

The probability density <math>|\psi_n(x,t)|^2</math> is therefore independent of time: eigenstates are stationary states.
Two qualitatively different situations may occur depending on the form of the potential.

=== Discrete and continuous spectra ===

* '''Discrete spectrum (bound states)'''

The energies take isolated values <math>E_n</math>.
This typically happens when the particle is confined in a finite region
(for instance in a potential well).

The corresponding eigenfunctions are normalizable,

<math display="block">
\int dx\, |\psi_n(x)|^2 = 1,
</math>

and the particle remains localized in space. These states are called
bound states.

* '''Continuous spectrum (continuum states)'''

The energy can take any value in a continuous interval.
This occurs for instance for a free particle or for a particle with
energy above the confining potential.

A simple example is provided by plane waves

<math display="block">
\psi_k(x)=\frac{1}{\sqrt{2\pi}} e^{ikx},
\qquad
E=\frac{\hbar^2 k^2}{2m}.
</math>
These states are not normalizable in the usual sense. They are instead
normalized using Dirac delta functions,

<math display="block">
\int dx\, \psi_k^*(x)\psi_{k'}(x)=\delta(k-k').
</math>

They form a continuous basis that can be used to construct physical
wave packets.

=== Superposition principle ===

The Schrödinger equation is linear. As a consequence, any linear combination
of solutions is again a solution. This property is known as the
superposition principle.

If the spectrum is discrete, an arbitrary wavefunction can be expanded
in the basis of eigenstates

<math display="block">
\psi(x,t)=\sum_n c_n e^{-iE_n t/\hbar}\psi_n(x).
</math>

If the spectrum is continuous the expansion becomes an integral

<math display="block">
\psi(x,t)=\int dk\, c(k)\psi_k(x,t),
\qquad
\int dk\, |c(k)|^2=1.
</math>

By choosing the coefficients <math>c(k)</math> appropriately one can construct
a localized '''wave packet''' describing a particle initially confined in space.

=== Probability current ===

Besides the probability density one can define a '''probability current'''

<math display="block">
J(x,t)=\frac{\hbar}{2mi}
\left(
\psi^* \frac{d\psi}{dx}
-
\psi \frac{d\psi^*}{dx}
\right).
</math>

This quantity measures the '''flow of probability''' across a point in space.

Its expression follows from combining the Schrödinger equation with the
conservation of probability. Indeed the probability density

<math>
\rho(x,t)=|\psi(x,t)|^2
</math>

satisfies the continuity equation

<math display="block">
\partial_t \rho(x,t) + \partial_x J(x,t)=0,
</math>

which has the same structure as a conservation law in hydrodynamics.

For a plane wave

<math display="block">
\psi(x)=e^{ikx}
</math>

one finds

<math display="block">
J=\frac{\hbar k}{m}.
</math>

Thus plane waves describe particles propagating through space and carrying
a non–zero probability current.

If the wavefunction is real the probability current vanishes, since the two
terms in the expression of <math>J</math> cancel. This is the case for bound
states in one dimension: for a real potential the eigenfunctions can be
chosen real, and bound states in 1D are non–degenerate. Physically this
corresponds to a '''standing wave''' rather than a propagating wave.

=== Scattering states ===

Transport problems often involve a localized potential (for instance a sample
or a potential barrier) surrounded by free space. Outside this region the
solutions of the Schrödinger equation are plane waves.

When a particle interacts with the sample, the wavefunction generally contains
three contributions:

* an incoming wave,
* a reflected wave,
* a transmitted wave.

The corresponding solutions are called '''scattering states'''.

For example, a particle incoming from the left is described asymptotically by

<math display="block">
\psi_{k,L}(x)=
\begin{cases}
e^{ikx}+r\, e^{-ikx} & x\to -\infty \\
t\, e^{ikx} & x\to +\infty
\end{cases}
</math>

The first term represents the incoming wave, the second the reflected wave,
while the transmitted wave propagates to the right with amplitude <math>t</math>.

Using the expression of the probability current, one finds that a plane wave
<math>e^{ikx}</math> carries a current

<math display="block">
J=\frac{\hbar k}{m}.
</math>

Thus the incoming, reflected and transmitted waves correspond to incoming,
reflected and transmitted probability currents.

Since probability is conserved, the current must be the same on both sides
of the sample. This implies the relation

<math display="block">
R + T = 1,
</math>

where

<math display="block">
R = |r|^2, \qquad T = |t|^2
</math>

are the reflection and transmission probabilities.

== Free particles and ballistic behaviour ==

We now illustrate how a localized quantum particle evolves in the absence of disorder.

For a free particle the Hamiltonian reads

<math display="block">
H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}.
</math>

As discussed above, the eigenstates of this Hamiltonian are plane waves with a continuous spectrum

<math display="block">
\psi_k(x,t)=\frac{1}{\sqrt{2\pi}}e^{ikx}e^{-iE_k t/\hbar},
\qquad
E_k=\frac{\hbar^2k^2}{2m}.
</math>

Since plane waves are completely delocalized, a physical particle must be described by a superposition of eigenstates, forming a '''wave packet'''

<math display="block">
\psi(x,t)=\int_{-\infty}^{\infty}dk\,c(k)\psi_k(x,t),
\qquad
\int_{-\infty}^{\infty}dk\,|c(k)|^2=1.
</math>

=== Evolution of a Gaussian wave packet ===

* '''Initial state'''

At time <math>t=0</math> consider a Gaussian wave packet

<math display="block">
\psi(x,0)=\frac{e^{-x^2/(4a^2)}}{(2\pi a^2)^{1/4}},
\qquad
|\psi(x,0)|^2=\frac{e^{-x^2/(2a^2)}}{\sqrt{2\pi a^2}}.
</math>

Show that the coefficients of the plane-wave decomposition are

<math display="block">
c(k)=\left(\frac{2a^2}{\pi}\right)^{1/4}e^{-a^2k^2}.
</math>

* '''Time evolution'''

Define the spreading velocity

<math display="block">
v_s=\frac{\hbar}{2ma^2}.
</math>

Show that the time evolution of the packet is

<math display="block">
\psi(x,t)=
\frac{e^{-x^2/(4a^2(1+i v_s t))}}
{[2\pi a^2(1+i v_s t)]^{1/4}}.
</math>

* '''Ballistic spreading'''

The probability density becomes

<math display="block">
|\psi(x,t)|^2=
\frac{e^{-x^2/(2a^2(1+v_s^2 t^2))}}
{\sqrt{2\pi a^2(1+v_s^2 t^2)}}.
</math>

Hence

<math display="block">
\sqrt{\langle x^2\rangle}
=
\left(\int dx\,x^2|\psi(x,t)|^2\right)^{1/2}
=
a\sqrt{1+v_s^2 t^2}.
</math>

At long times

<math display="block">
\sqrt{\langle x^2\rangle}\sim v_s t.
</math>

This linear growth is called '''ballistic spreading'''.

It should be contrasted with two other possible transport regimes:

* '''Diffusive motion'''
<math>\sqrt{\langle x^2\rangle}\sim \sqrt{t}</math>

* '''Localized regime'''
<math>\sqrt{\langle x^2\rangle}</math> saturates at long times.

Understanding how disorder modifies ballistic motion and eventually suppresses transport is the main goal of the following sections.

== Localization of the packet: general idea and experiment ==

[[File:Localization1DB.png|thumb|left|x240px|BEC condensate expanding in a 1D disordered potential. Billy et al., Nature (2008).]]

In 1958, P. Anderson proposed that disorder can suppress transport in quantum systems. This phenomenon, known as '''Anderson localization''', has since been observed both numerically and experimentally.

In the experiment shown here, a Gaussian packet of a Bose–Einstein condensate is prepared in a harmonic trap. When the trap is removed the cloud initially expands but quickly stops spreading and remains localized.

To understand this behaviour we must solve the eigenstate problem of the Hamiltonian in the presence of disorder.

[[File:Localization1dB.png|thumb|left|x140px|Semilog plot of the particle density. Billy et al., Nature (2008).]]

In a disordered potential an eigenstate of energy <math>E_k</math> has the form

<math display="block">
\psi_k(x,t)=\psi_k(x)e^{-iE_k t/\hbar}.
</math>

The spatial part of the wavefunction is localized around some position <math>\bar{x}</math> and decays exponentially

<math display="block">
\psi_k(x)\sim
e^{-|x-\bar{x}|/\xi_{\text{loc}}}.
</math>

Here <math>\xi_{\text{loc}}</math> is the localization length.

Because the eigenstates are localized, the wave packet is built from states localized near its initial position. Eigenstates far from the packet contribute exponentially little, and states composing the packet decay exponentially away from their center.

As a consequence transport far from the initial position of the particle is '''exponentially suppressed'''.
== Diffusive transport ==

In most materials weak disorder does not lead to localization but to diffusive transport. A simple microscopic description is provided by the Drude model. In this classical picture electrons move freely between collisions with impurities or lattice defects. These scattering events occur on average after a characteristic time <math>\tau</math>, called the mean free time.

Under the action of an electric field <math>E</math>, an electron of charge <math>-e</math> obeys

<math display="block">
m\frac{dv}{dt}=-eE.
</math>

Between two collisions the electron accelerates, but collisions randomize its velocity. Averaging over many such events leads to a stationary drift velocity

<math display="block">
v_d=-\frac{eE\tau}{m}.
</math>

If the electron density is <math>n</math>, the electric current density is

<math display="block">
j=-ne\,v_d.
</math>

Substituting the drift velocity gives

<math display="block">
j=\frac{ne^2\tau}{m}E.
</math>

This is the microscopic origin of Ohm's law

<math display="block">
j=\sigma E,
\qquad
\sigma=\frac{ne^2\tau}{m}.
</math>

It is important to stress that the Drude model is purely '''classical'''. It ignores the wave nature of electrons and quantum interference effects. A fully quantum derivation of Ohm's law is much more subtle and typically relies on Green functions and diagrammatic methods. Here we will instead focus on the regime where quantum interference becomes crucial and can eventually suppress diffusion altogether.
== Conductance ==

The conductance <math>G</math> is the inverse of the resistance. Understanding how it depends on the system size will be the starting point for the scaling theory of localization.

=== Diffusive transport ===

Consider a wire of length <math>L</math> and cross section <math>S</math>. The total current is

<math display="block">
I=jS,
</math>

while the voltage drop is

<math display="block">
V=EL.
</math>

Using Ohm's law <math>j=\sigma E</math> we obtain

<math display="block">
I=\sigma \frac{S}{L} V.
</math>

Therefore

<math display="block">
G=\frac{I}{V}=\sigma \frac{S}{L}.
</math>

For a sample of linear size <math>L</math> in spatial dimension <math>d</math>, the cross section scales as

<math display="block">
S\sim L^{d-1}.
</math>

Hence

<math display="block">
G(L)\sim \sigma L^{d-2}.
</math>

This power-law scaling is the characteristic signature of diffusive transport.

=== Localized regime ===

When disorder is strong diffusion is suppressed and the system becomes insulating.

In Exercise 15 we studied transport using the Landauer picture. In this approach the conductance is proportional to the transmission probability through the sample:

<math display="block">
G=\frac{e^2}{\hbar}|t(E_F)|^2.
</math>

The factor <math>e^2/\hbar</math> sets the natural quantum scale of conductance for a single transport channel.

In a localized system the transmission probability typically decays exponentially with the system size

<math display="block">
|t(E_F)|^2 \sim e^{-2L/\xi_{\text{loc}}},
</math>

where <math>\xi_{\text{loc}}</math> is the localization length.

This leads to

<math display="block">
G(L)\sim e^{-2L/\xi_{\text{loc}}}.
</math>

Thus in the localized phase the conductance decreases exponentially with the system size.

== The “Gang of Four” scaling theory ==

The two behaviors discussed above raise a natural question: how does the conductance of a disordered system evolve when the system size is increased?

In a famous PRL (1979), Abrahams, Anderson, Licciardello and Ramakrishnan proposed a scaling theory of localization that addresses this question.

The key quantity is the '''dimensionless conductance'''

<math display="block">
g=\frac{G\hbar}{e^2}.
</math>

The factor <math>e^2/\hbar</math> is the natural quantum unit of conductance (see the Landauer formula). The quantity <math>g</math> therefore measures the effective number of conducting channels.

Instead of computing the conductance microscopically, the scaling theory studies how <math>g</math> evolves when the system size <math>L</math> is increased.

This evolution is described by the scaling equation

<math display="block">
\frac{d\ln g}{d\ln L}=\beta(g).
</math>

The function <math>\beta(g)</math> depends only on <math>g</math> and on the spatial dimension.

Two limiting regimes are known.

* '''Metallic regime''' (<math>g\gg1</math>)

Transport is diffusive. From the Drude result

<math display="block">
G(L)\sim L^{d-2},
</math>

we obtain

<math display="block">
\beta(g)\to d-2.
</math>

* '''Localized regime''' (<math>g\ll1</math>)

The conductance decreases exponentially

<math display="block">
g(L)\sim e^{-L/\xi_{\text{loc}}},
</math>

which implies

<math display="block">
\beta(g)\sim \ln g.
</math>

The simplest scenario is that the beta function is monotonic.

This leads to a striking dimensional prediction:

* for <math>d>2</math> the beta function changes sign and a '''metal–insulator transition''' exists
* for <math>d\le2</math> the beta function remains negative and '''all states are localized'''.

The scaling theory therefore predicts that in low dimension disorder inevitably destroys diffusion.

To understand how localization emerges microscopically, we must now study the spectrum and eigenstates of quantum particles moving in a random potential.

L-7

2026-03-08T11:28:20Z

Rosso:

'''Goal.''' This lecture introduces the phenomenon of localization. Localization is a '''wave phenomenon induced by disorder''' that suppresses transport in a system.

== Short recap: wavefunctions and eigenstates ==

Before discussing localization we briefly recall a few basic notions of quantum mechanics.

A quantum particle in one dimension is described by a '''wavefunction''' <math>\psi(x,t)</math>.
The quantity <math>|\psi(x,t)|^2</math> is the probability density of finding the particle at position <math>x</math> at time <math>t</math>. The wavefunction therefore satisfies the normalization condition

<math display="block">
\int_{-\infty}^{\infty} dx\, |\psi(x,t)|^2 = 1 .
</math>

The time evolution of the wavefunction is governed by the Schrödinger equation

<math display="block">
i\hbar \partial_t \psi(x,t) = H\psi(x,t),
</math>

where <math>H</math> is the Hamiltonian of the system. For a particle moving in one dimension in a potential <math>V(x)</math>, the Hamiltonian reads

<math display="block">
H = -\frac{\hbar^2}{2m}\partial_x^2 + V(x).
</math>

=== Eigenstates ===

A particularly important class of solutions are the '''eigenstates''' of the Hamiltonian

<math display="block">
H\psi_n(x) = E_n \psi_n(x).
</math>

If the particle is in an eigenstate the full solution reads

<math display="block">
\psi_n(x,t)=\psi_n(x)e^{-iE_n t/\hbar}.
</math>

The probability density <math>|\psi_n(x,t)|^2</math> is therefore independent of time: eigenstates are stationary states.
Two qualitatively different situations may occur depending on the form of the potential.

In many physical situations the Hamiltonian has a mixed spectrum, containing both discrete bound states and continuum states. In this case a general wavefunction can be expressed as a superposition involving contributions from both sets of eigenstates.
A simple example is a finite potential well: it supports a finite number of bound states with discrete energies, while particles with higher energies belong to the continuum and correspond to scattering states.

=== Discrete and continuous spectra ===

* '''Discrete spectrum (bound states)'''

The energies take isolated values <math>E_n</math>.
This typically happens when the particle is confined in a finite region
(for instance in a potential well).

The corresponding eigenfunctions are normalizable,

<math display="block">
\int dx\, |\psi_n(x)|^2 = 1,
</math>

and the particle remains localized in space. These states are called
bound states.

* '''Continuous spectrum (continuum states)'''

The energy can take any value in a continuous interval.
This occurs for instance for a free particle or for a particle with
energy above the confining potential.

A simple example is provided by plane waves

<math display="block">
\psi_k(x)=\frac{1}{\sqrt{2\pi}} e^{ikx},
\qquad
E=\frac{\hbar^2 k^2}{2m}.
</math>
These states are not normalizable in the usual sense. They are instead
normalized using Dirac delta functions,

<math display="block">
\int dx\, \psi_k^*(x)\psi_{k'}(x)=\delta(k-k').
</math>

They form a continuous basis that can be used to construct physical
wave packets.

=== Superposition principle ===

The Schrödinger equation is linear. As a consequence, any linear combination
of solutions is again a solution. This property is known as the
superposition principle.

If the spectrum is discrete, an arbitrary wavefunction can be expanded
in the basis of eigenstates

<math display="block">
\psi(x,t)=\sum_n c_n e^{-iE_n t/\hbar}\psi_n(x).
</math>

If the spectrum is continuous the expansion becomes an integral

<math display="block">
\psi(x,t)=\int dk\, c(k)\psi_k(x,t),
\qquad
\int dk\, |c(k)|^2=1.
</math>

By choosing the coefficients <math>c(k)</math> appropriately one can construct
a localized '''wave packet''' describing a particle initially confined in space.

=== Probability current ===

Besides the probability density one can define a '''probability current'''

<math display="block">
J(x,t)=\frac{\hbar}{2mi}
\left(
\psi^* \frac{d\psi}{dx}
-
\psi \frac{d\psi^*}{dx}
\right).
</math>

This quantity measures the '''flow of probability''' across a point in space.

Its expression follows from combining the Schrödinger equation with the
conservation of probability. Indeed the probability density

<math>
\rho(x,t)=|\psi(x,t)|^2
</math>

satisfies the continuity equation

<math display="block">
\partial_t \rho(x,t) + \partial_x J(x,t)=0,
</math>

which has the same structure as a conservation law in hydrodynamics.

For a plane wave

<math display="block">
\psi(x)=e^{ikx}
</math>

one finds

<math display="block">
J=\frac{\hbar k}{m}.
</math>

Thus plane waves describe particles propagating through space and carrying
a non–zero probability current.

If the wavefunction is real the probability current vanishes, since the two
terms in the expression of <math>J</math> cancel. This is the case for bound
states in one dimension: for a real potential the eigenfunctions can be
chosen real, and bound states in 1D are non–degenerate. Physically this
corresponds to a '''standing wave''' rather than a propagating wave.

=== Scattering states ===

Transport problems often involve a localized potential (for instance a sample
or a potential barrier) surrounded by free space. Outside this region the
solutions of the Schrödinger equation are plane waves.

When a particle interacts with the sample, the wavefunction generally contains
three contributions:

* an incoming wave,
* a reflected wave,
* a transmitted wave.

The corresponding solutions are called '''scattering states'''.

For example, a particle incoming from the left is described asymptotically by

<math display="block">
\psi_{k,L}(x)=
\begin{cases}
e^{ikx}+r\, e^{-ikx} & x\to -\infty \\
t\, e^{ikx} & x\to +\infty
\end{cases}
</math>

The first term represents the incoming wave, the second the reflected wave,
while the transmitted wave propagates to the right with amplitude <math>t</math>.

Using the expression of the probability current, one finds that a plane wave
<math>e^{ikx}</math> carries a current

<math display="block">
J=\frac{\hbar k}{m}.
</math>

Thus the incoming, reflected and transmitted waves correspond to incoming,
reflected and transmitted probability currents.

Since probability is conserved, the current must be the same on both sides
of the sample. This implies the relation

<math display="block">
R + T = 1,
</math>

where

<math display="block">
R = |r|^2, \qquad T = |t|^2
</math>

are the reflection and transmission probabilities.

== Free particles and ballistic behaviour ==

We now illustrate how a localized quantum particle evolves in the absence of disorder.

For a free particle the Hamiltonian reads

<math display="block">
H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}.
</math>

As discussed above, the eigenstates of this Hamiltonian are plane waves with a continuous spectrum

<math display="block">
\psi_k(x,t)=\frac{1}{\sqrt{2\pi}}e^{ikx}e^{-iE_k t/\hbar},
\qquad
E_k=\frac{\hbar^2k^2}{2m}.
</math>

Since plane waves are completely delocalized, a physical particle must be described by a superposition of eigenstates, forming a '''wave packet'''

<math display="block">
\psi(x,t)=\int_{-\infty}^{\infty}dk\,c(k)\psi_k(x,t),
\qquad
\int_{-\infty}^{\infty}dk\,|c(k)|^2=1.
</math>

=== Evolution of a Gaussian wave packet ===

* '''Initial state'''

At time <math>t=0</math> consider a Gaussian wave packet

<math display="block">
\psi(x,0)=\frac{e^{-x^2/(4a^2)}}{(2\pi a^2)^{1/4}},
\qquad
|\psi(x,0)|^2=\frac{e^{-x^2/(2a^2)}}{\sqrt{2\pi a^2}}.
</math>

Show that the coefficients of the plane-wave decomposition are

<math display="block">
c(k)=\left(\frac{2a^2}{\pi}\right)^{1/4}e^{-a^2k^2}.
</math>

* '''Time evolution'''

Define the spreading velocity

<math display="block">
v_s=\frac{\hbar}{2ma^2}.
</math>

Show that the time evolution of the packet is

<math display="block">
\psi(x,t)=
\frac{e^{-x^2/(4a^2(1+i v_s t))}}
{[2\pi a^2(1+i v_s t)]^{1/4}}.
</math>

* '''Ballistic spreading'''

The probability density becomes

<math display="block">
|\psi(x,t)|^2=
\frac{e^{-x^2/(2a^2(1+v_s^2 t^2))}}
{\sqrt{2\pi a^2(1+v_s^2 t^2)}}.
</math>

Hence

<math display="block">
\sqrt{\langle x^2\rangle}
=
\left(\int dx\,x^2|\psi(x,t)|^2\right)^{1/2}
=
a\sqrt{1+v_s^2 t^2}.
</math>

At long times

<math display="block">
\sqrt{\langle x^2\rangle}\sim v_s t.
</math>

This linear growth is called '''ballistic spreading'''.

It should be contrasted with two other possible transport regimes:

* '''Diffusive motion'''
<math>\sqrt{\langle x^2\rangle}\sim \sqrt{t}</math>

* '''Localized regime'''
<math>\sqrt{\langle x^2\rangle}</math> saturates at long times.

Understanding how disorder modifies ballistic motion and eventually suppresses transport is the main goal of the following sections.

== Localization of the packet: general idea and experiment ==

[[File:Localization1DB.png|thumb|left|x240px|BEC condensate expanding in a 1D disordered potential. Billy et al., Nature (2008).]]

In 1958, P. Anderson proposed that disorder can suppress transport in quantum systems. This phenomenon, known as '''Anderson localization''', has since been observed both numerically and experimentally.

In the experiment shown here, a Gaussian packet of a Bose–Einstein condensate is prepared in a harmonic trap. When the trap is removed the cloud initially expands but quickly stops spreading and remains localized.

To understand this behaviour we must solve the eigenstate problem of the Hamiltonian in the presence of disorder.

[[File:Localization1dB.png|thumb|left|x140px|Semilog plot of the particle density. Billy et al., Nature (2008).]]

In a disordered potential an eigenstate of energy <math>E_k</math> has the form

<math display="block">
\psi_k(x,t)=\psi_k(x)e^{-iE_k t/\hbar}.
</math>

The spatial part of the wavefunction is localized around some position <math>\bar{x}</math> and decays exponentially

<math display="block">
\psi_k(x)\sim
e^{-|x-\bar{x}|/\xi_{\text{loc}}}.
</math>

Here <math>\xi_{\text{loc}}</math> is the localization length.

Because the eigenstates are localized, the wave packet is built from states localized near its initial position. Eigenstates far from the packet contribute exponentially little, and states composing the packet decay exponentially away from their center.

As a consequence transport far from the initial position of the particle is '''exponentially suppressed'''.
== Diffusive transport ==

In most materials weak disorder does not lead to localization but to diffusive transport. A simple microscopic description is provided by the Drude model. In this classical picture electrons move freely between collisions with impurities or lattice defects. These scattering events occur on average after a characteristic time <math>\tau</math>, called the mean free time.

Under the action of an electric field <math>E</math>, an electron of charge <math>-e</math> obeys

<math display="block">
m\frac{dv}{dt}=-eE.
</math>

Between two collisions the electron accelerates, but collisions randomize its velocity. Averaging over many such events leads to a stationary drift velocity

<math display="block">
v_d=-\frac{eE\tau}{m}.
</math>

If the electron density is <math>n</math>, the electric current density is

<math display="block">
j=-ne\,v_d.
</math>

Substituting the drift velocity gives

<math display="block">
j=\frac{ne^2\tau}{m}E.
</math>

This is the microscopic origin of Ohm's law

<math display="block">
j=\sigma E,
\qquad
\sigma=\frac{ne^2\tau}{m}.
</math>

It is important to stress that the Drude model is purely '''classical'''. It ignores the wave nature of electrons and quantum interference effects. A fully quantum derivation of Ohm's law is much more subtle and typically relies on Green functions and diagrammatic methods. Here we will instead focus on the regime where quantum interference becomes crucial and can eventually suppress diffusion altogether.
== Conductance ==

The conductance <math>G</math> is the inverse of the resistance. Understanding how it depends on the system size will be the starting point for the scaling theory of localization.

=== Diffusive transport ===

Consider a wire of length <math>L</math> and cross section <math>S</math>. The total current is

<math display="block">
I=jS,
</math>

while the voltage drop is

<math display="block">
V=EL.
</math>

Using Ohm's law <math>j=\sigma E</math> we obtain

<math display="block">
I=\sigma \frac{S}{L} V.
</math>

Therefore

<math display="block">
G=\frac{I}{V}=\sigma \frac{S}{L}.
</math>

For a sample of linear size <math>L</math> in spatial dimension <math>d</math>, the cross section scales as

<math display="block">
S\sim L^{d-1}.
</math>

Hence

<math display="block">
G(L)\sim \sigma L^{d-2}.
</math>

This power-law scaling is the characteristic signature of diffusive transport.

=== Localized regime ===

When disorder is strong diffusion is suppressed and the system becomes insulating.

In Exercise 15 we studied transport using the Landauer picture. In this approach the conductance is proportional to the transmission probability through the sample:

<math display="block">
G=\frac{e^2}{\hbar}|t(E_F)|^2.
</math>

The factor <math>e^2/\hbar</math> sets the natural quantum scale of conductance for a single transport channel.

In a localized system the transmission probability typically decays exponentially with the system size

<math display="block">
|t(E_F)|^2 \sim e^{-2L/\xi_{\text{loc}}},
</math>

where <math>\xi_{\text{loc}}</math> is the localization length.

This leads to

<math display="block">
G(L)\sim e^{-2L/\xi_{\text{loc}}}.
</math>

Thus in the localized phase the conductance decreases exponentially with the system size.

== The “Gang of Four” scaling theory ==

The two behaviors discussed above raise a natural question: how does the conductance of a disordered system evolve when the system size is increased?

In a famous PRL (1979), Abrahams, Anderson, Licciardello and Ramakrishnan proposed a scaling theory of localization that addresses this question.

The key quantity is the '''dimensionless conductance'''

<math display="block">
g=\frac{G\hbar}{e^2}.
</math>

The factor <math>e^2/\hbar</math> is the natural quantum unit of conductance (see the Landauer formula). The quantity <math>g</math> therefore measures the effective number of conducting channels.

Instead of computing the conductance microscopically, the scaling theory studies how <math>g</math> evolves when the system size <math>L</math> is increased.

This evolution is described by the scaling equation

<math display="block">
\frac{d\ln g}{d\ln L}=\beta(g).
</math>

The function <math>\beta(g)</math> depends only on <math>g</math> and on the spatial dimension.

Two limiting regimes are known.

* '''Metallic regime''' (<math>g\gg1</math>)

Transport is diffusive. From the Drude result

<math display="block">
G(L)\sim L^{d-2},
</math>

we obtain

<math display="block">
\beta(g)\to d-2.
</math>

* '''Localized regime''' (<math>g\ll1</math>)

The conductance decreases exponentially

<math display="block">
g(L)\sim e^{-L/\xi_{\text{loc}}},
</math>

which implies

<math display="block">
\beta(g)\sim \ln g.
</math>

The simplest scenario is that the beta function is monotonic.

This leads to a striking dimensional prediction:

* for <math>d>2</math> the beta function changes sign and a '''metal–insulator transition''' exists
* for <math>d\le2</math> the beta function remains negative and '''all states are localized'''.

The scaling theory therefore predicts that in low dimension disorder inevitably destroys diffusion.

To understand how localization emerges microscopically, we must now study the spectrum and eigenstates of quantum particles moving in a random potential.

L-7

2026-03-08T11:19:34Z

Rosso: /* Conductance and diffusive transport */

'''Goal.''' This lecture introduces the phenomenon of localization. Localization is a '''wave phenomenon induced by disorder''' that suppresses transport in a system.

== Short recap: wavefunctions and eigenstates ==

Before discussing localization we briefly recall a few basic notions of quantum mechanics.

A quantum particle in one dimension is described by a '''wavefunction''' <math>\psi(x,t)</math>.
The quantity <math>|\psi(x,t)|^2</math> is the probability density of finding the particle at position <math>x</math> at time <math>t</math>. The wavefunction therefore satisfies the normalization condition

<math display="block">
\int_{-\infty}^{\infty} dx\, |\psi(x,t)|^2 = 1 .
</math>

The time evolution of the wavefunction is governed by the Schrödinger equation

<math display="block">
i\hbar \partial_t \psi(x,t) = H\psi(x,t),
</math>

where <math>H</math> is the Hamiltonian of the system. For a particle moving in one dimension in a potential <math>V(x)</math>, the Hamiltonian reads

<math display="block">
H = -\frac{\hbar^2}{2m}\partial_x^2 + V(x).
</math>

=== Eigenstates ===

A particularly important class of solutions are the '''eigenstates''' of the Hamiltonian

<math display="block">
H\psi_n(x) = E_n \psi_n(x).
</math>

If the particle is in an eigenstate the full solution reads

<math display="block">
\psi_n(x,t)=\psi_n(x)e^{-iE_n t/\hbar}.
</math>

The probability density <math>|\psi_n(x,t)|^2</math> is therefore independent of time: eigenstates are stationary states.
Two qualitatively different situations may occur depending on the form of the potential.

In many physical situations the Hamiltonian has a mixed spectrum, containing both discrete bound states and continuum states. In this case a general wavefunction can be expressed as a superposition involving contributions from both sets of eigenstates.
A simple example is a finite potential well: it supports a finite number of bound states with discrete energies, while particles with higher energies belong to the continuum and correspond to scattering states.

=== Discrete and continuous spectra ===

* '''Discrete spectrum (bound states)'''

The energies take isolated values <math>E_n</math>.
This typically happens when the particle is confined in a finite region
(for instance in a potential well).

The corresponding eigenfunctions are normalizable,

<math display="block">
\int dx\, |\psi_n(x)|^2 = 1,
</math>

and the particle remains localized in space. These states are called
bound states.

* '''Continuous spectrum (continuum states)'''

The energy can take any value in a continuous interval.
This occurs for instance for a free particle or for a particle with
energy above the confining potential.

A simple example is provided by plane waves

<math display="block">
\psi_k(x)=\frac{1}{\sqrt{2\pi}} e^{ikx},
\qquad
E=\frac{\hbar^2 k^2}{2m}.
</math>
These states are not normalizable in the usual sense. They are instead
normalized using Dirac delta functions,

<math display="block">
\int dx\, \psi_k^*(x)\psi_{k'}(x)=\delta(k-k').
</math>

They form a continuous basis that can be used to construct physical
wave packets.

=== Superposition principle ===

The Schrödinger equation is linear. As a consequence, any linear combination
of solutions is again a solution. This property is known as the
superposition principle.

If the spectrum is discrete, an arbitrary wavefunction can be expanded
in the basis of eigenstates

<math display="block">
\psi(x,t)=\sum_n c_n e^{-iE_n t/\hbar}\psi_n(x).
</math>

If the spectrum is continuous the expansion becomes an integral

<math display="block">
\psi(x,t)=\int dk\, c(k)\psi_k(x,t),
\qquad
\int dk\, |c(k)|^2=1.
</math>

By choosing the coefficients <math>c(k)</math> appropriately one can construct
a localized '''wave packet''' describing a particle initially confined in space.

=== Probability current ===

Besides the probability density one can define a '''probability current'''

<math display="block">
J(x,t)=\frac{\hbar}{2mi}
\left(
\psi^* \frac{d\psi}{dx}
-
\psi \frac{d\psi^*}{dx}
\right).
</math>

This quantity measures the '''flow of probability''' across a point in space.

Its expression follows from combining the Schrödinger equation with the
conservation of probability. Indeed the probability density

<math>
\rho(x,t)=|\psi(x,t)|^2
</math>

satisfies the continuity equation

<math display="block">
\partial_t \rho(x,t) + \partial_x J(x,t)=0,
</math>

which has the same structure as a conservation law in hydrodynamics.

For a plane wave

<math display="block">
\psi(x)=e^{ikx}
</math>

one finds

<math display="block">
J=\frac{\hbar k}{m}.
</math>

Thus plane waves describe particles propagating through space and carrying
a non–zero probability current.

If the wavefunction is real the probability current vanishes, since the two
terms in the expression of <math>J</math> cancel. This is the case for bound
states in one dimension: for a real potential the eigenfunctions can be
chosen real, and bound states in 1D are non–degenerate. Physically this
corresponds to a '''standing wave''' rather than a propagating wave.

=== Scattering states ===

Transport problems often involve a localized potential (for instance a sample
or a potential barrier) surrounded by free space. Outside this region the
solutions of the Schrödinger equation are plane waves.

When a particle interacts with the sample, the wavefunction generally contains
three contributions:

* an incoming wave,
* a reflected wave,
* a transmitted wave.

The corresponding solutions are called '''scattering states'''.

For example, a particle incoming from the left is described asymptotically by

<math display="block">
\psi_{k,L}(x)=
\begin{cases}
e^{ikx}+r\, e^{-ikx} & x\to -\infty \\
t\, e^{ikx} & x\to +\infty
\end{cases}
</math>

The first term represents the incoming wave, the second the reflected wave,
while the transmitted wave propagates to the right with amplitude <math>t</math>.

Using the expression of the probability current, one finds that a plane wave
<math>e^{ikx}</math> carries a current

<math display="block">
J=\frac{\hbar k}{m}.
</math>

Thus the incoming, reflected and transmitted waves correspond to incoming,
reflected and transmitted probability currents.

Since probability is conserved, the current must be the same on both sides
of the sample. This implies the relation

<math display="block">
R + T = 1,
</math>

where

<math display="block">
R = |r|^2, \qquad T = |t|^2
</math>

are the reflection and transmission probabilities.

== Free particles and ballistic behaviour ==

We now illustrate how a localized quantum particle evolves in the absence of disorder.

For a free particle the Hamiltonian reads

<math display="block">
H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}.
</math>

As discussed above, the eigenstates of this Hamiltonian are plane waves with a continuous spectrum

<math display="block">
\psi_k(x,t)=\frac{1}{\sqrt{2\pi}}e^{ikx}e^{-iE_k t/\hbar},
\qquad
E_k=\frac{\hbar^2k^2}{2m}.
</math>

Since plane waves are completely delocalized, a physical particle must be described by a superposition of eigenstates, forming a '''wave packet'''

<math display="block">
\psi(x,t)=\int_{-\infty}^{\infty}dk\,c(k)\psi_k(x,t),
\qquad
\int_{-\infty}^{\infty}dk\,|c(k)|^2=1.
</math>

=== Evolution of a Gaussian wave packet ===

* '''Initial state'''

At time <math>t=0</math> consider a Gaussian wave packet

<math display="block">
\psi(x,0)=\frac{e^{-x^2/(4a^2)}}{(2\pi a^2)^{1/4}},
\qquad
|\psi(x,0)|^2=\frac{e^{-x^2/(2a^2)}}{\sqrt{2\pi a^2}}.
</math>

Show that the coefficients of the plane-wave decomposition are

<math display="block">
c(k)=\left(\frac{2a^2}{\pi}\right)^{1/4}e^{-a^2k^2}.
</math>

* '''Time evolution'''

Define the spreading velocity

<math display="block">
v_s=\frac{\hbar}{2ma^2}.
</math>

Show that the time evolution of the packet is

<math display="block">
\psi(x,t)=
\frac{e^{-x^2/(4a^2(1+i v_s t))}}
{[2\pi a^2(1+i v_s t)]^{1/4}}.
</math>

* '''Ballistic spreading'''

The probability density becomes

<math display="block">
|\psi(x,t)|^2=
\frac{e^{-x^2/(2a^2(1+v_s^2 t^2))}}
{\sqrt{2\pi a^2(1+v_s^2 t^2)}}.
</math>

Hence

<math display="block">
\sqrt{\langle x^2\rangle}
=
\left(\int dx\,x^2|\psi(x,t)|^2\right)^{1/2}
=
a\sqrt{1+v_s^2 t^2}.
</math>

At long times

<math display="block">
\sqrt{\langle x^2\rangle}\sim v_s t.
</math>

This linear growth is called '''ballistic spreading'''.

It should be contrasted with two other possible transport regimes:

* '''Diffusive motion'''
<math>\sqrt{\langle x^2\rangle}\sim \sqrt{t}</math>

* '''Localized regime'''
<math>\sqrt{\langle x^2\rangle}</math> saturates at long times.

Understanding how disorder modifies ballistic motion and eventually suppresses transport is the main goal of the following sections.

== Localization of the packet: general idea and experiment ==

[[File:Localization1DB.png|thumb|left|x240px|BEC condensate expanding in a 1D disordered potential. Billy et al., Nature (2008).]]

In 1958, P. Anderson proposed that disorder can suppress transport in quantum systems. This phenomenon, known as '''Anderson localization''', has since been observed both numerically and experimentally.

In the experiment shown here, a Gaussian packet of a Bose–Einstein condensate is prepared in a harmonic trap. When the trap is removed the cloud initially expands but quickly stops spreading and remains localized.

To understand this behaviour we must solve the eigenstate problem of the Hamiltonian in the presence of disorder.

[[File:Localization1dB.png|thumb|left|x140px|Semilog plot of the particle density. Billy et al., Nature (2008).]]

In a disordered potential an eigenstate of energy <math>E_k</math> has the form

<math display="block">
\psi_k(x,t)=\psi_k(x)e^{-iE_k t/\hbar}.
</math>

The spatial part of the wavefunction is localized around some position <math>\bar{x}</math> and decays exponentially

<math display="block">
\psi_k(x)\sim
e^{-|x-\bar{x}|/\xi_{\text{loc}}}.
</math>

Here <math>\xi_{\text{loc}}</math> is the localization length.

Because the eigenstates are localized, the wave packet is built from states localized near its initial position. Eigenstates far from the packet contribute exponentially little, and states composing the packet decay exponentially away from their center.

As a consequence transport far from the initial position of the particle is '''exponentially suppressed'''.
== Diffusive transport ==

In most materials weak disorder does not lead to localization but to diffusive transport. A simple microscopic description is provided by the Drude model. In this classical picture electrons move freely between collisions with impurities or lattice defects. These scattering events occur on average after a characteristic time <math>\tau</math>, called the mean free time.

Under the action of an electric field <math>E</math>, an electron of charge <math>-e</math> obeys

<math display="block">
m\frac{dv}{dt}=-eE.
</math>

Between two collisions the electron accelerates, but collisions randomize its velocity. Averaging over many such events leads to a stationary drift velocity

<math display="block">
v_d=-\frac{eE\tau}{m}.
</math>

If the electron density is <math>n</math>, the electric current density is

<math display="block">
j=-ne\,v_d.
</math>

Substituting the drift velocity gives

<math display="block">
j=\frac{ne^2\tau}{m}E.
</math>

This is the microscopic origin of Ohm's law

<math display="block">
j=\sigma E,
\qquad
\sigma=\frac{ne^2\tau}{m}.
</math>

It is important to stress that the Drude model is purely '''classical'''. It ignores the wave nature of electrons and quantum interference effects. A fully quantum derivation of Ohm's law is much more subtle and typically relies on Green functions and diagrammatic methods. Here we will instead focus on the regime where quantum interference becomes crucial and can eventually suppress diffusion altogether.

==Conductance==

The conductance <math> G</math> is the inverse of the resistance. We will determine is scaling, it will be the starting point for the scaling theory of localization.

* Diffusive Transport: Consider a wire of length <math>L</math> and cross section <math>S</math>. The total current is
<math display="block">
I=jS,
</math>
while the voltage drop is
<math display="block">
V=EL.
</math>

Therefore

<math display="block">
I=\sigma \frac{S}{L} V.
</math>
Using <math>G=I/V</math>, one obtains
<math display="block">
G=\sigma \frac{S}{L}.
</math>

For a sample of linear size <math>L</math> in spatial dimension <math>d</math>, the cross section scales as <math>S\sim L^{d-1}</math>. Hence

<math display="block">
G(L)\sim \sigma L^{d-2}.
</math>

This scaling is the characteristic signature of diffusive transport.

== Conductance in the localized regime ==

When disorder is strong diffusion is suppressed and the system becomes insulating.

In Exercise 15 we derived, using the Landauer picture of transport, that the conductance is proportional to the transmission probability

<math display="block">
G=\frac{e^2}{\hbar}|t(E_F)|^2.
</math>

The factor <math>e^2/\hbar</math> therefore sets the natural quantum scale of conductance for a single channel.

In a localized system the transmission probability decays exponentially with the system size

<math display="block">
|t(E_F)|^2 \sim e^{-2L/\xi_{\text{loc}}},
</math>

which leads to

<math display="block">
G(L)\sim e^{-2L/\xi_{\text{loc}}}.
</math>

Thus the conductance decreases exponentially with the size of the sample.

== The “Gang of Four” scaling theory ==

In a famous PRL (1979), Abrahams, Anderson, Licciardello and Ramakrishnan proposed a scaling theory of localization.

The key quantity is the '''dimensionless conductance'''

<math display="block">
g=\frac{G\hbar}{e^2}.
</math>

In the Landauer picture this quantity measures how many quantum channels effectively contribute to transport.

The central question of the scaling theory is the following:

'''If we increase the size of a disordered sample, does it behave more and more like a metal or more and more like an insulator?'''

Instead of computing the conductance microscopically, the idea is to study how the conductance evolves when the system size <math>L</math> is increased.

This evolution is described by a renormalization group equation

<math display="block">
\frac{d\ln g}{d\ln L}=\beta(g).
</math>

The function <math>\beta(g)</math> depends only on <math>g</math> and on the spatial dimension.

Two limits are known:

* '''Metallic regime''' (<math>g\gg1</math>)

Transport is diffusive. Since

<math display="block">
G\sim L^{d-2},
</math>

we obtain

<math display="block">
\beta(g)\to d-2.
</math>

* '''Localized regime''' (<math>g\ll1</math>)

The conductance decays exponentially

<math display="block">
g\sim e^{-L/\xi_{\text{loc}}},
</math>

which implies

<math display="block">
\beta(g)\sim \ln g.
</math>

The simplest scenario is that the beta function is monotonic.

This leads to a striking dimensional prediction:

* for <math>d>2</math> the beta function changes sign → a '''metal–insulator transition''' exists
* for <math>d\le2</math> the beta function is always negative → '''all states are localized'''.

The scaling theory therefore predicts that in low dimension disorder inevitably destroys diffusion.

To understand how localization emerges microscopically we must now study the spectrum and eigenstates of quantum particles moving in a random potential.

L-7

2026-03-08T11:11:44Z

Rosso: /* Conductance and diffusive transport */

'''Goal.''' This lecture introduces the phenomenon of localization. Localization is a '''wave phenomenon induced by disorder''' that suppresses transport in a system.

== Short recap: wavefunctions and eigenstates ==

Before discussing localization we briefly recall a few basic notions of quantum mechanics.

A quantum particle in one dimension is described by a '''wavefunction''' <math>\psi(x,t)</math>.
The quantity <math>|\psi(x,t)|^2</math> is the probability density of finding the particle at position <math>x</math> at time <math>t</math>. The wavefunction therefore satisfies the normalization condition

<math display="block">
\int_{-\infty}^{\infty} dx\, |\psi(x,t)|^2 = 1 .
</math>

The time evolution of the wavefunction is governed by the Schrödinger equation

<math display="block">
i\hbar \partial_t \psi(x,t) = H\psi(x,t),
</math>

where <math>H</math> is the Hamiltonian of the system. For a particle moving in one dimension in a potential <math>V(x)</math>, the Hamiltonian reads

<math display="block">
H = -\frac{\hbar^2}{2m}\partial_x^2 + V(x).
</math>

=== Eigenstates ===

A particularly important class of solutions are the '''eigenstates''' of the Hamiltonian

<math display="block">
H\psi_n(x) = E_n \psi_n(x).
</math>

If the particle is in an eigenstate the full solution reads

<math display="block">
\psi_n(x,t)=\psi_n(x)e^{-iE_n t/\hbar}.
</math>

The probability density <math>|\psi_n(x,t)|^2</math> is therefore independent of time: eigenstates are stationary states.
Two qualitatively different situations may occur depending on the form of the potential.

In many physical situations the Hamiltonian has a mixed spectrum, containing both discrete bound states and continuum states. In this case a general wavefunction can be expressed as a superposition involving contributions from both sets of eigenstates.
A simple example is a finite potential well: it supports a finite number of bound states with discrete energies, while particles with higher energies belong to the continuum and correspond to scattering states.

=== Discrete and continuous spectra ===

* '''Discrete spectrum (bound states)'''

The energies take isolated values <math>E_n</math>.
This typically happens when the particle is confined in a finite region
(for instance in a potential well).

The corresponding eigenfunctions are normalizable,

<math display="block">
\int dx\, |\psi_n(x)|^2 = 1,
</math>

and the particle remains localized in space. These states are called
bound states.

* '''Continuous spectrum (continuum states)'''

The energy can take any value in a continuous interval.
This occurs for instance for a free particle or for a particle with
energy above the confining potential.

A simple example is provided by plane waves

<math display="block">
\psi_k(x)=\frac{1}{\sqrt{2\pi}} e^{ikx},
\qquad
E=\frac{\hbar^2 k^2}{2m}.
</math>
These states are not normalizable in the usual sense. They are instead
normalized using Dirac delta functions,

<math display="block">
\int dx\, \psi_k^*(x)\psi_{k'}(x)=\delta(k-k').
</math>

They form a continuous basis that can be used to construct physical
wave packets.

=== Superposition principle ===

The Schrödinger equation is linear. As a consequence, any linear combination
of solutions is again a solution. This property is known as the
superposition principle.

If the spectrum is discrete, an arbitrary wavefunction can be expanded
in the basis of eigenstates

<math display="block">
\psi(x,t)=\sum_n c_n e^{-iE_n t/\hbar}\psi_n(x).
</math>

If the spectrum is continuous the expansion becomes an integral

<math display="block">
\psi(x,t)=\int dk\, c(k)\psi_k(x,t),
\qquad
\int dk\, |c(k)|^2=1.
</math>

By choosing the coefficients <math>c(k)</math> appropriately one can construct
a localized '''wave packet''' describing a particle initially confined in space.

=== Probability current ===

Besides the probability density one can define a '''probability current'''

<math display="block">
J(x,t)=\frac{\hbar}{2mi}
\left(
\psi^* \frac{d\psi}{dx}
-
\psi \frac{d\psi^*}{dx}
\right).
</math>

This quantity measures the '''flow of probability''' across a point in space.

Its expression follows from combining the Schrödinger equation with the
conservation of probability. Indeed the probability density

<math>
\rho(x,t)=|\psi(x,t)|^2
</math>

satisfies the continuity equation

<math display="block">
\partial_t \rho(x,t) + \partial_x J(x,t)=0,
</math>

which has the same structure as a conservation law in hydrodynamics.

For a plane wave

<math display="block">
\psi(x)=e^{ikx}
</math>

one finds

<math display="block">
J=\frac{\hbar k}{m}.
</math>

Thus plane waves describe particles propagating through space and carrying
a non–zero probability current.

If the wavefunction is real the probability current vanishes, since the two
terms in the expression of <math>J</math> cancel. This is the case for bound
states in one dimension: for a real potential the eigenfunctions can be
chosen real, and bound states in 1D are non–degenerate. Physically this
corresponds to a '''standing wave''' rather than a propagating wave.

=== Scattering states ===

Transport problems often involve a localized potential (for instance a sample
or a potential barrier) surrounded by free space. Outside this region the
solutions of the Schrödinger equation are plane waves.

When a particle interacts with the sample, the wavefunction generally contains
three contributions:

* an incoming wave,
* a reflected wave,
* a transmitted wave.

The corresponding solutions are called '''scattering states'''.

For example, a particle incoming from the left is described asymptotically by

<math display="block">
\psi_{k,L}(x)=
\begin{cases}
e^{ikx}+r\, e^{-ikx} & x\to -\infty \\
t\, e^{ikx} & x\to +\infty
\end{cases}
</math>

The first term represents the incoming wave, the second the reflected wave,
while the transmitted wave propagates to the right with amplitude <math>t</math>.

Using the expression of the probability current, one finds that a plane wave
<math>e^{ikx}</math> carries a current

<math display="block">
J=\frac{\hbar k}{m}.
</math>

Thus the incoming, reflected and transmitted waves correspond to incoming,
reflected and transmitted probability currents.

Since probability is conserved, the current must be the same on both sides
of the sample. This implies the relation

<math display="block">
R + T = 1,
</math>

where

<math display="block">
R = |r|^2, \qquad T = |t|^2
</math>

are the reflection and transmission probabilities.

== Free particles and ballistic behaviour ==

We now illustrate how a localized quantum particle evolves in the absence of disorder.

For a free particle the Hamiltonian reads

<math display="block">
H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}.
</math>

As discussed above, the eigenstates of this Hamiltonian are plane waves with a continuous spectrum

<math display="block">
\psi_k(x,t)=\frac{1}{\sqrt{2\pi}}e^{ikx}e^{-iE_k t/\hbar},
\qquad
E_k=\frac{\hbar^2k^2}{2m}.
</math>

Since plane waves are completely delocalized, a physical particle must be described by a superposition of eigenstates, forming a '''wave packet'''

<math display="block">
\psi(x,t)=\int_{-\infty}^{\infty}dk\,c(k)\psi_k(x,t),
\qquad
\int_{-\infty}^{\infty}dk\,|c(k)|^2=1.
</math>

=== Evolution of a Gaussian wave packet ===

* '''Initial state'''

At time <math>t=0</math> consider a Gaussian wave packet

<math display="block">
\psi(x,0)=\frac{e^{-x^2/(4a^2)}}{(2\pi a^2)^{1/4}},
\qquad
|\psi(x,0)|^2=\frac{e^{-x^2/(2a^2)}}{\sqrt{2\pi a^2}}.
</math>

Show that the coefficients of the plane-wave decomposition are

<math display="block">
c(k)=\left(\frac{2a^2}{\pi}\right)^{1/4}e^{-a^2k^2}.
</math>

* '''Time evolution'''

Define the spreading velocity

<math display="block">
v_s=\frac{\hbar}{2ma^2}.
</math>

Show that the time evolution of the packet is

<math display="block">
\psi(x,t)=
\frac{e^{-x^2/(4a^2(1+i v_s t))}}
{[2\pi a^2(1+i v_s t)]^{1/4}}.
</math>

* '''Ballistic spreading'''

The probability density becomes

<math display="block">
|\psi(x,t)|^2=
\frac{e^{-x^2/(2a^2(1+v_s^2 t^2))}}
{\sqrt{2\pi a^2(1+v_s^2 t^2)}}.
</math>

Hence

<math display="block">
\sqrt{\langle x^2\rangle}
=
\left(\int dx\,x^2|\psi(x,t)|^2\right)^{1/2}
=
a\sqrt{1+v_s^2 t^2}.
</math>

At long times

<math display="block">
\sqrt{\langle x^2\rangle}\sim v_s t.
</math>

This linear growth is called '''ballistic spreading'''.

It should be contrasted with two other possible transport regimes:

* '''Diffusive motion'''
<math>\sqrt{\langle x^2\rangle}\sim \sqrt{t}</math>

* '''Localized regime'''
<math>\sqrt{\langle x^2\rangle}</math> saturates at long times.

Understanding how disorder modifies ballistic motion and eventually suppresses transport is the main goal of the following sections.

== Localization of the packet: general idea and experiment ==

[[File:Localization1DB.png|thumb|left|x240px|BEC condensate expanding in a 1D disordered potential. Billy et al., Nature (2008).]]

In 1958, P. Anderson proposed that disorder can suppress transport in quantum systems. This phenomenon, known as '''Anderson localization''', has since been observed both numerically and experimentally.

In the experiment shown here, a Gaussian packet of a Bose–Einstein condensate is prepared in a harmonic trap. When the trap is removed the cloud initially expands but quickly stops spreading and remains localized.

To understand this behaviour we must solve the eigenstate problem of the Hamiltonian in the presence of disorder.

[[File:Localization1dB.png|thumb|left|x140px|Semilog plot of the particle density. Billy et al., Nature (2008).]]

In a disordered potential an eigenstate of energy <math>E_k</math> has the form

<math display="block">
\psi_k(x,t)=\psi_k(x)e^{-iE_k t/\hbar}.
</math>

The spatial part of the wavefunction is localized around some position <math>\bar{x}</math> and decays exponentially

<math display="block">
\psi_k(x)\sim
e^{-|x-\bar{x}|/\xi_{\text{loc}}}.
</math>

Here <math>\xi_{\text{loc}}</math> is the localization length.

Because the eigenstates are localized, the wave packet is built from states localized near its initial position. Eigenstates far from the packet contribute exponentially little, and states composing the packet decay exponentially away from their center.

As a consequence transport far from the initial position of the particle is '''exponentially suppressed'''.
== Conductance and diffusive transport ==

In most materials weak disorder does not lead to localization but to '''diffusive transport'''.

A simple microscopic description is provided by the '''Drude model'''. In this classical picture electrons move freely between collisions with impurities or lattice defects. These scattering events occur on average after a characteristic time <math>\tau</math>, called the '''mean free time'''.

Under the action of an electric field <math>E</math>, an electron of charge <math>-e</math> obeys

<math display="block">
m\frac{dv}{dt}=-eE.
</math>

Between two collisions the electron accelerates, but collisions randomize its velocity. Averaging over many such events leads to a stationary drift velocity

<math display="block">
v_d=-\frac{eE\tau}{m}.
</math>

If the electron density is <math>n</math>, the electric current density is

<math display="block">
j=-ne\,v_d.
</math>

Substituting the drift velocity gives

<math display="block">
j=\frac{ne^2\tau}{m}E.
</math>

This is the microscopic origin of Ohm's law

<math display="block">
j=\sigma E,
\qquad
\sigma=\frac{ne^2\tau}{m}.
</math>

To relate this to the conductance of a sample, consider a wire of length <math>L</math> and cross section <math>S</math>. The total current is

<math display="block">
I=jS,
</math>

while the voltage drop is

<math display="block">
V=EL.
</math>

Therefore

<math display="block">
I=\sigma \frac{S}{L} V.
</math>

Using <math>G=I/V</math>, one obtains

<math display="block">
G=\sigma \frac{S}{L}.
</math>

Equivalently,

<math display="block">
R=\frac{1}{G}=\rho\frac{L}{S},
\qquad
\rho=\frac{1}{\sigma}.
</math>

For a sample of linear size <math>L</math> in spatial dimension <math>d</math>, the cross section scales as <math>S\sim L^{d-1}</math>. Hence

<math display="block">
G(L)\sim \sigma L^{d-2}.
</math>

This scaling is the characteristic signature of diffusive transport, and it will be the starting point for the scaling theory of localization.

It is important to stress that the Drude model is purely '''classical'''. It ignores the wave nature of electrons and quantum interference effects. A fully quantum derivation of Ohm's law is much more subtle and typically relies on Green functions and diagrammatic methods. Here we will instead focus on the regime where quantum interference becomes crucial and can eventually suppress diffusion altogether.

== Conductance in the localized regime ==

When disorder is strong diffusion is suppressed and the system becomes insulating.

In Exercise 15 we derived, using the Landauer picture of transport, that the conductance is proportional to the transmission probability

<math display="block">
G=\frac{e^2}{\hbar}|t(E_F)|^2.
</math>

The factor <math>e^2/\hbar</math> therefore sets the natural quantum scale of conductance for a single channel.

In a localized system the transmission probability decays exponentially with the system size

<math display="block">
|t(E_F)|^2 \sim e^{-2L/\xi_{\text{loc}}},
</math>

which leads to

<math display="block">
G(L)\sim e^{-2L/\xi_{\text{loc}}}.
</math>

Thus the conductance decreases exponentially with the size of the sample.

== The “Gang of Four” scaling theory ==

In a famous PRL (1979), Abrahams, Anderson, Licciardello and Ramakrishnan proposed a scaling theory of localization.

The key quantity is the '''dimensionless conductance'''

<math display="block">
g=\frac{G\hbar}{e^2}.
</math>

In the Landauer picture this quantity measures how many quantum channels effectively contribute to transport.

The central question of the scaling theory is the following:

'''If we increase the size of a disordered sample, does it behave more and more like a metal or more and more like an insulator?'''

Instead of computing the conductance microscopically, the idea is to study how the conductance evolves when the system size <math>L</math> is increased.

This evolution is described by a renormalization group equation

<math display="block">
\frac{d\ln g}{d\ln L}=\beta(g).
</math>

The function <math>\beta(g)</math> depends only on <math>g</math> and on the spatial dimension.

Two limits are known:

* '''Metallic regime''' (<math>g\gg1</math>)

Transport is diffusive. Since

<math display="block">
G\sim L^{d-2},
</math>

we obtain

<math display="block">
\beta(g)\to d-2.
</math>

* '''Localized regime''' (<math>g\ll1</math>)

The conductance decays exponentially

<math display="block">
g\sim e^{-L/\xi_{\text{loc}}},
</math>

which implies

<math display="block">
\beta(g)\sim \ln g.
</math>

The simplest scenario is that the beta function is monotonic.

This leads to a striking dimensional prediction:

* for <math>d>2</math> the beta function changes sign → a '''metal–insulator transition''' exists
* for <math>d\le2</math> the beta function is always negative → '''all states are localized'''.

The scaling theory therefore predicts that in low dimension disorder inevitably destroys diffusion.

To understand how localization emerges microscopically we must now study the spectrum and eigenstates of quantum particles moving in a random potential.

L-7

2026-03-08T10:25:30Z

Rosso: /* Short recap: wavefunctions and eigenstates */

'''Goal.''' This lecture introduces the phenomenon of localization. Localization is a '''wave phenomenon induced by disorder''' that suppresses transport in a system.

== Short recap: wavefunctions and eigenstates ==

Before discussing localization we briefly recall a few basic notions of quantum mechanics.

A quantum particle in one dimension is described by a '''wavefunction''' <math>\psi(x,t)</math>.
The quantity <math>|\psi(x,t)|^2</math> is the probability density of finding the particle at position <math>x</math> at time <math>t</math>. The wavefunction therefore satisfies the normalization condition

<math display="block">
\int_{-\infty}^{\infty} dx\, |\psi(x,t)|^2 = 1 .
</math>

The time evolution of the wavefunction is governed by the Schrödinger equation

<math display="block">
i\hbar \partial_t \psi(x,t) = H\psi(x,t),
</math>

where <math>H</math> is the Hamiltonian of the system. For a particle moving in one dimension in a potential <math>V(x)</math>, the Hamiltonian reads

<math display="block">
H = -\frac{\hbar^2}{2m}\partial_x^2 + V(x).
</math>

=== Eigenstates ===

A particularly important class of solutions are the '''eigenstates''' of the Hamiltonian

<math display="block">
H\psi_n(x) = E_n \psi_n(x).
</math>

If the particle is in an eigenstate the full solution reads

<math display="block">
\psi_n(x,t)=\psi_n(x)e^{-iE_n t/\hbar}.
</math>

The probability density <math>|\psi_n(x,t)|^2</math> is therefore independent of time: eigenstates are stationary states.
Two qualitatively different situations may occur depending on the form of the potential.

In many physical situations the Hamiltonian has a mixed spectrum, containing both discrete bound states and continuum states. In this case a general wavefunction can be expressed as a superposition involving contributions from both sets of eigenstates.
A simple example is a finite potential well: it supports a finite number of bound states with discrete energies, while particles with higher energies belong to the continuum and correspond to scattering states.

=== Discrete and continuous spectra ===

* '''Discrete spectrum (bound states)'''

The energies take isolated values <math>E_n</math>.
This typically happens when the particle is confined in a finite region
(for instance in a potential well).

The corresponding eigenfunctions are normalizable,

<math display="block">
\int dx\, |\psi_n(x)|^2 = 1,
</math>

and the particle remains localized in space. These states are called
bound states.

* '''Continuous spectrum (continuum states)'''

The energy can take any value in a continuous interval.
This occurs for instance for a free particle or for a particle with
energy above the confining potential.

A simple example is provided by plane waves

<math display="block">
\psi_k(x)=\frac{1}{\sqrt{2\pi}} e^{ikx},
\qquad
E=\frac{\hbar^2 k^2}{2m}.
</math>
These states are not normalizable in the usual sense. They are instead
normalized using Dirac delta functions,

<math display="block">
\int dx\, \psi_k^*(x)\psi_{k'}(x)=\delta(k-k').
</math>

They form a continuous basis that can be used to construct physical
wave packets.

=== Superposition principle ===

The Schrödinger equation is linear. As a consequence, any linear combination
of solutions is again a solution. This property is known as the
superposition principle.

If the spectrum is discrete, an arbitrary wavefunction can be expanded
in the basis of eigenstates

<math display="block">
\psi(x,t)=\sum_n c_n e^{-iE_n t/\hbar}\psi_n(x).
</math>

If the spectrum is continuous the expansion becomes an integral

<math display="block">
\psi(x,t)=\int dk\, c(k)\psi_k(x,t),
\qquad
\int dk\, |c(k)|^2=1.
</math>

By choosing the coefficients <math>c(k)</math> appropriately one can construct
a localized '''wave packet''' describing a particle initially confined in space.

=== Probability current ===

Besides the probability density one can define a '''probability current'''

<math display="block">
J(x,t)=\frac{\hbar}{2mi}
\left(
\psi^* \frac{d\psi}{dx}
-
\psi \frac{d\psi^*}{dx}
\right).
</math>

This quantity measures the '''flow of probability''' across a point in space.

Its expression follows from combining the Schrödinger equation with the
conservation of probability. Indeed the probability density

<math>
\rho(x,t)=|\psi(x,t)|^2
</math>

satisfies the continuity equation

<math display="block">
\partial_t \rho(x,t) + \partial_x J(x,t)=0,
</math>

which has the same structure as a conservation law in hydrodynamics.

For a plane wave

<math display="block">
\psi(x)=e^{ikx}
</math>

one finds

<math display="block">
J=\frac{\hbar k}{m}.
</math>

Thus plane waves describe particles propagating through space and carrying
a non–zero probability current.

If the wavefunction is real the probability current vanishes, since the two
terms in the expression of <math>J</math> cancel. This is the case for bound
states in one dimension: for a real potential the eigenfunctions can be
chosen real, and bound states in 1D are non–degenerate. Physically this
corresponds to a '''standing wave''' rather than a propagating wave.

=== Scattering states ===

Transport problems often involve a localized potential (for instance a sample
or a potential barrier) surrounded by free space. Outside this region the
solutions of the Schrödinger equation are plane waves.

When a particle interacts with the sample, the wavefunction generally contains
three contributions:

* an incoming wave,
* a reflected wave,
* a transmitted wave.

The corresponding solutions are called '''scattering states'''.

For example, a particle incoming from the left is described asymptotically by

<math display="block">
\psi_{k,L}(x)=
\begin{cases}
e^{ikx}+r\, e^{-ikx} & x\to -\infty \\
t\, e^{ikx} & x\to +\infty
\end{cases}
</math>

The first term represents the incoming wave, the second the reflected wave,
while the transmitted wave propagates to the right with amplitude <math>t</math>.

Using the expression of the probability current, one finds that a plane wave
<math>e^{ikx}</math> carries a current

<math display="block">
J=\frac{\hbar k}{m}.
</math>

Thus the incoming, reflected and transmitted waves correspond to incoming,
reflected and transmitted probability currents.

Since probability is conserved, the current must be the same on both sides
of the sample. This implies the relation

<math display="block">
R + T = 1,
</math>

where

<math display="block">
R = |r|^2, \qquad T = |t|^2
</math>

are the reflection and transmission probabilities.

== Free particles and ballistic behaviour ==

We now illustrate how a localized quantum particle evolves in the absence of disorder.

For a free particle the Hamiltonian reads

<math display="block">
H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}.
</math>

As discussed above, the eigenstates of this Hamiltonian are plane waves with a continuous spectrum

<math display="block">
\psi_k(x,t)=\frac{1}{\sqrt{2\pi}}e^{ikx}e^{-iE_k t/\hbar},
\qquad
E_k=\frac{\hbar^2k^2}{2m}.
</math>

Since plane waves are completely delocalized, a physical particle must be described by a superposition of eigenstates, forming a '''wave packet'''

<math display="block">
\psi(x,t)=\int_{-\infty}^{\infty}dk\,c(k)\psi_k(x,t),
\qquad
\int_{-\infty}^{\infty}dk\,|c(k)|^2=1.
</math>

=== Evolution of a Gaussian wave packet ===

* '''Initial state'''

At time <math>t=0</math> consider a Gaussian wave packet

<math display="block">
\psi(x,0)=\frac{e^{-x^2/(4a^2)}}{(2\pi a^2)^{1/4}},
\qquad
|\psi(x,0)|^2=\frac{e^{-x^2/(2a^2)}}{\sqrt{2\pi a^2}}.
</math>

Show that the coefficients of the plane-wave decomposition are

<math display="block">
c(k)=\left(\frac{2a^2}{\pi}\right)^{1/4}e^{-a^2k^2}.
</math>

* '''Time evolution'''

Define the spreading velocity

<math display="block">
v_s=\frac{\hbar}{2ma^2}.
</math>

Show that the time evolution of the packet is

<math display="block">
\psi(x,t)=
\frac{e^{-x^2/(4a^2(1+i v_s t))}}
{[2\pi a^2(1+i v_s t)]^{1/4}}.
</math>

* '''Ballistic spreading'''

The probability density becomes

<math display="block">
|\psi(x,t)|^2=
\frac{e^{-x^2/(2a^2(1+v_s^2 t^2))}}
{\sqrt{2\pi a^2(1+v_s^2 t^2)}}.
</math>

Hence

<math display="block">
\sqrt{\langle x^2\rangle}
=
\left(\int dx\,x^2|\psi(x,t)|^2\right)^{1/2}
=
a\sqrt{1+v_s^2 t^2}.
</math>

At long times

<math display="block">
\sqrt{\langle x^2\rangle}\sim v_s t.
</math>

This linear growth is called '''ballistic spreading'''.

It should be contrasted with two other possible transport regimes:

* '''Diffusive motion'''
<math>\sqrt{\langle x^2\rangle}\sim \sqrt{t}</math>

* '''Localized regime'''
<math>\sqrt{\langle x^2\rangle}</math> saturates at long times.

Understanding how disorder modifies ballistic motion and eventually suppresses transport is the main goal of the following sections.

== Localization of the packet: general idea and experiment ==

[[File:Localization1DB.png|thumb|left|x240px|BEC condensate expanding in a 1D disordered potential. Billy et al., Nature (2008).]]

In 1958, P. Anderson proposed that disorder can suppress transport in quantum systems. This phenomenon, known as '''Anderson localization''', has since been observed both numerically and experimentally.

In the experiment shown here, a Gaussian packet of a Bose–Einstein condensate is prepared in a harmonic trap. When the trap is removed the cloud initially expands but quickly stops spreading and remains localized.

To understand this behaviour we must solve the eigenstate problem of the Hamiltonian in the presence of disorder.

[[File:Localization1dB.png|thumb|left|x140px|Semilog plot of the particle density. Billy et al., Nature (2008).]]

In a disordered potential an eigenstate of energy <math>E_k</math> has the form

<math display="block">
\psi_k(x,t)=\psi_k(x)e^{-iE_k t/\hbar}.
</math>

The spatial part of the wavefunction is localized around some position <math>\bar{x}</math> and decays exponentially

<math display="block">
\psi_k(x)\sim
e^{-|x-\bar{x}|/\xi_{\text{loc}}}.
</math>

Here <math>\xi_{\text{loc}}</math> is the localization length.

Because the eigenstates are localized, the wave packet is built from states localized near its initial position. Eigenstates far from the packet contribute exponentially little, and states composing the packet decay exponentially away from their center.

As a consequence transport far from the initial position of the particle is '''exponentially suppressed'''.
== Conductance and diffusive transport ==

In most materials weak disorder does not lead to localization but to '''diffusive transport'''.

In the Drude picture electrons scatter randomly on impurities. After many scattering events their motion becomes a random walk. Beyond the mean free path the motion is therefore diffusive.

In this regime Ohm's laws hold.

* First law

<math display="block">
\frac{V}{I}=R,
\qquad
\frac{I}{V}=G.
</math>

* Second law

<math display="block">
R=\rho\frac{L}{S},
\qquad
G=\sigma\frac{S}{L}.
</math>

Here <math>\rho</math> and <math>\sigma</math> are the resistivity and conductivity of the material.

For a sample of linear size <math>L</math> in spatial dimension <math>d</math>, the cross section scales as <math>S\sim L^{d-1}</math>. Therefore

<math display="block">
G \sim \sigma L^{d-2}.
</math>

This scaling behavior is the characteristic signature of diffusive transport.

== Conductance in the localized regime ==

When disorder is strong diffusion is suppressed and the system becomes insulating.

In Exercise 15 we derived, using the Landauer picture of transport, that the conductance is proportional to the transmission probability

<math display="block">
G=\frac{e^2}{\hbar}|t(E_F)|^2.
</math>

The factor <math>e^2/\hbar</math> therefore sets the natural quantum scale of conductance for a single channel.

In a localized system the transmission probability decays exponentially with the system size

<math display="block">
|t(E_F)|^2 \sim e^{-2L/\xi_{\text{loc}}},
</math>

which leads to

<math display="block">
G(L)\sim e^{-2L/\xi_{\text{loc}}}.
</math>

Thus the conductance decreases exponentially with the size of the sample.

== The “Gang of Four” scaling theory ==

In a famous PRL (1979), Abrahams, Anderson, Licciardello and Ramakrishnan proposed a scaling theory of localization.

The key quantity is the '''dimensionless conductance'''

<math display="block">
g=\frac{G\hbar}{e^2}.
</math>

In the Landauer picture this quantity measures how many quantum channels effectively contribute to transport.

The central question of the scaling theory is the following:

'''If we increase the size of a disordered sample, does it behave more and more like a metal or more and more like an insulator?'''

Instead of computing the conductance microscopically, the idea is to study how the conductance evolves when the system size <math>L</math> is increased.

This evolution is described by a renormalization group equation

<math display="block">
\frac{d\ln g}{d\ln L}=\beta(g).
</math>

The function <math>\beta(g)</math> depends only on <math>g</math> and on the spatial dimension.

Two limits are known:

* '''Metallic regime''' (<math>g\gg1</math>)

Transport is diffusive. Since

<math display="block">
G\sim L^{d-2},
</math>

we obtain

<math display="block">
\beta(g)\to d-2.
</math>

* '''Localized regime''' (<math>g\ll1</math>)

The conductance decays exponentially

<math display="block">
g\sim e^{-L/\xi_{\text{loc}}},
</math>

which implies

<math display="block">
\beta(g)\sim \ln g.
</math>

The simplest scenario is that the beta function is monotonic.

This leads to a striking dimensional prediction:

* for <math>d>2</math> the beta function changes sign → a '''metal–insulator transition''' exists
* for <math>d\le2</math> the beta function is always negative → '''all states are localized'''.

The scaling theory therefore predicts that in low dimension disorder inevitably destroys diffusion.

To understand how localization emerges microscopically we must now study the spectrum and eigenstates of quantum particles moving in a random potential.

L-7

2026-03-08T10:17:01Z

Rosso:

'''Goal.''' This lecture introduces the phenomenon of localization. Localization is a '''wave phenomenon induced by disorder''' that suppresses transport in a system.

== Short recap: wavefunctions and eigenstates ==

Before discussing localization we briefly recall a few basic notions of quantum mechanics.

A quantum particle in one dimension is described by a '''wavefunction''' <math>\psi(x,t)</math>.
The quantity <math>|\psi(x,t)|^2</math> is the probability density of finding the particle at position <math>x</math> at time <math>t</math>. The wavefunction therefore satisfies the normalization condition

<math display="block">
\int_{-\infty}^{\infty} dx\, |\psi(x,t)|^2 = 1 .
</math>

The time evolution of the wavefunction is governed by the Schrödinger equation

<math display="block">
i\hbar \partial_t \psi(x,t) = H\psi(x,t),
</math>

where <math>H</math> is the Hamiltonian of the system.

=== Eigenstates ===

A particularly important class of solutions are the '''eigenstates''' of the Hamiltonian

<math display="block">
H\psi_n(x) = E_n \psi_n(x).
</math>

If the particle is in an eigenstate the full solution reads

<math display="block">
\psi_n(x,t)=\psi_n(x)e^{-iE_n t/\hbar}.
</math>

The probability density <math>|\psi_n(x,t)|^2</math> is therefore independent of time: eigenstates are '''stationary states'''.
Two qualitatively different situations may occur depending on the form of the potential.

In many physical situations the Hamiltonian has a mixed spectrum, containing both discrete bound states and continuum states. In this case a general wavefunction can be expressed as a superposition involving contributions from both sets of eigenstates.
A simple example is a finite potential well: it supports a finite number of bound states with discrete energies, while particles with higher energies belong to the continuum and correspond to scattering states.

=== Discrete and continuous spectra ===

* '''Discrete spectrum (bound states)'''

The energies take isolated values <math>E_n</math>.
This typically happens when the particle is confined in a finite region
(for instance in a potential well).

The corresponding eigenfunctions are normalizable,

<math display="block">
\int dx\, |\psi_n(x)|^2 = 1,
</math>

and the particle remains localized in space. These states are called
'''bound states'''.

* '''Continuous spectrum (continuum states)'''

The energy can take any value in a continuous interval.
This occurs for instance for a free particle or for a particle with
energy above the confining potential.

A simple example is provided by plane waves

<math display="block">
\psi_k(x)=\frac{1}{\sqrt{2\pi}} e^{ikx},
\qquad
E=\frac{\hbar^2 k^2}{2m}.
</math>

These states are not normalizable in the usual sense. They are instead
normalized using Dirac delta functions and serve as a basis for
constructing physical wave packets.

=== Superposition principle ===

The Schrödinger equation is linear. As a consequence, any linear combination
of solutions is again a solution. This property is known as the
'''superposition principle'''.

If the spectrum is discrete, an arbitrary wavefunction can be expanded
in the basis of eigenstates

<math display="block">
\psi(x,t)=\sum_n c_n e^{-iE_n t/\hbar}\psi_n(x).
</math>

If the spectrum is continuous the expansion becomes an integral

<math display="block">
\psi(x,t)=\int dk\, c(k)\psi_k(x,t),
\qquad
\int dk\, |c(k)|^2=1.
</math>

By choosing the coefficients <math>c(k)</math> appropriately one can construct
a localized '''wave packet''' describing a particle initially confined in space.

=== Probability current ===

Besides the probability density one can define a '''probability current'''

<math display="block">
J(x,t)=\frac{\hbar}{2mi}
\left(
\psi^* \frac{d\psi}{dx}
-
\psi \frac{d\psi^*}{dx}
\right).
</math>

This quantity measures the '''flow of probability''' across a point in space.

Its expression follows from combining the Schrödinger equation with the
conservation of probability. Indeed the probability density

<math>
\rho(x,t)=|\psi(x,t)|^2
</math>

satisfies the continuity equation

<math display="block">
\partial_t \rho(x,t) + \partial_x J(x,t)=0,
</math>

which has the same structure as a conservation law in hydrodynamics.

For a plane wave

<math display="block">
\psi(x)=e^{ikx}
</math>

one finds

<math display="block">
J=\frac{\hbar k}{m}.
</math>

Thus plane waves describe particles propagating through space and carrying
a non–zero probability current.

If the wavefunction is real the probability current vanishes, since the two
terms in the expression of <math>J</math> cancel. This is the case for bound
states in one dimension: for a real potential the eigenfunctions can be
chosen real, and bound states in 1D are non–degenerate. Physically this
corresponds to a '''standing wave''' rather than a propagating wave.

=== Scattering states ===

Transport problems often involve a localized potential (for instance a sample
or a potential barrier) surrounded by free space. Outside this region the
solutions of the Schrödinger equation are plane waves.

When a particle interacts with the sample, the wavefunction generally contains
three contributions:

* an incoming wave,
* a reflected wave,
* a transmitted wave.

The corresponding solutions are called '''scattering states'''.

For example, a particle incoming from the left is described asymptotically by

<math display="block">
\psi_{k,L}(x)=
\begin{cases}
e^{ikx}+r\, e^{-ikx} & x\to -\infty \\
t\, e^{ikx} & x\to +\infty
\end{cases}
</math>

The first term represents the incoming wave, the second the reflected wave,
while the transmitted wave propagates to the right with amplitude <math>t</math>.

Using the expression of the probability current, one finds that a plane wave
<math>e^{ikx}</math> carries a current

<math display="block">
J=\frac{\hbar k}{m}.
</math>

Thus the incoming, reflected and transmitted waves correspond to incoming,
reflected and transmitted probability currents.

Since probability is conserved, the current must be the same on both sides
of the sample. This implies the relation

<math display="block">
R + T = 1,
</math>

where

<math display="block">
R = |r|^2, \qquad T = |t|^2
</math>

are the reflection and transmission probabilities.

== Free particles and ballistic behaviour ==

We now illustrate how a localized quantum particle evolves in the absence of disorder.

For a free particle the Hamiltonian reads

<math display="block">
H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}.
</math>

As discussed above, the eigenstates of this Hamiltonian are plane waves with a continuous spectrum

<math display="block">
\psi_k(x,t)=\frac{1}{\sqrt{2\pi}}e^{ikx}e^{-iE_k t/\hbar},
\qquad
E_k=\frac{\hbar^2k^2}{2m}.
</math>

Since plane waves are completely delocalized, a physical particle must be described by a superposition of eigenstates, forming a '''wave packet'''

<math display="block">
\psi(x,t)=\int_{-\infty}^{\infty}dk\,c(k)\psi_k(x,t),
\qquad
\int_{-\infty}^{\infty}dk\,|c(k)|^2=1.
</math>

=== Evolution of a Gaussian wave packet ===

* '''Initial state'''

At time <math>t=0</math> consider a Gaussian wave packet

<math display="block">
\psi(x,0)=\frac{e^{-x^2/(4a^2)}}{(2\pi a^2)^{1/4}},
\qquad
|\psi(x,0)|^2=\frac{e^{-x^2/(2a^2)}}{\sqrt{2\pi a^2}}.
</math>

Show that the coefficients of the plane-wave decomposition are

<math display="block">
c(k)=\left(\frac{2a^2}{\pi}\right)^{1/4}e^{-a^2k^2}.
</math>

* '''Time evolution'''

Define the spreading velocity

<math display="block">
v_s=\frac{\hbar}{2ma^2}.
</math>

Show that the time evolution of the packet is

<math display="block">
\psi(x,t)=
\frac{e^{-x^2/(4a^2(1+i v_s t))}}
{[2\pi a^2(1+i v_s t)]^{1/4}}.
</math>

* '''Ballistic spreading'''

The probability density becomes

<math display="block">
|\psi(x,t)|^2=
\frac{e^{-x^2/(2a^2(1+v_s^2 t^2))}}
{\sqrt{2\pi a^2(1+v_s^2 t^2)}}.
</math>

Hence

<math display="block">
\sqrt{\langle x^2\rangle}
=
\left(\int dx\,x^2|\psi(x,t)|^2\right)^{1/2}
=
a\sqrt{1+v_s^2 t^2}.
</math>

At long times

<math display="block">
\sqrt{\langle x^2\rangle}\sim v_s t.
</math>

This linear growth is called '''ballistic spreading'''.

It should be contrasted with two other possible transport regimes:

* '''Diffusive motion'''
<math>\sqrt{\langle x^2\rangle}\sim \sqrt{t}</math>

* '''Localized regime'''
<math>\sqrt{\langle x^2\rangle}</math> saturates at long times.

Understanding how disorder modifies ballistic motion and eventually suppresses transport is the main goal of the following sections.

== Localization of the packet: general idea and experiment ==

[[File:Localization1DB.png|thumb|left|x240px|BEC condensate expanding in a 1D disordered potential. Billy et al., Nature (2008).]]

In 1958, P. Anderson proposed that disorder can suppress transport in quantum systems. This phenomenon, known as '''Anderson localization''', has since been observed both numerically and experimentally.

In the experiment shown here, a Gaussian packet of a Bose–Einstein condensate is prepared in a harmonic trap. When the trap is removed the cloud initially expands but quickly stops spreading and remains localized.

To understand this behaviour we must solve the eigenstate problem of the Hamiltonian in the presence of disorder.

[[File:Localization1dB.png|thumb|left|x140px|Semilog plot of the particle density. Billy et al., Nature (2008).]]

In a disordered potential an eigenstate of energy <math>E_k</math> has the form

<math display="block">
\psi_k(x,t)=\psi_k(x)e^{-iE_k t/\hbar}.
</math>

The spatial part of the wavefunction is localized around some position <math>\bar{x}</math> and decays exponentially

<math display="block">
\psi_k(x)\sim
e^{-|x-\bar{x}|/\xi_{\text{loc}}}.
</math>

Here <math>\xi_{\text{loc}}</math> is the localization length.

Because the eigenstates are localized, the wave packet is built from states localized near its initial position. Eigenstates far from the packet contribute exponentially little, and states composing the packet decay exponentially away from their center.

As a consequence transport far from the initial position of the particle is '''exponentially suppressed'''.
== Conductance and diffusive transport ==

In most materials weak disorder does not lead to localization but to '''diffusive transport'''.

In the Drude picture electrons scatter randomly on impurities. After many scattering events their motion becomes a random walk. Beyond the mean free path the motion is therefore diffusive.

In this regime Ohm's laws hold.

* First law

<math display="block">
\frac{V}{I}=R,
\qquad
\frac{I}{V}=G.
</math>

* Second law

<math display="block">
R=\rho\frac{L}{S},
\qquad
G=\sigma\frac{S}{L}.
</math>

Here <math>\rho</math> and <math>\sigma</math> are the resistivity and conductivity of the material.

For a sample of linear size <math>L</math> in spatial dimension <math>d</math>, the cross section scales as <math>S\sim L^{d-1}</math>. Therefore

<math display="block">
G \sim \sigma L^{d-2}.
</math>

This scaling behavior is the characteristic signature of diffusive transport.

== Conductance in the localized regime ==

When disorder is strong diffusion is suppressed and the system becomes insulating.

In Exercise 15 we derived, using the Landauer picture of transport, that the conductance is proportional to the transmission probability

<math display="block">
G=\frac{e^2}{\hbar}|t(E_F)|^2.
</math>

The factor <math>e^2/\hbar</math> therefore sets the natural quantum scale of conductance for a single channel.

In a localized system the transmission probability decays exponentially with the system size

<math display="block">
|t(E_F)|^2 \sim e^{-2L/\xi_{\text{loc}}},
</math>

which leads to

<math display="block">
G(L)\sim e^{-2L/\xi_{\text{loc}}}.
</math>

Thus the conductance decreases exponentially with the size of the sample.

== The “Gang of Four” scaling theory ==

In a famous PRL (1979), Abrahams, Anderson, Licciardello and Ramakrishnan proposed a scaling theory of localization.

The key quantity is the '''dimensionless conductance'''

<math display="block">
g=\frac{G\hbar}{e^2}.
</math>

In the Landauer picture this quantity measures how many quantum channels effectively contribute to transport.

The central question of the scaling theory is the following:

'''If we increase the size of a disordered sample, does it behave more and more like a metal or more and more like an insulator?'''

Instead of computing the conductance microscopically, the idea is to study how the conductance evolves when the system size <math>L</math> is increased.

This evolution is described by a renormalization group equation

<math display="block">
\frac{d\ln g}{d\ln L}=\beta(g).
</math>

The function <math>\beta(g)</math> depends only on <math>g</math> and on the spatial dimension.

Two limits are known:

* '''Metallic regime''' (<math>g\gg1</math>)

Transport is diffusive. Since

<math display="block">
G\sim L^{d-2},
</math>

we obtain

<math display="block">
\beta(g)\to d-2.
</math>

* '''Localized regime''' (<math>g\ll1</math>)

The conductance decays exponentially

<math display="block">
g\sim e^{-L/\xi_{\text{loc}}},
</math>

which implies

<math display="block">
\beta(g)\sim \ln g.
</math>

The simplest scenario is that the beta function is monotonic.

This leads to a striking dimensional prediction:

* for <math>d>2</math> the beta function changes sign → a '''metal–insulator transition''' exists
* for <math>d\le2</math> the beta function is always negative → '''all states are localized'''.

The scaling theory therefore predicts that in low dimension disorder inevitably destroys diffusion.

To understand how localization emerges microscopically we must now study the spectrum and eigenstates of quantum particles moving in a random potential.

L-7

2026-03-07T15:33:40Z

Rosso:

'''Goal.''' This lecture introduces the phenomenon of localization. Localization is a '''wave phenomenon induced by disorder''' that suppresses transport in a system.

== Short recap: wavefunctions and eigenstates ==

Before discussing localization we briefly recall a few basic notions of quantum mechanics.

A quantum particle in one dimension is described by a '''wavefunction''' <math>\psi(x,t)</math>.
The quantity <math>|\psi(x,t)|^2</math> is the probability density of finding the particle at position <math>x</math> at time <math>t</math>. The wavefunction therefore satisfies the normalization condition

<math display="block">
\int_{-\infty}^{\infty} dx\, |\psi(x,t)|^2 = 1 .
</math>

The time evolution of the wavefunction is governed by the Schrödinger equation

<math display="block">
i\hbar \partial_t \psi(x,t) = H\psi(x,t),
</math>

where <math>H</math> is the Hamiltonian of the system.

=== Eigenstates ===

A particularly important class of solutions are the '''eigenstates''' of the Hamiltonian

<math display="block">
H\psi_n(x) = E_n \psi_n(x).
</math>

If the particle is in an eigenstate the full solution reads

<math display="block">
\psi_n(x,t)=\psi_n(x)e^{-iE_n t/\hbar}.
</math>

The probability density <math>|\psi_n(x,t)|^2</math> is therefore independent of time: eigenstates are '''stationary states'''.

=== Discrete and continuous spectra ===

Two situations may occur.

* '''Discrete spectrum'''

The energies take isolated values <math>E_n</math>.
This happens when the particle is confined in a finite region (for instance in a potential well).
The eigenstates are normalizable and labeled by an integer <math>n</math>.

* '''Continuous spectrum'''

The energy can take any value in a continuous interval.
This happens for instance for a free particle. The eigenstates are plane waves

<math display="block">
\psi_k(x)=\frac{1}{\sqrt{2\pi}} e^{ikx},
\qquad
E=\frac{\hbar^2 k^2}{2m}.
</math>

These states are not normalizable in the usual sense. They are instead normalized using Dirac delta functions and serve as a basis for constructing physical wave packets.

=== Probability current ===

Besides the probability density one can define a '''probability current'''

<math display="block">
J(x,t)=\frac{\hbar}{2mi}
\left(
\psi^* \frac{d\psi}{dx}
-
\psi \frac{d\psi^*}{dx}
\right).
</math>

This quantity measures the '''flow of probability''' across a point in space.

Its expression follows from combining the Schrödinger equation with the conservation of probability. Indeed the probability density

<math>\rho(x,t)=|\psi(x,t)|^2</math>

satisfies a continuity equation

<math display="block">
\partial_t \rho(x,t) + \partial_x J(x,t)=0,
</math>

which has the same structure as a conservation law in hydrodynamics.
The current originates from the kinetic term of the Hamiltonian, which contains spatial derivatives.

For a plane wave

<math display="block">
\psi(x)=e^{ikx}
</math>

one finds

<math display="block">
J=\frac{\hbar k}{m}.
</math>

Thus plane waves describe particles propagating through space and carrying a non–zero current. If the wavefunction is real the probability current vanishes, since the two terms in the expression of <math>J</math> cancel.
This is the case for bound states in one dimension: for a real potential the eigenfunctions can be chosen real, and bound states in 1D are non–degenerate. Physically this corresponds to a '''standing wave''' rather than a propagating wave.

=== Scattering states ===

Transport problems often involve a localized potential (for instance a disordered region or a sample) surrounded by free space. Outside this region the solutions of the Schrödinger equation are plane waves.

When a particle interacts with the sample, the wavefunction generally contains three contributions:

* an incoming wave,
* a reflected wave,
* a transmitted wave.

The corresponding solutions are called '''scattering states'''.

For example, a particle incoming from the left is described asymptotically by

<math display="block">
\psi_{k,L}(x)=
\begin{cases}
e^{ikx}+r\, e^{-ikx} & x\to -\infty \\
t\, e^{ikx} & x\to +\infty
\end{cases}
</math>

The first term represents the incoming wave, the second the reflected wave, while the transmitted wave propagates to the right with amplitude <math>t</math>.

Using the expression of the probability current, one finds that a plane wave <math>e^{ikx}</math> carries a current

<math display="block">
J=\frac{\hbar k}{m}.
</math>

Thus the incoming, reflected and transmitted waves correspond to incoming, reflected and transmitted probability currents. Since probability is conserved, the current is the same on both sides of the sample. This implies the relation

<math display="block">
R + T = 1,
</math>

where

<math display="block">
R = |r|^2, \qquad T = |t|^2
</math>

are the reflection and transmission probabilities.

These quantities will play a central role in transport problems, where the electrical current is determined by the transmission probability of the sample.

=== Superposition principle ===

The Schrödinger equation is linear. Any linear combination of eigenstates is therefore again a solution.

For a continuous spectrum one writes

<math display="block">
\psi(x,t)=\int dk\, c(k)\psi_k(x,t),
\qquad
\int dk\, |c(k)|^2=1.
</math>

By choosing the coefficients <math>c(k)</math> appropriately one can construct a localized '''wave packet''' describing a particle initially confined in space.

In the next section we study the time evolution of such a packet.

== Free particles and ballistic behaviour ==

We now illustrate how a localized quantum particle evolves in the absence of disorder.

For a free particle the Hamiltonian reads

<math display="block">
H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}.
</math>

As discussed above, the eigenstates of this Hamiltonian are plane waves with a continuous spectrum

<math display="block">
\psi_k(x,t)=\frac{1}{\sqrt{2\pi}}e^{ikx}e^{-iE_k t/\hbar},
\qquad
E_k=\frac{\hbar^2k^2}{2m}.
</math>

Since plane waves are completely delocalized, a physical particle must be described by a superposition of eigenstates, forming a '''wave packet'''

<math display="block">
\psi(x,t)=\int_{-\infty}^{\infty}dk\,c(k)\psi_k(x,t),
\qquad
\int_{-\infty}^{\infty}dk\,|c(k)|^2=1.
</math>

=== Evolution of a Gaussian wave packet ===

* '''Initial state'''

At time <math>t=0</math> consider a Gaussian wave packet

<math display="block">
\psi(x,0)=\frac{e^{-x^2/(4a^2)}}{(2\pi a^2)^{1/4}},
\qquad
|\psi(x,0)|^2=\frac{e^{-x^2/(2a^2)}}{\sqrt{2\pi a^2}}.
</math>

Show that the coefficients of the plane-wave decomposition are

<math display="block">
c(k)=\left(\frac{2a^2}{\pi}\right)^{1/4}e^{-a^2k^2}.
</math>

* '''Time evolution'''

Define the spreading velocity

<math display="block">
v_s=\frac{\hbar}{2ma^2}.
</math>

Show that the time evolution of the packet is

<math display="block">
\psi(x,t)=
\frac{e^{-x^2/(4a^2(1+i v_s t))}}
{[2\pi a^2(1+i v_s t)]^{1/4}}.
</math>

* '''Ballistic spreading'''

The probability density becomes

<math display="block">
|\psi(x,t)|^2=
\frac{e^{-x^2/(2a^2(1+v_s^2 t^2))}}
{\sqrt{2\pi a^2(1+v_s^2 t^2)}}.
</math>

Hence

<math display="block">
\sqrt{\langle x^2\rangle}
=
\left(\int dx\,x^2|\psi(x,t)|^2\right)^{1/2}
=
a\sqrt{1+v_s^2 t^2}.
</math>

At long times

<math display="block">
\sqrt{\langle x^2\rangle}\sim v_s t.
</math>

This linear growth is called '''ballistic spreading'''.

It should be contrasted with two other possible transport regimes:

* '''Diffusive motion'''
<math>\sqrt{\langle x^2\rangle}\sim \sqrt{t}</math>

* '''Localized regime'''
<math>\sqrt{\langle x^2\rangle}</math> saturates at long times.

Understanding how disorder modifies ballistic motion and eventually suppresses transport is the main goal of the following sections.

== Localization of the packet: general idea and experiment ==

[[File:Localization1DB.png|thumb|left|x240px|BEC condensate expanding in a 1D disordered potential. Billy et al., Nature (2008).]]

In 1958, P. Anderson proposed that disorder can suppress transport in quantum systems. This phenomenon, known as '''Anderson localization''', has since been observed both numerically and experimentally.

In the experiment shown here, a Gaussian packet of a Bose–Einstein condensate is prepared in a harmonic trap. When the trap is removed the cloud initially expands but quickly stops spreading and remains localized.

To understand this behaviour we must solve the eigenstate problem of the Hamiltonian in the presence of disorder.

[[File:Localization1dB.png|thumb|left|x140px|Semilog plot of the particle density. Billy et al., Nature (2008).]]

In a disordered potential an eigenstate of energy <math>E_k</math> has the form

<math display="block">
\psi_k(x,t)=\psi_k(x)e^{-iE_k t/\hbar}.
</math>

The spatial part of the wavefunction is localized around some position <math>\bar{x}</math> and decays exponentially

<math display="block">
\psi_k(x)\sim
e^{-|x-\bar{x}|/\xi_{\text{loc}}}.
</math>

Here <math>\xi_{\text{loc}}</math> is the localization length.

Because the eigenstates are localized, the wave packet is built from states localized near its initial position. Eigenstates far from the packet contribute exponentially little, and states composing the packet decay exponentially away from their center.

As a consequence transport far from the initial position of the particle is '''exponentially suppressed'''.
== Conductance and diffusive transport ==

In most materials weak disorder does not lead to localization but to '''diffusive transport'''.

In the Drude picture electrons scatter randomly on impurities. After many scattering events their motion becomes a random walk. Beyond the mean free path the motion is therefore diffusive.

In this regime Ohm's laws hold.

* First law

<math display="block">
\frac{V}{I}=R,
\qquad
\frac{I}{V}=G.
</math>

* Second law

<math display="block">
R=\rho\frac{L}{S},
\qquad
G=\sigma\frac{S}{L}.
</math>

Here <math>\rho</math> and <math>\sigma</math> are the resistivity and conductivity of the material.

For a sample of linear size <math>L</math> in spatial dimension <math>d</math>, the cross section scales as <math>S\sim L^{d-1}</math>. Therefore

<math display="block">
G \sim \sigma L^{d-2}.
</math>

This scaling behavior is the characteristic signature of diffusive transport.

== Conductance in the localized regime ==

When disorder is strong diffusion is suppressed and the system becomes insulating.

In Exercise 15 we derived, using the Landauer picture of transport, that the conductance is proportional to the transmission probability

<math display="block">
G=\frac{e^2}{\hbar}|t(E_F)|^2.
</math>

The factor <math>e^2/\hbar</math> therefore sets the natural quantum scale of conductance for a single channel.

In a localized system the transmission probability decays exponentially with the system size

<math display="block">
|t(E_F)|^2 \sim e^{-2L/\xi_{\text{loc}}},
</math>

which leads to

<math display="block">
G(L)\sim e^{-2L/\xi_{\text{loc}}}.
</math>

Thus the conductance decreases exponentially with the size of the sample.

== The “Gang of Four” scaling theory ==

In a famous PRL (1979), Abrahams, Anderson, Licciardello and Ramakrishnan proposed a scaling theory of localization.

The key quantity is the '''dimensionless conductance'''

<math display="block">
g=\frac{G\hbar}{e^2}.
</math>

In the Landauer picture this quantity measures how many quantum channels effectively contribute to transport.

The central question of the scaling theory is the following:

'''If we increase the size of a disordered sample, does it behave more and more like a metal or more and more like an insulator?'''

Instead of computing the conductance microscopically, the idea is to study how the conductance evolves when the system size <math>L</math> is increased.

This evolution is described by a renormalization group equation

<math display="block">
\frac{d\ln g}{d\ln L}=\beta(g).
</math>

The function <math>\beta(g)</math> depends only on <math>g</math> and on the spatial dimension.

Two limits are known:

* '''Metallic regime''' (<math>g\gg1</math>)

Transport is diffusive. Since

<math display="block">
G\sim L^{d-2},
</math>

we obtain

<math display="block">
\beta(g)\to d-2.
</math>

* '''Localized regime''' (<math>g\ll1</math>)

The conductance decays exponentially

<math display="block">
g\sim e^{-L/\xi_{\text{loc}}},
</math>

which implies

<math display="block">
\beta(g)\sim \ln g.
</math>

The simplest scenario is that the beta function is monotonic.

This leads to a striking dimensional prediction:

* for <math>d>2</math> the beta function changes sign → a '''metal–insulator transition''' exists
* for <math>d\le2</math> the beta function is always negative → '''all states are localized'''.

The scaling theory therefore predicts that in low dimension disorder inevitably destroys diffusion.

To understand how localization emerges microscopically we must now study the spectrum and eigenstates of quantum particles moving in a random potential.

L-7

2026-03-07T15:23:29Z

Rosso: /* Free particles and ballistic behaviour */

L-7

2026-03-07T15:04:17Z

Rosso: /* Short recap: wavefunctions and eigenstates */

L-7

2026-03-07T14:17:26Z

Rosso: /* Scattering states */

L-7

2026-03-07T14:15:54Z

Rosso: /* Probability current */

L-7

2026-03-05T21:16:26Z

Rosso:

L-7

2026-03-05T17:43:52Z

Rosso:

'''Goal.''' This lecture introduces the phenomenon of localization. Localization is a '''wave phenomenon induced by disorder''' that suppresses transport in a system.

== Free particles and ballistic behaviour ==

The Schrödinger equation governs the evolution of the quantum state of a particle in one dimension:

<math display="block">
i\hbar \partial_t \psi(x,t)=H\psi(x,t),
\qquad
H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x).
</math>

Here <math>H</math> is the Hamiltonian. For a free particle the potential vanishes, <math>V(x)=0</math>.

One first looks for separable solutions, also called eigenstates or stationary solutions. If the particle is in an eigenstate, all physical observables are time independent.

For a free particle the stationary solutions are plane waves

<math display="block">
\psi_k(x,t)=\frac{1}{\sqrt{2\pi}}e^{ikx}e^{-iE_k t/\hbar},
\qquad
E_k=\frac{\hbar^2k^2}{2m}.
</math>

Here <math>k</math> is a real number and the spectrum is continuous. These solutions are completely delocalized over the real axis and cannot be normalized.

Physical states are obtained as superpositions of eigenstates. By the superposition principle any linear combination of stationary solutions is again a solution of the Schrödinger equation. Hence we can construct a localized wave packet:

<math display="block">
\psi(x,t)=\int_{-\infty}^{\infty}dk\,c(k)\psi_k(x,t),
\qquad
\int_{-\infty}^{\infty}dk\,|c(k)|^2=1.
</math>

=== Evolution of a Gaussian wave packet ===

* '''Initial state.'''
At time <math>t=0</math> consider the Gaussian packet

<math display="block">
\psi(x,0)=\frac{e^{-x^2/(4a^2)}}{(2\pi a^2)^{1/4}},
\qquad
|\psi(x,0)|^2=\frac{e^{-x^2/(2a^2)}}{\sqrt{2\pi a^2}}.
</math>

Show that the coefficients of the plane wave decomposition are

<math display="block">
c(k)=\left(\frac{2a^2}{\pi}\right)^{1/4}e^{-a^2k^2}.
</math>

* '''Time evolution.'''

Define the spreading velocity

<math display="block">
v_s=\frac{\hbar}{2ma^2}.
</math>

Show that the time evolution of the packet is

<math display="block">
\psi(x,t)=
\frac{e^{-x^2/(4a^2(1+i v_s t))}}
{[2\pi a^2(1+i v_s t)]^{1/4}}.
</math>

* '''Ballistic spreading.'''

The probability density becomes

<math display="block">
|\psi(x,t)|^2=
\frac{e^{-x^2/(2a^2(1+v_s^2 t^2))}}
{\sqrt{2\pi a^2(1+v_s^2 t^2)}}.
</math>

Hence

<math display="block">
\sqrt{\langle x^2\rangle}
=
\left(\int dx\,x^2|\psi(x,t)|^2\right)^{1/2}
=
a\sqrt{1+v_s^2 t^2}.
</math>

At long times

<math display="block">
\sqrt{\langle x^2\rangle}\sim v_s t.
</math>

This behaviour is called '''ballistic spreading'''.

It should be contrasted with the two other possible transport regimes:

* '''Diffusive motion''' (random walk):
<math>\sqrt{\langle x^2\rangle}\sim \sqrt{t}</math>

* '''Localized regime''':
<math>\sqrt{\langle x^2\rangle}</math> saturates at long times.

Understanding how disorder changes ballistic motion into diffusion and eventually localization is the main goal of the following lectures.

== Localization of the packet: general idea and experiment ==

[[File:Localization1DB.png|thumb|left|x240px|BEC condensate expanding in a 1D disordered potential. Billy et al., Nature (2008).]]

In 1958, P. Anderson proposed that disorder can suppress transport in quantum systems. This phenomenon, called '''Anderson localization''', has since been observed both numerically and experimentally.

In the experiment shown here, a Gaussian packet of a Bose–Einstein condensate is prepared in a harmonic trap. When the trap is removed the cloud initially starts to expand but quickly stops spreading and remains localized.

To understand this behaviour we must solve the eigenstate problem of the Hamiltonian in the presence of disorder.

[[File:Localization1dB.png|thumb|left|x140px|Semilog plot of the particle density. Billy et al., Nature (2008).]]

In a disordered potential an eigenstate of energy <math>E_k</math> has the form

<math display="block">
\psi_k(x,t)=\psi_k(x)e^{-iE_k t/\hbar}.
</math>

The spatial part of the wavefunction is localized around some position <math>\bar{x}</math> and decays exponentially:

<math display="block">
\psi_k(x)\sim
e^{-|x-\bar{x}|/\xi_{\text{loc}}(E_k)}.
</math>

Here <math>\xi_{\text{loc}}</math> is the localization length.

Because the eigenstates are localized, each of them has support only in a finite spatial region. The wave packet is therefore built from eigenstates localized near its initial position.

Two important observations follow:

* eigenstates localized far from the packet contribute exponentially little to the wavefunction near the packet,
* eigenstates contributing to the packet decay exponentially far from their center.

As a consequence, transport far away from the initial position of the particle is '''exponentially suppressed'''.

Localization therefore does not mean that the particle is frozen. The wave packet can still evolve in time, but it cannot propagate arbitrarily far.

== Conductance and diffusive transport ==

In most materials the effect of weak disorder is not localization but diffusion.

In the Drude picture electrons scatter randomly on impurities. Beyond the mean free path the motion of electrons becomes diffusive.

In this regime Ohm's laws hold.

* First law

<math display="block">
\frac{V}{I}=R,
\qquad
\frac{I}{V}=G.
</math>

Here <math>R</math> is the resistance and <math>G</math> the conductance.

* Second law

<math display="block">
R=\rho\frac{L}{S}\sim \rho L^{2-d},
\qquad
G=\sigma\frac{S}{L}\sim \sigma L^{d-2}.
</math>

Here <math>\rho</math> and <math>\sigma</math> are the resistivity and conductivity. These are material properties independent of the geometry of the sample.

These phenomenological laws are the macroscopic manifestation of diffusive transport.

== Conductance in the localized regime ==

When disorder is strong, diffusion is suppressed and the system becomes insulating.

In the localized phase the conductance decays exponentially with the system size:

<math display="block">
G\sim e^{-2L/\xi_{\text{loc}}}.
</math>

The localization length <math>\xi_{\text{loc}}</math> characterizes the spatial decay of the eigenstates.

== The “Gang of Four” scaling theory ==

In a famous PRL (1979), Abrahams, Anderson, Licciardello and Ramakrishnan proposed a scaling theory of localization.

It is based on the idea that the relevant quantity is the dimensionless conductance

<math display="block">
g=\frac{G\hbar}{e^2}.
</math>

The scaling equation reads

<math display="block">
\frac{d\ln g}{d\ln L}=\beta(g).
</math>

The function <math>\beta(g)</math> depends only on <math>g</math> and the spatial dimension.

The asymptotic behaviours are

<math display="block">
\beta(g)=
\begin{cases}
d-2 & g\to\infty \\
\sim \ln g & g\to 0
\end{cases}
</math>

The second relation reflects the exponential suppression of conductance in the localized regime.

If the beta function is monotonic, the scaling theory predicts:

* a metal–insulator transition for <math>d>2</math>,
* complete localization for <math>d\le2</math>.

L-8

2026-03-03T17:44:42Z

Rosso:

Goal: we will introduce the Anderson model, discuss the behaviour as a function of the dimension. In 1d localization can be connected to the product of random matrices.

= Anderson model (tight binding model)=

We consider disordered non-interacting particles hopping between nearest neighbors sites on a lattice. The hamiltonian reads:
<center> <math>
H= - t \sum_{ <i, j> } (c_i^\dagger c_j +c_j^\dagger c_i) \sum_i V_i c_i^\dagger c_i
</math></center>
The single particle hamiltonian in 1d reads
<center> <math>
H =
\begin{bmatrix}
V_1 & -t & 0 & 0 & 0 & 0 \\
-t & V_2 & -t & 0 & 0 & 0 \\
0 & -t & V_3 & -t & 0 & 0 \\
0 & 0 & -t & \ldots &-t & 0\\
0 & 0 & 0 & -t & \ldots & -t\\
0 & 0 & 0 & 0 & -t & V_L
\end{bmatrix}
</math></center>

For simplicity we set the hopping <math>t=1 </math>. The disorder are iid random variables drawn, uniformly from the box <math>(-\frac{W}{2},\frac{W}{2})</math>.

The final goal is to study the statistical properties of eigensystem
<center> <math>
H \psi=\epsilon \psi, \quad \text{with} \sum_{n=1}^L |\psi_n|^2=1
</math></center>

== Density of states (DOS)==

In 1d and in absence of disorder, the dispersion relation is
<math> \epsilon(k) = -2 \cos k, \quad k \in (-\pi, \pi), -2< \epsilon(k)< 2 </math>. From the dispersion relation, we compute the density of states (DOS) :
<center><math>
\rho(\epsilon) =\int_{-\pi}^\pi \frac{d k}{2 \pi} \delta(\epsilon-\epsilon(k))=\frac{1}{\pi } \frac{1}{\sqrt{4-\epsilon^2}} \quad \text{for } \epsilon \in (-2,2)</math></center>

In presence of disorder the DOS becomes larger, and display sample to sample fluctuations. One can consider its mean value, avergaed over disorder realization.

==Transfer matrices and Lyapunov exponents==

== Product of random matrices==

Let's consider again the Anderson Model in 1d. The eigensystem is well defined in a box of size L with Dirichelet boundary condition on the extremeties of the box.

Here we will solve the second order differential equation imposing instead Cauchy boundaries on one side of the box. Let's rewrite the previous eigensystem in the following form
<center> <math>
\begin{bmatrix}
\psi_{n+1} \\
\psi_{n}
\end{bmatrix}
=
\begin{bmatrix}
V_n -\epsilon & -1 \\
1 & 0
\end{bmatrix} \begin{bmatrix}
\psi_{n} \\
\psi_{n-1}
\end{bmatrix}
</math></center>
We can continue the recursion
<center> <math>
\begin{bmatrix}
\psi_{n+1} \\
\psi_{n}
\end{bmatrix}
=
\begin{bmatrix}
V_n -\epsilon & -1 \\
1 & 0
\end{bmatrix} \begin{bmatrix}
V_{n-1} -\epsilon & -1 \\
1 & 0
\end{bmatrix} \begin{bmatrix}
\psi_{n-1} \\
\psi_{n-2}
\end{bmatrix}
</math></center>
It is useful to introduce the transfer matrix and their product
<center> <math>
T_n =\begin{bmatrix}
V_n -\epsilon & -1 \\
1 & 0
\end{bmatrix}, \quad \Pi_n= T_n \cdot T_{n-1} \cdot\ldots T_1
</math></center>
The Schrodinger equation can be written as
<center> <math>
\begin{bmatrix}
\psi_{n+1} \\
\psi_{n}
\end{bmatrix}
=
\Pi_n \begin{bmatrix}
\psi_{1} \\
\psi_{0}
\end{bmatrix}
=
\begin{bmatrix}
\pi_{11} & \pi_{12} \\
\pi_{21} & \pi_{22}
\end{bmatrix} \begin{bmatrix}
\psi_{1} \\
\psi_{0}
\end{bmatrix}
</math></center>
==== Fustenberg Theorem ====
We define the norm of a 2x2 matrix:
<center> <math>
\|\Pi_n\|^2 =\frac{\pi_{11}^2+\pi_{21}^2+\pi_{12}^2+\pi_{22}^2}{2}
</math></center>
For large N, the Fustenberg theorem ensures the existence of the non-negative Lyapunov exponent, namely
<center> <math>
\lim_{n\to \infty} \frac{\ln \|\Pi_n\|}{n} = \gamma \ge 0
</math></center>
In absence of disorder <math> \gamma =0 </math> for <math>\epsilon \in (-2,2)</math>. Generically the Lyapunov is positive, <math> \gamma >0 </math>, and depends on <math>\epsilon </math> and on the distribution of <math>V_i </math>.

====Consequences====

 Localization length

Together with the norm, also <math> |\psi_n|^2</math> grows exponentially with n. We can write
<center> <math>
\ln |\psi_n| \sim \gamma n + \gamma_2 \chi \sqrt{n}</math>
</center>
which means that <math> \ln |\psi_n| </math> is performing a random walk with a drift.

However, our initial goal is a properly normalized eigenstate at energy <math>\epsilon </math>. We need to switch from Cauchy, where you set the initial condition, to Dirichelet or vonNeuman, where you set the behaviour at the two boundaries. The true eigenstate is obtained by matching two "Cauchy" solutions on the half box and imposing the normalization. Hence, we obtain a localized eigenstate and we can identify
<center> <math>
\xi_{\text{loc}}(\epsilon) \equiv \gamma^{-1}(\epsilon)
</math>
</center>

 Fluctuations

We expect strong fluctuations on quantites like <math> |\psi_n|, \|\Pi_n\|, G, \ldots </math>, while their logarithm is self averaging.

L-9

2026-03-03T17:01:36Z

Rosso:

= Eigenstates =

Without disorder, the eigenstates are delocalized plane waves.

In the presence of disorder, three scenarios can arise: delocalized eigenstates, where the wavefunction remains extended; localized eigenstates, where the wavefunction is exponentially confined to a finite region; multifractal eigenstates, occurring at the mobility edge, where the wavefunction exhibits a complex, scale-dependent structure.

To distinguish these regimes, it is useful to introduce the inverse participation ratio (IPR).
<math display="block">
\mathrm{IPR}(q)=\sum_n |\psi_n|^{2 q} \sim L^{-\tau_q}.
</math>

== Delocalized eigenstates ==

In this case, <math>|\psi_n|^{2} \approx L^{-d}</math>. Hence, we expect
<math display="block">
\mathrm{IPR}(q)=L^{d(1-q)},
\quad
\tau_q=d(1-q).
</math>

== Localized eigenstates ==

In this case, <math>|\psi_n|^{2} \approx 1/\xi_{\text{loc}}^{d}</math> on <math>\xi_{\text{loc}}^{d}</math> sites and almost zero elsewhere. Hence, we expect
<math display="block">
\mathrm{IPR}(q)=\text{const},
\quad
\tau_q=0.
</math>

== Multifractal eigenstates ==

The exponent <math>\tau_q</math> is called the '''multifractal exponent'''. It is a non-decreasing function of <math>q</math> with some special points:

* <math>\tau_0 = -d</math>, since the wavefunction is defined on all sites. In general, <math>\tau_0</math> represents the fractal dimension of the object under consideration and is purely a geometric property.
* <math>\tau_1 = 0</math>, imposed by normalization.

To observe multifractal behavior, we expect:
<math display="block">
|\psi_n|^{2} \approx L^{-\alpha}
\quad \text{for}\;
L^{f(\alpha)} \; \text{sites}.
</math>

The exponent <math>\alpha</math> is positive, and <math>f(\alpha)</math> is called the '''multifractal spectrum'''. Its maximum corresponds to the fractal dimension of the object, which in our case is <math>d</math>.

The relation between the multifractal spectrum <math>f(\alpha)</math> and the exponent <math>\tau_q</math> is given by:
<math display="block">
\mathrm{IPR}(q) = \sum_n |\psi_n|^{2q}
\sim \int d\alpha \, L^{-\alpha q} L^{f(\alpha)}
</math>
for large <math>L</math>. From this, we obtain:
<math display="block">
\tau(q) = \min_{\alpha} (\alpha q - f(\alpha)).
</math>

This implies that for <math>\alpha^*(q)</math>, which satisfies
<math display="block">f'(\alpha^*(q)) = q,</math>
we have
<math display="block">\tau(q) = \alpha^*(q) q - f(\alpha^*(q)).</math>

'''Delocalized wavefunctions''' have a simple spectrum: for <math>\alpha = d</math>, we find <math>f(\alpha = d) = d</math> and <math>f(\alpha \neq d) = -\infty</math>. This means that <math>\alpha^*(q) = d</math> is independent of <math>q</math>.

'''Multifractal wavefunctions''' exhibit a smoother dependence, leading to a continuous spectrum with a maximum at <math>\alpha_0</math>, where <math>f(\alpha_0) = d</math>. At <math>q = 1</math>, we have <math>f'(\alpha_1) = 1</math> and <math>f(\alpha_1) = \alpha_1</math>.

= Larkin model =

In your homework you solved a toy model for the interface:
<math display="block">
\partial_t h(r,t) = \nabla^2 h(r,t) + F(r).
</math>
For simplicity, we assume Gaussian disorder <math>\overline{F(r)}=0</math>, <math>\overline{F(r)F(r')}=\sigma^2 \delta^d(r-r')</math>.

You proved that:

* the roughness exponent of this model is <math>\zeta_L=\frac{4-d}{2}</math> below dimension 4
* the force per unit length acting on the center of the interface is <math>f= \sigma/\sqrt{L^d}</math>
* at long times the interface shape is
<math display="block">
\overline{h(q)h(-q)} = \frac{\sigma^2}{q^{d+2\zeta_L}}.
</math>

In the real depinning model the disorder is, however, a non-linear function of <math>h</math>. The idea of Larkin is that this linearization is correct up to <math>r_f</math>, the correlation length of the disorder along the <math>h</math> direction. This defines a Larkin length.

Indeed, from
<math display="block">
\overline{(h(r)-h(0))^2}
= \int d^d q \,\overline{h(q)h(-q)}\,(1-\cos(q r))
\sim \sigma^2 r^{2\zeta_L},
</math>
you get
<math display="block">
\overline{(h(\ell_L)-h(0))^2}= r_f^2,
\qquad
\ell_L=\left(\frac{r_f}{\sigma}\right)^{1/\zeta_L}.
</math>

Above this scale, roughness changes and pinning starts with a critical force
<math display="block">
f_c= \frac{\sigma}{\ell_L^{d/(2 \zeta_L)}}.
</math>

In <math>d=1</math> we have <math>\ell_L=\left(\frac{r_f}{\sigma}\right)^{2/3}</math>.

LBan-V

2026-03-03T08:07:30Z

Rosso: /* First generation */

= Avalanches at the Depinning Transition =

In the previous lesson, we studied the dynamics of an interface in a disordered medium under a uniform external force <math>F</math>. In a fully connected model, we derived the force–velocity characteristic and identified the critical depinning force <math>F_c</math>.

In this lesson, we focus on the avalanches that occur precisely at the depinning transition. To do so, we introduce a new driving protocol: instead of controlling the external force <math>F</math>, we control the position of the interface by coupling it to a parabolic potential. Each block is attracted toward a prescribed position <math>w</math> through a spring of stiffness <math>k_0</math>.

For simplicity, we restrict to the fully connected model, where the distance of block <math>i</math> from its local instability threshold is
<math display="block">
x_i = 1 - (h_{CM} - h_i) - k_0 (w - h_i).
</math>

The first term represents the elastic pull exerted by the rest of the interface, while the second encodes the external driving through the spring.

Here <math>h_{CM}</math> is the center-of-mass position of the interface. Because the sum of all internal elastic forces vanishes, the total external force is balanced solely by the pinning forces. The effective external force can thus be written as a function of <math>w</math>:
<math display="block">
F(w) = k_0 (w - h_{CM}).
</math>

As <math>w</math> is increased quasistatically, the force <math>F(w)</math> would increase if <math>h_{CM}</math> were fixed. When an avalanche takes place, <math>h_{CM}</math> jumps forward and <math>F(w)</math> suddenly decreases. However, in the steady state and in the thermodynamic limit <math>N \to \infty</math>, the force recovers a well-defined value. In the limit <math>k_0 \to 0</math>, this force tends to the critical depinning force <math>F_c</math>; at finite <math>k_0</math> it lies slightly below <math>F_c</math>.

== Quasi-Static Protocol and Avalanche Definition ==

To study avalanches, the position <math>w</math> is increased '''quasi-statically''' so that the block closest to its instability threshold reaches it,
<math display="block">x_i = 0.</math>

This block is the '''epicenter''' of the avalanche: it becomes unstable and jumps to the next well.

When block <math>i</math> jumps by <math>\Delta</math>, both the elastic contribution and the driving spring relax. This gives
<math display="block">
\begin{cases}
x_i = 0 \;\longrightarrow\; x_i = \Delta (1 + k_0), \\[6pt]
x_j \;\longrightarrow\; x_j - \dfrac{\Delta}{N} \quad (j \neq i).
\end{cases}
</math>

The key feature of the quasi-static protocol is that <math>w</math> does not evolve during the avalanche: all subsequent destabilizations are triggered exclusively by previously unstable blocks.

It is convenient to organize the avalanche into generations of unstable sites:

* First generation: the epicenter.
* Second generation: sites destabilized by it.
* Third generation: sites destabilized by generation two.
* And so on.

This hierarchical construction allows us to compute avalanche amplification step by step.

== Derivation of the Evolution Equation ==

Our goal is to determine the distribution <math>P_w(x)</math> of distances to threshold at fixed <math>w</math>.

We shift the parabola by <math>w \to w + \mathrm{d}w</math>. Before the shift:
<math display="block">
x_i(w) = 1 - k_0(w - h_i(w)) - (h_{CM}(w) - h_i(w)).
</math>

We now follow the dynamics generation by generation.

=== First generation ===

During the shift, the center of mass has not yet moved.

* Stable blocks (<math>x_i > k_0dw</math>):
<math display="block">
x_i^{t=1} = x_i - k_0dw.
</math>

* Blocks with <math>0 < x_i < k_0dw</math> are unstable. Since <math>dw</math> is infinitesimal, their fraction is
<math display="block">
P_w(0)\,k_0dw.
</math>

Unstable blocks first relax to ($x_i=0$) and are
then reinjected with a random kick $\Delta$ drawn from $g(\Delta)$,
leading to
\[
x_i^{t=1} = (1+k_0)\Delta.
\]
Let <math>p(x)\,dx</math> be the probability that an unstable block stabilizes in the interval <math>(x, x+dx)</math>.
By change of variables we have
<math display="block">
p(x)\,dx = g(\Delta)\,d\Delta.
</math>
Since <math>x = (1+k_0)\Delta</math>, it follows that <math>\Delta = \frac{x}{1+k_0}</math> and
<math display="block">
d\Delta = \frac{dx}{1+k_0}.
</math>
Therefore,
<math display="block">
p(x) = \frac{1}{1+k_0}\, g\!\left(\frac{x}{1+k_0}\right).
</math>

=== Second generation ===

The parabola is now fixed, but the center of mass has advanced:
<math display="block">
h_{CM} \to h_{CM} + \overline{\Delta}\,P_w(0)\,k_0dw.
</math>

Thus all sites shift again toward instability.

* Stable sites:
<math display="block">
x_i^{t=2}
=
x_i
-
\left(1+\overline{\Delta}P_w(0)\right)k_0dw.
</math>

* Newly unstable fraction:
<math display="block">
P_w(0)k_0dw + \left(\overline{\Delta}P_w(0)\right)P_w(0)k_0dw .
</math>

=== Higher generations ===

Iterating produces a geometric amplification:
<math display="block">
1+\overline{\Delta}P_w(0)
+(\overline{\Delta}P_w(0))^2+\dots
=
\frac{1}{1-\overline{\Delta}P_w(0)}.
</math>

The quantity <math>\overline{\Delta}P_w(0)</math> plays the role of a '''branching ratio''': it measures the average number of sites destabilized by one instability.

For stable sites
<math display="block">
x_i(w+dw) = x_i(w) - \frac{k_0}{1-\overline{\Delta}P_w(0)}\,dw.
</math>
while unstable sites are reinjected at a random location <math>\Delta(1+k_0)</math>. The fraction of unstable sites is
<math display="block">
\frac{P_w(0)}{1-\overline{\Delta}P_w(0)}k_0dw
</math>
is
This yields
<math display="block">
\partial_w P_w(x)
=
\frac{k_0}{1-\overline{\Delta}P_w(0)}
\left[
\partial_x P_w(x)
+
\frac{P_w(0)}{1+k_0}
g\!\left(\frac{x}{1+k_0}\right)
\right].
</math>

== Stationary solution ==

At large <math>w</math>:
<math display="block">
0=
\partial_x P_{\text{stat}}(x)
+
\frac{P_{\text{stat}}(0)}{1+k_0}
g\!\left(\frac{x}{1+k_0}\right).
</math>

Solving:
<math display="block">
P_{\text{stat}}(x)
=
\frac{1}{\overline{\Delta}(1+k_0)}
\int_{x/(1+k_0)}^\infty g(z)\,dz.
</math>

=== Critical Force ===

The average distance from the threshold gives a simple relation for the force acting on the system, namely
<math display="block">
F(k_0)= 1- \overline{x} = 1-(1+k_0) \frac{\overline{\Delta^2}}{2 \overline{\Delta}}.
</math>

In the limit <math>k_0\to 0</math> we obtain:
<math display="block">
F_c = F(k_0\to 0)= 1 - \frac{1}{2}\frac{\overline{\Delta^2}}{\overline{\Delta}}.
</math>

== Avalanches ==

We consider an avalanche starting from a single unstable site <math>x_0=0</math>.

Ordering sites by stability:
<math display="block">x_1<x_2<x_3<\dots</math>

From order statistics:
<math display="block">
\int_0^{x_1}P_w(t)dt=\frac1N.
</math>

Thus
<math display="block">
x_n \sim \frac{n}{NP_w(0)}.
</math>

Each instability gives kicks <math>\Delta/N</math>.

Compare mean kick and mean gap:
<math display="block">
\frac{\overline{\Delta}}{N}
\quad \text{vs}\quad
\frac{1}{NP_w(0)}.
</math>

Criticality occurs when
<math display="block">\overline{\Delta}P_w(0)=1.</math>

Using the stationary solution:
<math display="block">\overline{\Delta}P_w(0)=\frac{1}{1+k_0}.</math>

Hence:

* <math>k_0>0</math> → subcritical.
* <math>k_0=0</math> → critical.

== Mapping to a Random Walk ==

Define the random increments
<math display="block">
\eta_1 = \frac{\Delta_1}{N}- x_1,
\quad
\eta_2 = \frac{\Delta_2}{N}- (x_2-x_1),
\quad
\eta_3 = \frac{\Delta_3}{N}- (x_3-x_2),
\ldots
</math>
and the associated random walk
<math display="block">
X_n = \sum_{i=1}^n \eta_i.
</math>

The mean increment is
<math display="block">
\overline{\eta}
=
\frac{\overline{\Delta}}{N}
-
\frac{1}{N P_w(0)}.
</math>

An avalanche remains active as long as
<math display="block">X_n > 0.</math>

The avalanche size <math>S</math> is therefore the first-passage time of the walk to zero.

=== Critical case (k₀ = 0) ===

At criticality,
<math display="block">
\overline{\eta}=0,
\qquad
\overline{\Delta}P_w(0)=1.
</math>

The jump distribution is symmetric and has zero drift. We set <math>X_0=0</math>.

Let
<math display="block">
Q(n)=\text{Prob}\left(X_1>0,\dots,X_n>0\right)
</math>
be the survival probability of the walk.

By the Sparre–Andersen theorem, for large <math>n</math>,
<math display="block">
Q(n)\sim \frac{1}{\sqrt{\pi n}}.
</math>

The avalanche-size distribution is the first-passage probability:
<math display="block">
P(S)=Q(S)-Q(S+1).
</math>

Using the asymptotic form,
<math display="block">
P(S)
\sim
\frac{1}{\sqrt{\pi S}}
-
\frac{1}{\sqrt{\pi (S+1)}}
\sim
\frac{1}{2\sqrt{\pi}}\, S^{-3/2}.
</math>

Thus, at criticality,
<math display="block">
P(S)=\frac{1}{2\sqrt{\pi}}\,S^{-3/2}
\quad (S\gg1).
</math>

The universal exponent is
<math display="block">\tau=\frac{3}{2}.</math>

This power law is of Gutenberg–Richter type.

=== Finite k₀ > 0 (Subcritical case) ===

Using the stationary solution,
<math display="block">
\overline{\Delta}P_{\text{stat}}(0)
=
\frac{1}{1+k_0},
</math>
the mean drift becomes
<math display="block">
\overline{\eta}
=
-
k_0\,\frac{\overline{\Delta}}{N}.
</math>

The random walk is weakly biased toward negative values. For small <math>k_0</math>, the walk is only slightly tilted.

In this case the distribution retains the critical form
<math display="block">
P(S)\sim S^{-3/2}
</math>
up to a cutoff set by the inverse squared drift:
<math display="block">
S_{\max}\sim k_0^{-2}.
</math>

LBan-V

2026-03-03T07:54:07Z

Rosso: /* First generation */

= Avalanches at the Depinning Transition =

In the previous lesson, we studied the dynamics of an interface in a disordered medium under a uniform external force <math>F</math>. In a fully connected model, we derived the force–velocity characteristic and identified the critical depinning force <math>F_c</math>.

In this lesson, we focus on the avalanches that occur precisely at the depinning transition. To do so, we introduce a new driving protocol: instead of controlling the external force <math>F</math>, we control the position of the interface by coupling it to a parabolic potential. Each block is attracted toward a prescribed position <math>w</math> through a spring of stiffness <math>k_0</math>.

For simplicity, we restrict to the fully connected model, where the distance of block <math>i</math> from its local instability threshold is
<math display="block">
x_i = 1 - (h_{CM} - h_i) - k_0 (w - h_i).
</math>

The first term represents the elastic pull exerted by the rest of the interface, while the second encodes the external driving through the spring.

Here <math>h_{CM}</math> is the center-of-mass position of the interface. Because the sum of all internal elastic forces vanishes, the total external force is balanced solely by the pinning forces. The effective external force can thus be written as a function of <math>w</math>:
<math display="block">
F(w) = k_0 (w - h_{CM}).
</math>

As <math>w</math> is increased quasistatically, the force <math>F(w)</math> would increase if <math>h_{CM}</math> were fixed. When an avalanche takes place, <math>h_{CM}</math> jumps forward and <math>F(w)</math> suddenly decreases. However, in the steady state and in the thermodynamic limit <math>N \to \infty</math>, the force recovers a well-defined value. In the limit <math>k_0 \to 0</math>, this force tends to the critical depinning force <math>F_c</math>; at finite <math>k_0</math> it lies slightly below <math>F_c</math>.

== Quasi-Static Protocol and Avalanche Definition ==

To study avalanches, the position <math>w</math> is increased '''quasi-statically''' so that the block closest to its instability threshold reaches it,
<math display="block">x_i = 0.</math>

This block is the '''epicenter''' of the avalanche: it becomes unstable and jumps to the next well.

When block <math>i</math> jumps by <math>\Delta</math>, both the elastic contribution and the driving spring relax. This gives
<math display="block">
\begin{cases}
x_i = 0 \;\longrightarrow\; x_i = \Delta (1 + k_0), \\[6pt]
x_j \;\longrightarrow\; x_j - \dfrac{\Delta}{N} \quad (j \neq i).
\end{cases}
</math>

The key feature of the quasi-static protocol is that <math>w</math> does not evolve during the avalanche: all subsequent destabilizations are triggered exclusively by previously unstable blocks.

It is convenient to organize the avalanche into generations of unstable sites:

* First generation: the epicenter.
* Second generation: sites destabilized by it.
* Third generation: sites destabilized by generation two.
* And so on.

This hierarchical construction allows us to compute avalanche amplification step by step.

== Derivation of the Evolution Equation ==

Our goal is to determine the distribution <math>P_w(x)</math> of distances to threshold at fixed <math>w</math>.

We shift the parabola by <math>w \to w + \mathrm{d}w</math>. Before the shift:
<math display="block">
x_i(w) = 1 - k_0(w - h_i(w)) - (h_{CM}(w) - h_i(w)).
</math>

We now follow the dynamics generation by generation.

=== First generation ===

During the shift, the center of mass has not yet moved.

* Stable sites (<math>x_i > k_0dw</math>):
<math display="block">
x_i^{t=1} = x_i - k_0dw.
</math>

* Sites with <math>0 < x_i < k_0dw</math> become unstable.

Since <math>dw</math> is infinitesimal, their fraction is
<math display="block">
P_w(0)\,k_0dw.
</math>

These unstable blocks jump and stabilize at
<math display="block">
x_i^{t=1} = \Delta (1+k_0).
</math>

Here <math>\Delta</math> is a random variable drawn from the distribution <math>g(\Delta)</math>.
Let <math>p(x)\,dx</math> be the probability that an unstable block stabilizes in the interval <math>(x, x+dx)</math>.
By change of variables we have
<math display="block">
p(x)\,dx = g(\Delta)\,d\Delta.
</math>
Since <math>x = (1+k_0)\Delta</math>, it follows that <math>\Delta = \frac{x}{1+k_0}</math> and
<math display="block">
d\Delta = \frac{dx}{1+k_0}.
</math>
Therefore,
<math display="block">
p(x) = \frac{1}{1+k_0}\, g\!\left(\frac{x}{1+k_0}\right).
</math>

=== Second generation ===

The parabola is now fixed, but the center of mass has advanced:
<math display="block">
h_{CM} \to h_{CM} + \overline{\Delta}\,P_w(0)\,k_0dw.
</math>

Thus all sites shift again toward instability.

* Stable sites:
<math display="block">
x_i^{t=2}
=
x_i
-
\left(1+\overline{\Delta}P_w(0)\right)k_0dw.
</math>

* Newly unstable fraction:
<math display="block">
P_w(0)k_0dw + \left(\overline{\Delta}P_w(0)\right)P_w(0)k_0dw .
</math>

=== Higher generations ===

Iterating produces a geometric amplification:
<math display="block">
1+\overline{\Delta}P_w(0)
+(\overline{\Delta}P_w(0))^2+\dots
=
\frac{1}{1-\overline{\Delta}P_w(0)}.
</math>

The quantity <math>\overline{\Delta}P_w(0)</math> plays the role of a '''branching ratio''': it measures the average number of sites destabilized by one instability.

For stable sites
<math display="block">
x_i(w+dw) = x_i(w) - \frac{k_0}{1-\overline{\Delta}P_w(0)}\,dw.
</math>
while unstable sites are reinjected at a random location <math>\Delta(1+k_0)</math>. The fraction of unstable sites is
<math display="block">
\frac{P_w(0)}{1-\overline{\Delta}P_w(0)}k_0dw
</math>
is
This yields
<math display="block">
\partial_w P_w(x)
=
\frac{k_0}{1-\overline{\Delta}P_w(0)}
\left[
\partial_x P_w(x)
+
\frac{P_w(0)}{1+k_0}
g\!\left(\frac{x}{1+k_0}\right)
\right].
</math>

== Stationary solution ==

At large <math>w</math>:
<math display="block">
0=
\partial_x P_{\text{stat}}(x)
+
\frac{P_{\text{stat}}(0)}{1+k_0}
g\!\left(\frac{x}{1+k_0}\right).
</math>

Solving:
<math display="block">
P_{\text{stat}}(x)
=
\frac{1}{\overline{\Delta}(1+k_0)}
\int_{x/(1+k_0)}^\infty g(z)\,dz.
</math>

=== Critical Force ===

The average distance from the threshold gives a simple relation for the force acting on the system, namely
<math display="block">
F(k_0)= 1- \overline{x} = 1-(1+k_0) \frac{\overline{\Delta^2}}{2 \overline{\Delta}}.
</math>

In the limit <math>k_0\to 0</math> we obtain:
<math display="block">
F_c = F(k_0\to 0)= 1 - \frac{1}{2}\frac{\overline{\Delta^2}}{\overline{\Delta}}.
</math>

== Avalanches ==

We consider an avalanche starting from a single unstable site <math>x_0=0</math>.

Ordering sites by stability:
<math display="block">x_1<x_2<x_3<\dots</math>

From order statistics:
<math display="block">
\int_0^{x_1}P_w(t)dt=\frac1N.
</math>

Thus
<math display="block">
x_n \sim \frac{n}{NP_w(0)}.
</math>

Each instability gives kicks <math>\Delta/N</math>.

Compare mean kick and mean gap:
<math display="block">
\frac{\overline{\Delta}}{N}
\quad \text{vs}\quad
\frac{1}{NP_w(0)}.
</math>

Criticality occurs when
<math display="block">\overline{\Delta}P_w(0)=1.</math>

Using the stationary solution:
<math display="block">\overline{\Delta}P_w(0)=\frac{1}{1+k_0}.</math>

Hence:

* <math>k_0>0</math> → subcritical.
* <math>k_0=0</math> → critical.

== Mapping to a Random Walk ==

Define the random increments
<math display="block">
\eta_1 = \frac{\Delta_1}{N}- x_1,
\quad
\eta_2 = \frac{\Delta_2}{N}- (x_2-x_1),
\quad
\eta_3 = \frac{\Delta_3}{N}- (x_3-x_2),
\ldots
</math>
and the associated random walk
<math display="block">
X_n = \sum_{i=1}^n \eta_i.
</math>

The mean increment is
<math display="block">
\overline{\eta}
=
\frac{\overline{\Delta}}{N}
-
\frac{1}{N P_w(0)}.
</math>

An avalanche remains active as long as
<math display="block">X_n > 0.</math>

The avalanche size <math>S</math> is therefore the first-passage time of the walk to zero.

=== Critical case (k₀ = 0) ===

At criticality,
<math display="block">
\overline{\eta}=0,
\qquad
\overline{\Delta}P_w(0)=1.
</math>

The jump distribution is symmetric and has zero drift. We set <math>X_0=0</math>.

Let
<math display="block">
Q(n)=\text{Prob}\left(X_1>0,\dots,X_n>0\right)
</math>
be the survival probability of the walk.

By the Sparre–Andersen theorem, for large <math>n</math>,
<math display="block">
Q(n)\sim \frac{1}{\sqrt{\pi n}}.
</math>

The avalanche-size distribution is the first-passage probability:
<math display="block">
P(S)=Q(S)-Q(S+1).
</math>

Using the asymptotic form,
<math display="block">
P(S)
\sim
\frac{1}{\sqrt{\pi S}}
-
\frac{1}{\sqrt{\pi (S+1)}}
\sim
\frac{1}{2\sqrt{\pi}}\, S^{-3/2}.
</math>

Thus, at criticality,
<math display="block">
P(S)=\frac{1}{2\sqrt{\pi}}\,S^{-3/2}
\quad (S\gg1).
</math>

The universal exponent is
<math display="block">\tau=\frac{3}{2}.</math>

This power law is of Gutenberg–Richter type.

=== Finite k₀ > 0 (Subcritical case) ===

Using the stationary solution,
<math display="block">
\overline{\Delta}P_{\text{stat}}(0)
=
\frac{1}{1+k_0},
</math>
the mean drift becomes
<math display="block">
\overline{\eta}
=
-
k_0\,\frac{\overline{\Delta}}{N}.
</math>

The random walk is weakly biased toward negative values. For small <math>k_0</math>, the walk is only slightly tilted.

In this case the distribution retains the critical form
<math display="block">
P(S)\sim S^{-3/2}
</math>
up to a cutoff set by the inverse squared drift:
<math display="block">
S_{\max}\sim k_0^{-2}.
</math>

LBan-V

2026-03-03T07:53:27Z

Rosso: /* First generation */

= Avalanches at the Depinning Transition =

In the previous lesson, we studied the dynamics of an interface in a disordered medium under a uniform external force <math>F</math>. In a fully connected model, we derived the force–velocity characteristic and identified the critical depinning force <math>F_c</math>.

In this lesson, we focus on the avalanches that occur precisely at the depinning transition. To do so, we introduce a new driving protocol: instead of controlling the external force <math>F</math>, we control the position of the interface by coupling it to a parabolic potential. Each block is attracted toward a prescribed position <math>w</math> through a spring of stiffness <math>k_0</math>.

For simplicity, we restrict to the fully connected model, where the distance of block <math>i</math> from its local instability threshold is
<math display="block">
x_i = 1 - (h_{CM} - h_i) - k_0 (w - h_i).
</math>

The first term represents the elastic pull exerted by the rest of the interface, while the second encodes the external driving through the spring.

Here <math>h_{CM}</math> is the center-of-mass position of the interface. Because the sum of all internal elastic forces vanishes, the total external force is balanced solely by the pinning forces. The effective external force can thus be written as a function of <math>w</math>:
<math display="block">
F(w) = k_0 (w - h_{CM}).
</math>

As <math>w</math> is increased quasistatically, the force <math>F(w)</math> would increase if <math>h_{CM}</math> were fixed. When an avalanche takes place, <math>h_{CM}</math> jumps forward and <math>F(w)</math> suddenly decreases. However, in the steady state and in the thermodynamic limit <math>N \to \infty</math>, the force recovers a well-defined value. In the limit <math>k_0 \to 0</math>, this force tends to the critical depinning force <math>F_c</math>; at finite <math>k_0</math> it lies slightly below <math>F_c</math>.

== Quasi-Static Protocol and Avalanche Definition ==

To study avalanches, the position <math>w</math> is increased '''quasi-statically''' so that the block closest to its instability threshold reaches it,
<math display="block">x_i = 0.</math>

This block is the '''epicenter''' of the avalanche: it becomes unstable and jumps to the next well.

When block <math>i</math> jumps by <math>\Delta</math>, both the elastic contribution and the driving spring relax. This gives
<math display="block">
\begin{cases}
x_i = 0 \;\longrightarrow\; x_i = \Delta (1 + k_0), \\[6pt]
x_j \;\longrightarrow\; x_j - \dfrac{\Delta}{N} \quad (j \neq i).
\end{cases}
</math>

The key feature of the quasi-static protocol is that <math>w</math> does not evolve during the avalanche: all subsequent destabilizations are triggered exclusively by previously unstable blocks.

It is convenient to organize the avalanche into generations of unstable sites:

* First generation: the epicenter.
* Second generation: sites destabilized by it.
* Third generation: sites destabilized by generation two.
* And so on.

This hierarchical construction allows us to compute avalanche amplification step by step.

== Derivation of the Evolution Equation ==

Our goal is to determine the distribution <math>P_w(x)</math> of distances to threshold at fixed <math>w</math>.

We shift the parabola by <math>w \to w + \mathrm{d}w</math>. Before the shift:
<math display="block">
x_i(w) = 1 - k_0(w - h_i(w)) - (h_{CM}(w) - h_i(w)).
</math>

We now follow the dynamics generation by generation.

=== First generation ===

During the shift, the center of mass has not yet moved.

* Stable sites (<math>x_i > k_0dw</math>):
<math display="block">
x_i^{t=1} = x_i - k_0dw.
</math>

* Sites with <math>0 < x_i < k_0dw</math> become unstable.

Since <math>dw</math> is infinitesimal, their fraction is
<math display="block">
P_w(0)\,k_0dw.
</math>

These unstable blocks jump and stabilize at
<math display="block">
x_i^{t=1} = \Delta (1+k_0).
</math>

Here <math>\Delta</math> is a random variable drawn from the distribution <math>g(\Delta)</math>.
Let <math>p(x)\,dx</math> be the probability that an unstable block stabilizes in the interval <math>(x, x+dx)</math>.
By change of variables we have
<math display="block">
p(x)\,dx = g(\Delta)\,d\Delta.
</math>
Since <math>x = (1+k_0)\Delta</math>, it follows that <math>\Delta = \frac{x}{1+k_0}</math> and
<math display="block">
d\Delta = \frac{dx}{1+k_0}.
</math>
Therefore,
<math display="block">
p(x) = \frac{1}{1+k_0}\, g\!\left(\frac{x}{1+k_0}\right).
\]

=== Second generation ===

The parabola is now fixed, but the center of mass has advanced:
<math display="block">
h_{CM} \to h_{CM} + \overline{\Delta}\,P_w(0)\,k_0dw.
</math>

Thus all sites shift again toward instability.

* Stable sites:
<math display="block">
x_i^{t=2}
=
x_i
-
\left(1+\overline{\Delta}P_w(0)\right)k_0dw.
</math>

* Newly unstable fraction:
<math display="block">
P_w(0)k_0dw + \left(\overline{\Delta}P_w(0)\right)P_w(0)k_0dw .
</math>

=== Higher generations ===

Iterating produces a geometric amplification:
<math display="block">
1+\overline{\Delta}P_w(0)
+(\overline{\Delta}P_w(0))^2+\dots
=
\frac{1}{1-\overline{\Delta}P_w(0)}.
</math>

The quantity <math>\overline{\Delta}P_w(0)</math> plays the role of a '''branching ratio''': it measures the average number of sites destabilized by one instability.

For stable sites
<math display="block">
x_i(w+dw) = x_i(w) - \frac{k_0}{1-\overline{\Delta}P_w(0)}\,dw.
</math>
while unstable sites are reinjected at a random location <math>\Delta(1+k_0)</math>. The fraction of unstable sites is
<math display="block">
\frac{P_w(0)}{1-\overline{\Delta}P_w(0)}k_0dw
</math>
is
This yields
<math display="block">
\partial_w P_w(x)
=
\frac{k_0}{1-\overline{\Delta}P_w(0)}
\left[
\partial_x P_w(x)
+
\frac{P_w(0)}{1+k_0}
g\!\left(\frac{x}{1+k_0}\right)
\right].
</math>

== Stationary solution ==

At large <math>w</math>:
<math display="block">
0=
\partial_x P_{\text{stat}}(x)
+
\frac{P_{\text{stat}}(0)}{1+k_0}
g\!\left(\frac{x}{1+k_0}\right).
</math>

Solving:
<math display="block">
P_{\text{stat}}(x)
=
\frac{1}{\overline{\Delta}(1+k_0)}
\int_{x/(1+k_0)}^\infty g(z)\,dz.
</math>

=== Critical Force ===

The average distance from the threshold gives a simple relation for the force acting on the system, namely
<math display="block">
F(k_0)= 1- \overline{x} = 1-(1+k_0) \frac{\overline{\Delta^2}}{2 \overline{\Delta}}.
</math>

In the limit <math>k_0\to 0</math> we obtain:
<math display="block">
F_c = F(k_0\to 0)= 1 - \frac{1}{2}\frac{\overline{\Delta^2}}{\overline{\Delta}}.
</math>

== Avalanches ==

We consider an avalanche starting from a single unstable site <math>x_0=0</math>.

Ordering sites by stability:
<math display="block">x_1<x_2<x_3<\dots</math>

From order statistics:
<math display="block">
\int_0^{x_1}P_w(t)dt=\frac1N.
</math>

Thus
<math display="block">
x_n \sim \frac{n}{NP_w(0)}.
</math>

Each instability gives kicks <math>\Delta/N</math>.

Compare mean kick and mean gap:
<math display="block">
\frac{\overline{\Delta}}{N}
\quad \text{vs}\quad
\frac{1}{NP_w(0)}.
</math>

Criticality occurs when
<math display="block">\overline{\Delta}P_w(0)=1.</math>

Using the stationary solution:
<math display="block">\overline{\Delta}P_w(0)=\frac{1}{1+k_0}.</math>

Hence:

* <math>k_0>0</math> → subcritical.
* <math>k_0=0</math> → critical.

== Mapping to a Random Walk ==

Define the random increments
<math display="block">
\eta_1 = \frac{\Delta_1}{N}- x_1,
\quad
\eta_2 = \frac{\Delta_2}{N}- (x_2-x_1),
\quad
\eta_3 = \frac{\Delta_3}{N}- (x_3-x_2),
\ldots
</math>
and the associated random walk
<math display="block">
X_n = \sum_{i=1}^n \eta_i.
</math>

The mean increment is
<math display="block">
\overline{\eta}
=
\frac{\overline{\Delta}}{N}
-
\frac{1}{N P_w(0)}.
</math>

An avalanche remains active as long as
<math display="block">X_n > 0.</math>

The avalanche size <math>S</math> is therefore the first-passage time of the walk to zero.

=== Critical case (k₀ = 0) ===

At criticality,
<math display="block">
\overline{\eta}=0,
\qquad
\overline{\Delta}P_w(0)=1.
</math>

The jump distribution is symmetric and has zero drift. We set <math>X_0=0</math>.

Let
<math display="block">
Q(n)=\text{Prob}\left(X_1>0,\dots,X_n>0\right)
</math>
be the survival probability of the walk.

By the Sparre–Andersen theorem, for large <math>n</math>,
<math display="block">
Q(n)\sim \frac{1}{\sqrt{\pi n}}.
</math>

The avalanche-size distribution is the first-passage probability:
<math display="block">
P(S)=Q(S)-Q(S+1).
</math>

Using the asymptotic form,
<math display="block">
P(S)
\sim
\frac{1}{\sqrt{\pi S}}
-
\frac{1}{\sqrt{\pi (S+1)}}
\sim
\frac{1}{2\sqrt{\pi}}\, S^{-3/2}.
</math>

Thus, at criticality,
<math display="block">
P(S)=\frac{1}{2\sqrt{\pi}}\,S^{-3/2}
\quad (S\gg1).
</math>

The universal exponent is
<math display="block">\tau=\frac{3}{2}.</math>

This power law is of Gutenberg–Richter type.

=== Finite k₀ > 0 (Subcritical case) ===

Using the stationary solution,
<math display="block">
\overline{\Delta}P_{\text{stat}}(0)
=
\frac{1}{1+k_0},
</math>
the mean drift becomes
<math display="block">
\overline{\eta}
=
-
k_0\,\frac{\overline{\Delta}}{N}.
</math>

The random walk is weakly biased toward negative values. For small <math>k_0</math>, the walk is only slightly tilted.

In this case the distribution retains the critical form
<math display="block">
P(S)\sim S^{-3/2}
</math>
up to a cutoff set by the inverse squared drift:
<math display="block">
S_{\max}\sim k_0^{-2}.
</math>

LBan-V

2026-03-03T07:52:54Z

Rosso: /* First generation */

= Avalanches at the Depinning Transition =

In the previous lesson, we studied the dynamics of an interface in a disordered medium under a uniform external force <math>F</math>. In a fully connected model, we derived the force–velocity characteristic and identified the critical depinning force <math>F_c</math>.

In this lesson, we focus on the avalanches that occur precisely at the depinning transition. To do so, we introduce a new driving protocol: instead of controlling the external force <math>F</math>, we control the position of the interface by coupling it to a parabolic potential. Each block is attracted toward a prescribed position <math>w</math> through a spring of stiffness <math>k_0</math>.

For simplicity, we restrict to the fully connected model, where the distance of block <math>i</math> from its local instability threshold is
<math display="block">
x_i = 1 - (h_{CM} - h_i) - k_0 (w - h_i).
</math>

The first term represents the elastic pull exerted by the rest of the interface, while the second encodes the external driving through the spring.

Here <math>h_{CM}</math> is the center-of-mass position of the interface. Because the sum of all internal elastic forces vanishes, the total external force is balanced solely by the pinning forces. The effective external force can thus be written as a function of <math>w</math>:
<math display="block">
F(w) = k_0 (w - h_{CM}).
</math>

As <math>w</math> is increased quasistatically, the force <math>F(w)</math> would increase if <math>h_{CM}</math> were fixed. When an avalanche takes place, <math>h_{CM}</math> jumps forward and <math>F(w)</math> suddenly decreases. However, in the steady state and in the thermodynamic limit <math>N \to \infty</math>, the force recovers a well-defined value. In the limit <math>k_0 \to 0</math>, this force tends to the critical depinning force <math>F_c</math>; at finite <math>k_0</math> it lies slightly below <math>F_c</math>.

== Quasi-Static Protocol and Avalanche Definition ==

To study avalanches, the position <math>w</math> is increased '''quasi-statically''' so that the block closest to its instability threshold reaches it,
<math display="block">x_i = 0.</math>

This block is the '''epicenter''' of the avalanche: it becomes unstable and jumps to the next well.

When block <math>i</math> jumps by <math>\Delta</math>, both the elastic contribution and the driving spring relax. This gives
<math display="block">
\begin{cases}
x_i = 0 \;\longrightarrow\; x_i = \Delta (1 + k_0), \\[6pt]
x_j \;\longrightarrow\; x_j - \dfrac{\Delta}{N} \quad (j \neq i).
\end{cases}
</math>

The key feature of the quasi-static protocol is that <math>w</math> does not evolve during the avalanche: all subsequent destabilizations are triggered exclusively by previously unstable blocks.

It is convenient to organize the avalanche into generations of unstable sites:

* First generation: the epicenter.
* Second generation: sites destabilized by it.
* Third generation: sites destabilized by generation two.
* And so on.

This hierarchical construction allows us to compute avalanche amplification step by step.

== Derivation of the Evolution Equation ==

Our goal is to determine the distribution <math>P_w(x)</math> of distances to threshold at fixed <math>w</math>.

We shift the parabola by <math>w \to w + \mathrm{d}w</math>. Before the shift:
<math display="block">
x_i(w) = 1 - k_0(w - h_i(w)) - (h_{CM}(w) - h_i(w)).
</math>

We now follow the dynamics generation by generation.

=== First generation ===

During the shift, the center of mass has not yet moved.

* Stable sites (<math>x_i > k_0dw</math>):
<math display="block">
x_i^{t=1} = x_i - k_0dw.
</math>

* Sites with <math>0 < x_i < k_0dw</math> become unstable.

Since <math>dw</math> is infinitesimal, their fraction is
<math display="block">
P_w(0)\,k_0dw.
</math>

These unstable blocks jump and stabilize at
<math display="block">
x_i^{t=1} = \Delta (1+k_0).
</math>

Here <math>\Delta</math> is a random variable drawn from the distribution <math>g(\Delta)</math>.
Let <math>p(x)\,dx<math> be the probability that an unstable block stabilizes in the interval <math>(x, x+dx)</math>.
By change of variables we have
<math display="block">
p(x)\,dx = g(\Delta)\,d\Delta.
</math>
Since <math>x = (1+k_0)\Delta</math>, it follows that <math>\Delta = \frac{x}{1+k_0}</math> and
<math display="block">
d\Delta = \frac{dx}{1+k_0}.
</math>
Therefore,
<math display="block">
p(x) = \frac{1}{1+k_0}\, g\!\left(\frac{x}{1+k_0}\right).
\]

=== Second generation ===

The parabola is now fixed, but the center of mass has advanced:
<math display="block">
h_{CM} \to h_{CM} + \overline{\Delta}\,P_w(0)\,k_0dw.
</math>

Thus all sites shift again toward instability.

* Stable sites:
<math display="block">
x_i^{t=2}
=
x_i
-
\left(1+\overline{\Delta}P_w(0)\right)k_0dw.
</math>

* Newly unstable fraction:
<math display="block">
P_w(0)k_0dw + \left(\overline{\Delta}P_w(0)\right)P_w(0)k_0dw .
</math>

=== Higher generations ===

Iterating produces a geometric amplification:
<math display="block">
1+\overline{\Delta}P_w(0)
+(\overline{\Delta}P_w(0))^2+\dots
=
\frac{1}{1-\overline{\Delta}P_w(0)}.
</math>

The quantity <math>\overline{\Delta}P_w(0)</math> plays the role of a '''branching ratio''': it measures the average number of sites destabilized by one instability.

For stable sites
<math display="block">
x_i(w+dw) = x_i(w) - \frac{k_0}{1-\overline{\Delta}P_w(0)}\,dw.
</math>
while unstable sites are reinjected at a random location <math>\Delta(1+k_0)</math>. The fraction of unstable sites is
<math display="block">
\frac{P_w(0)}{1-\overline{\Delta}P_w(0)}k_0dw
</math>
is
This yields
<math display="block">
\partial_w P_w(x)
=
\frac{k_0}{1-\overline{\Delta}P_w(0)}
\left[
\partial_x P_w(x)
+
\frac{P_w(0)}{1+k_0}
g\!\left(\frac{x}{1+k_0}\right)
\right].
</math>

== Stationary solution ==

At large <math>w</math>:
<math display="block">
0=
\partial_x P_{\text{stat}}(x)
+
\frac{P_{\text{stat}}(0)}{1+k_0}
g\!\left(\frac{x}{1+k_0}\right).
</math>

Solving:
<math display="block">
P_{\text{stat}}(x)
=
\frac{1}{\overline{\Delta}(1+k_0)}
\int_{x/(1+k_0)}^\infty g(z)\,dz.
</math>

=== Critical Force ===

The average distance from the threshold gives a simple relation for the force acting on the system, namely
<math display="block">
F(k_0)= 1- \overline{x} = 1-(1+k_0) \frac{\overline{\Delta^2}}{2 \overline{\Delta}}.
</math>

In the limit <math>k_0\to 0</math> we obtain:
<math display="block">
F_c = F(k_0\to 0)= 1 - \frac{1}{2}\frac{\overline{\Delta^2}}{\overline{\Delta}}.
</math>

== Avalanches ==

We consider an avalanche starting from a single unstable site <math>x_0=0</math>.

Ordering sites by stability:
<math display="block">x_1<x_2<x_3<\dots</math>

From order statistics:
<math display="block">
\int_0^{x_1}P_w(t)dt=\frac1N.
</math>

Thus
<math display="block">
x_n \sim \frac{n}{NP_w(0)}.
</math>

Each instability gives kicks <math>\Delta/N</math>.

Compare mean kick and mean gap:
<math display="block">
\frac{\overline{\Delta}}{N}
\quad \text{vs}\quad
\frac{1}{NP_w(0)}.
</math>

Criticality occurs when
<math display="block">\overline{\Delta}P_w(0)=1.</math>

Using the stationary solution:
<math display="block">\overline{\Delta}P_w(0)=\frac{1}{1+k_0}.</math>

Hence:

* <math>k_0>0</math> → subcritical.
* <math>k_0=0</math> → critical.

== Mapping to a Random Walk ==

Define the random increments
<math display="block">
\eta_1 = \frac{\Delta_1}{N}- x_1,
\quad
\eta_2 = \frac{\Delta_2}{N}- (x_2-x_1),
\quad
\eta_3 = \frac{\Delta_3}{N}- (x_3-x_2),
\ldots
</math>
and the associated random walk
<math display="block">
X_n = \sum_{i=1}^n \eta_i.
</math>

The mean increment is
<math display="block">
\overline{\eta}
=
\frac{\overline{\Delta}}{N}
-
\frac{1}{N P_w(0)}.
</math>

An avalanche remains active as long as
<math display="block">X_n > 0.</math>

The avalanche size <math>S</math> is therefore the first-passage time of the walk to zero.

=== Critical case (k₀ = 0) ===

At criticality,
<math display="block">
\overline{\eta}=0,
\qquad
\overline{\Delta}P_w(0)=1.
</math>

The jump distribution is symmetric and has zero drift. We set <math>X_0=0</math>.

Let
<math display="block">
Q(n)=\text{Prob}\left(X_1>0,\dots,X_n>0\right)
</math>
be the survival probability of the walk.

By the Sparre–Andersen theorem, for large <math>n</math>,
<math display="block">
Q(n)\sim \frac{1}{\sqrt{\pi n}}.
</math>

The avalanche-size distribution is the first-passage probability:
<math display="block">
P(S)=Q(S)-Q(S+1).
</math>

Using the asymptotic form,
<math display="block">
P(S)
\sim
\frac{1}{\sqrt{\pi S}}
-
\frac{1}{\sqrt{\pi (S+1)}}
\sim
\frac{1}{2\sqrt{\pi}}\, S^{-3/2}.
</math>

Thus, at criticality,
<math display="block">
P(S)=\frac{1}{2\sqrt{\pi}}\,S^{-3/2}
\quad (S\gg1).
</math>

The universal exponent is
<math display="block">\tau=\frac{3}{2}.</math>

This power law is of Gutenberg–Richter type.

=== Finite k₀ > 0 (Subcritical case) ===

Using the stationary solution,
<math display="block">
\overline{\Delta}P_{\text{stat}}(0)
=
\frac{1}{1+k_0},
</math>
the mean drift becomes
<math display="block">
\overline{\eta}
=
-
k_0\,\frac{\overline{\Delta}}{N}.
</math>

The random walk is weakly biased toward negative values. For small <math>k_0</math>, the walk is only slightly tilted.

In this case the distribution retains the critical form
<math display="block">
P(S)\sim S^{-3/2}
</math>
up to a cutoff set by the inverse squared drift:
<math display="block">
S_{\max}\sim k_0^{-2}.
</math>

LBan-V

2026-03-02T14:47:25Z

Rosso: /* Derivation of the Evolution Equation */

= Avalanches at the Depinning Transition =

In the previous lesson, we studied the dynamics of an interface in a disordered medium under a uniform external force <math>F</math>. In a fully connected model, we derived the force–velocity characteristic and identified the critical depinning force <math>F_c</math>.

In this lesson, we focus on the avalanches that occur precisely at the depinning transition. To do so, we introduce a new driving protocol: instead of controlling the external force <math>F</math>, we control the position of the interface by coupling it to a parabolic potential. Each block is attracted toward a prescribed position <math>w</math> through a spring of stiffness <math>k_0</math>.

For simplicity, we restrict to the fully connected model, where the distance of block <math>i</math> from its local instability threshold is
<math display="block">
x_i = 1 - (h_{CM} - h_i) - k_0 (w - h_i).
</math>

The first term represents the elastic pull exerted by the rest of the interface, while the second encodes the external driving through the spring.

Here <math>h_{CM}</math> is the center-of-mass position of the interface. Because the sum of all internal elastic forces vanishes, the total external force is balanced solely by the pinning forces. The effective external force can thus be written as a function of <math>w</math>:
<math display="block">
F(w) = k_0 (w - h_{CM}).
</math>

As <math>w</math> is increased quasistatically, the force <math>F(w)</math> would increase if <math>h_{CM}</math> were fixed. When an avalanche takes place, <math>h_{CM}</math> jumps forward and <math>F(w)</math> suddenly decreases. However, in the steady state and in the thermodynamic limit <math>N \to \infty</math>, the force recovers a well-defined value. In the limit <math>k_0 \to 0</math>, this force tends to the critical depinning force <math>F_c</math>; at finite <math>k_0</math> it lies slightly below <math>F_c</math>.

== Quasi-Static Protocol and Avalanche Definition ==

To study avalanches, the position <math>w</math> is increased '''quasi-statically''' so that the block closest to its instability threshold reaches it,
<math display="block">x_i = 0.</math>

This block is the '''epicenter''' of the avalanche: it becomes unstable and jumps to the next well.

When block <math>i</math> jumps by <math>\Delta</math>, both the elastic contribution and the driving spring relax. This gives
<math display="block">
\begin{cases}
x_i = 0 \;\longrightarrow\; x_i = \Delta (1 + k_0), \\[6pt]
x_j \;\longrightarrow\; x_j - \dfrac{\Delta}{N} \quad (j \neq i).
\end{cases}
</math>

The key feature of the quasi-static protocol is that <math>w</math> does not evolve during the avalanche: all subsequent destabilizations are triggered exclusively by previously unstable blocks.

It is convenient to organize the avalanche into generations of unstable sites:

* First generation: the epicenter.
* Second generation: sites destabilized by it.
* Third generation: sites destabilized by generation two.
* And so on.

This hierarchical construction allows us to compute avalanche amplification step by step.

== Derivation of the Evolution Equation ==

Our goal is to determine the distribution <math>P_w(x)</math> of distances to threshold at fixed <math>w</math>.

We shift the parabola by <math>w \to w + \mathrm{d}w</math>. Before the shift:
<math display="block">
x_i(w) = 1 - k_0(w - h_i(w)) - (h_{CM}(w) - h_i(w)).
</math>

We now follow the dynamics generation by generation.

=== First generation ===

During the shift, the center of mass has not yet moved.

* Stable sites (<math>x_i > k_0dw</math>):
<math display="block">
x_i^{t=1} = x_i - k_0dw.
</math>

* Sites with <math>0 < x_i < k_0dw</math> become unstable.

Since <math>dw</math> is infinitesimal, their fraction is
<math display="block">
P_w(0)\,k_0dw.
</math>

They jump and stabilize at
<math display="block">
x_i^{t=1} = \Delta(1+k_0).
</math>

=== Second generation ===

The parabola is now fixed, but the center of mass has advanced:
<math display="block">
h_{CM} \to h_{CM} + \overline{\Delta}\,P_w(0)\,k_0dw.
</math>

Thus all sites shift again toward instability.

* Stable sites:
<math display="block">
x_i^{t=2}
=
x_i
-
\left(1+\overline{\Delta}P_w(0)\right)k_0dw.
</math>

* Newly unstable fraction:
<math display="block">
P_w(0)k_0dw + \left(\overline{\Delta}P_w(0)\right)P_w(0)k_0dw .
</math>

=== Higher generations ===

Iterating produces a geometric amplification:
<math display="block">
1+\overline{\Delta}P_w(0)
+(\overline{\Delta}P_w(0))^2+\dots
=
\frac{1}{1-\overline{\Delta}P_w(0)}.
</math>

The quantity <math>\overline{\Delta}P_w(0)</math> plays the role of a '''branching ratio''': it measures the average number of sites destabilized by one instability.

For stable sites
<math display="block">
x_i(w+dw) = x_i(w) - \frac{k_0}{1-\overline{\Delta}P_w(0)}\,dw.
</math>
while unstable sites are reinjected at a random location <math>\Delta(1+k_0)</math>. The fraction of unstable sites is
<math display="block">
\frac{P_w(0)}{1-\overline{\Delta}P_w(0)}k_0dw
</math>
is
This yields
<math display="block">
\partial_w P_w(x)
=
\frac{k_0}{1-\overline{\Delta}P_w(0)}
\left[
\partial_x P_w(x)
+
\frac{P_w(0)}{1+k_0}
g\!\left(\frac{x}{1+k_0}\right)
\right].
</math>

== Stationary solution ==

At large <math>w</math>:
<math display="block">
0=
\partial_x P_{\text{stat}}(x)
+
\frac{P_{\text{stat}}(0)}{1+k_0}
g\!\left(\frac{x}{1+k_0}\right).
</math>

Solving:
<math display="block">
P_{\text{stat}}(x)
=
\frac{1}{\overline{\Delta}(1+k_0)}
\int_{x/(1+k_0)}^\infty g(z)\,dz.
</math>

=== Critical Force ===

The average distance from the threshold gives a simple relation for the force acting on the system, namely
<math display="block">
F(k_0)= 1- \overline{x} = 1-(1+k_0) \frac{\overline{\Delta^2}}{2 \overline{\Delta}}.
</math>

In the limit <math>k_0\to 0</math> we obtain:
<math display="block">
F_c = F(k_0\to 0)= 1 - \frac{1}{2}\frac{\overline{\Delta^2}}{\overline{\Delta}}.
</math>

== Avalanches ==

We consider an avalanche starting from a single unstable site <math>x_0=0</math>.

Ordering sites by stability:
<math display="block">x_1<x_2<x_3<\dots</math>

From order statistics:
<math display="block">
\int_0^{x_1}P_w(t)dt=\frac1N.
</math>

Thus
<math display="block">
x_n \sim \frac{n}{NP_w(0)}.
</math>

Each instability gives kicks <math>\Delta/N</math>.

Compare mean kick and mean gap:
<math display="block">
\frac{\overline{\Delta}}{N}
\quad \text{vs}\quad
\frac{1}{NP_w(0)}.
</math>

Criticality occurs when
<math display="block">\overline{\Delta}P_w(0)=1.</math>

Using the stationary solution:
<math display="block">\overline{\Delta}P_w(0)=\frac{1}{1+k_0}.</math>

Hence:

* <math>k_0>0</math> → subcritical.
* <math>k_0=0</math> → critical.

== Mapping to a Random Walk ==

Define the random increments
<math display="block">
\eta_1 = \frac{\Delta_1}{N}- x_1,
\quad
\eta_2 = \frac{\Delta_2}{N}- (x_2-x_1),
\quad
\eta_3 = \frac{\Delta_3}{N}- (x_3-x_2),
\ldots
</math>
and the associated random walk
<math display="block">
X_n = \sum_{i=1}^n \eta_i.
</math>

The mean increment is
<math display="block">
\overline{\eta}
=
\frac{\overline{\Delta}}{N}
-
\frac{1}{N P_w(0)}.
</math>

An avalanche remains active as long as
<math display="block">X_n > 0.</math>

The avalanche size <math>S</math> is therefore the first-passage time of the walk to zero.

=== Critical case (k₀ = 0) ===

At criticality,
<math display="block">
\overline{\eta}=0,
\qquad
\overline{\Delta}P_w(0)=1.
</math>

The jump distribution is symmetric and has zero drift. We set <math>X_0=0</math>.

Let
<math display="block">
Q(n)=\text{Prob}\left(X_1>0,\dots,X_n>0\right)
</math>
be the survival probability of the walk.

By the Sparre–Andersen theorem, for large <math>n</math>,
<math display="block">
Q(n)\sim \frac{1}{\sqrt{\pi n}}.
</math>

The avalanche-size distribution is the first-passage probability:
<math display="block">
P(S)=Q(S)-Q(S+1).
</math>

Using the asymptotic form,
<math display="block">
P(S)
\sim
\frac{1}{\sqrt{\pi S}}
-
\frac{1}{\sqrt{\pi (S+1)}}
\sim
\frac{1}{2\sqrt{\pi}}\, S^{-3/2}.
</math>

Thus, at criticality,
<math display="block">
P(S)=\frac{1}{2\sqrt{\pi}}\,S^{-3/2}
\quad (S\gg1).
</math>

The universal exponent is
<math display="block">\tau=\frac{3}{2}.</math>

This power law is of Gutenberg–Richter type.

=== Finite k₀ > 0 (Subcritical case) ===

Using the stationary solution,
<math display="block">
\overline{\Delta}P_{\text{stat}}(0)
=
\frac{1}{1+k_0},
</math>
the mean drift becomes
<math display="block">
\overline{\eta}
=
-
k_0\,\frac{\overline{\Delta}}{N}.
</math>

The random walk is weakly biased toward negative values. For small <math>k_0</math>, the walk is only slightly tilted.

In this case the distribution retains the critical form
<math display="block">
P(S)\sim S^{-3/2}
</math>
up to a cutoff set by the inverse squared drift:
<math display="block">
S_{\max}\sim k_0^{-2}.
</math>