L-7

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 $\psi (x,t)$. The quantity $|\psi (x,t){|}^2$ is the probability density of finding the particle at position $x$ at time $t$. The wavefunction therefore satisfies the normalization condition

$$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}dx|\psi (x,t){|}^2=1.$$

The time evolution of the wavefunction is governed by the Schrödinger equation

$$i\hslash {\partial }_t\psi (x,t)=H\psi (x,t),$$

where $H$ is the Hamiltonian of the system. For a particle moving in one dimension in a potential $V(x)$, the Hamiltonian reads

$$H=-\frac{{\hslash }^2}{2m}\underset{x}{\overset{2}{\partial }}+V(x).$$

Eigenstates

A particularly important class of solutions are the eigenstates of the Hamiltonian

$$H{\psi }_n(x)=E_n{\psi }_n(x).$$

If the particle is in an eigenstate the full solution reads

$${\psi }_n(x,t)={\psi }_n(x)e^{-i{E}_{n}t/\hslash }.$$

The probability density $|{\psi }_n(x,t){|}^2$ 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 $E_n$. This typically happens when the particle is confined in a finite region (for instance in a potential well).

The corresponding eigenfunctions are normalizable,

$$\int dx|{\psi }_n(x){|}^2=1,$$

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

$${\psi }_k(x)=\frac{1}{\sqrt{2\pi }}{e}^{ikx},E=\frac{{\hslash }^2{k}^2}{2m}.$$ These states are not normalizable in the usual sense. They are instead normalized using Dirac delta functions,

$$\int dx{\psi }_k^{*}(x){\psi }_{k^{\prime }}(x)=\delta (k-k^{\prime }).$$

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

$$\psi (x,t)=\sum _n{c}_n{e}^{-i{E}_{n}t/\hslash }{\psi }_n(x).$$

If the spectrum is continuous the expansion becomes an integral

$$\psi (x,t)=\int dkc(k){\psi }_k(x,t),\int dk|c(k){|}^2=1.$$

By choosing the coefficients $c(k)$ 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

$$J(x,t)=\frac{\hslash }{2mi}({\psi }^{*}\frac{d\psi }{dx}-\psi \frac{d{\psi }^{*}}{dx}).$$

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

$\rho (x,t)=|\psi (x,t){|}^2$

satisfies the continuity equation

$${\partial }_t\rho (x,t)+{\partial }_{x}J(x,t)=0,$$

which has the same structure as a conservation law in hydrodynamics.

For a plane wave

$$\psi (x)=e^{ikx}$$

one finds

$$J=\frac{\hslash k}{m}.$$

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 $J$ 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

$${\psi }_{k,L}(x)=\{\begin{array}{ll}{e}^{ikx}+r{e}^{-ikx}& x\to -\mathrm{\infty }\\ t{e}^{ikx}& x\to +\mathrm{\infty }\end{array}$$

The first term represents the incoming wave, the second the reflected wave, while the transmitted wave propagates to the right with amplitude $t$.

Using the expression of the probability current, one finds that a plane wave $e^{ikx}$ carries a current

$$J=\frac{\hslash k}{m}.$$

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

$$R+T=1,$$

where

$$R=|r{|}^2,T=|t{|}^2$$

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

$$H=-\frac{{\hslash }^2}{2m}\frac{d^2}{d{x}^2}.$$

As discussed above, the eigenstates of this Hamiltonian are plane waves with a continuous spectrum

$${\psi }_k(x,t)=\frac{1}{\sqrt{2\pi }}{e}^{ikx}{e}^{-i{E}_{k}t/\hslash },E_k=\frac{{\hslash }^2{k}^2}{2m}.$$

Since plane waves are completely delocalized, a physical particle must be described by a superposition of eigenstates, forming a wave packet

$$\psi (x,t)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}dkc(k){\psi }_k(x,t),\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}dk|c(k){|}^2=1.$$

Evolution of a Gaussian wave packet

• Initial state

At time $t=0$ consider a Gaussian wave packet

$$\psi (x,0)=\frac{e^{-x^2/(4a^2)}}{(2\pi a^2{)}^{1/4}},|\psi (x,0){|}^2=\frac{e^{-x^2/(2a^2)}}{\sqrt{2\pi a^2}}.$$

Show that the coefficients of the plane-wave decomposition are

$$c(k)={\left(\frac{2a^2}{\pi }\right)}^{1/4}{e}^{-a^2{k}^2}.$$

• Time evolution

Define the spreading velocity

$$v_s=\frac{\hslash }{2m{a}^2}.$$

Show that the time evolution of the packet is

$$\psi (x,t)=\frac{e^{-x^2/(4a^2(1+i{v}_{s}t))}}{[2\pi a^2(1+i{v}_{s}t){]}^{1/4}}.$$

• Ballistic spreading

The probability density becomes

$$|\psi (x,t){|}^2=\frac{e^{-x^2/(2a^2(1+v_s^2{t}^2))}}{\sqrt{2\pi a^2(1+v_s^2{t}^2)}}.$$

Hence

$$\sqrt{⟨x^2⟩}={(\int dx{x}^2|\psi (x,t){|}^2)}^{1/2}=a\sqrt{1+v_s^2{t}^2}.$$

At long times

$$\sqrt{⟨x^2⟩}\sim v_{s}t.$$

This linear growth is called ballistic spreading.

It should be contrasted with two other possible transport regimes:

• Diffusive motion

$\sqrt{⟨x^2⟩}\sim \sqrt{t}$

• Localized regime

$\sqrt{⟨x^2⟩}$ 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

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.

Semilog plot of the particle density. Billy et al., Nature (2008).

In a disordered potential an eigenstate of energy $E_k$ has the form

$${\psi }_k(x,t)={\psi }_k(x)e^{-i{E}_{k}t/\hslash }.$$

The spatial part of the wavefunction is localized around some position $\overline{x}$ and decays exponentially

$${\psi }_k(x)\sim e^{-|x-\overline{x}|/{\xi }_{\text{loc}}}.$$

Here ${\xi }_{\text{loc}}$ 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 $\tau$, called the mean free time.

Under the action of an electric field $E$, an electron of charge $-e$ obeys

$$m\frac{dv}{dt}=-eE.$$

Between two collisions the electron accelerates, but collisions randomize its velocity. Averaging over many such events leads to a stationary drift velocity

$$v_d=-\frac{eE\tau }{m}.$$

If the electron density is $n$, the electric current density is

$$j=-ne{v}_d.$$

Substituting the drift velocity gives

$$j=\frac{n{e}^2\tau }{m}E.$$

This is the microscopic origin of Ohm's law

$$j=\sigma E,\sigma =\frac{n{e}^2\tau }{m}.$$

To relate this to the conductance of a sample, consider a wire of length $L$ and cross section $S$. The total current is

$$I=jS,$$

while the voltage drop is

$$V=EL.$$

Therefore

$$I=\sigma \frac{S}{L}V.$$

Using $G=I/V$, one obtains

$$G=\sigma \frac{S}{L}.$$

Equivalently,

$$R=\frac{1}{G}=\rho \frac{L}{S},\rho =\frac{1}{\sigma }.$$

For a sample of linear size $L$ in spatial dimension $d$, the cross section scales as $S\sim L^{d-1}$. Hence

$$G(L)\sim \sigma L^{d-2}.$$

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

$$G=\frac{e^2}{\hslash }|t(E_F){|}^2.$$

The factor $e^2/\hslash$ 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

$$|t(E_F){|}^2\sim e^{-2L/{\xi }_{\text{loc}}},$$

which leads to

$$G(L)\sim e^{-2L/{\xi }_{\text{loc}}}.$$

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

$$g=\frac{G\hslash }{e^2}.$$

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 $L$ is increased.

This evolution is described by a renormalization group equation

$$\frac{d\ln g}{d\ln L}=\beta (g).$$

The function $\beta (g)$ depends only on $g$ and on the spatial dimension.

Two limits are known:

• Metallic regime ($g\gg 1$)

Transport is diffusive. Since

$$G\sim L^{d-2},$$

we obtain

$$\beta (g)\to d-2.$$

• Localized regime ($g\ll 1$)

The conductance decays exponentially

$$g\sim e^{-L/{\xi }_{\text{loc}}},$$

which implies

$$\beta (g)\sim \ln g.$$

The simplest scenario is that the beta function is monotonic.

This leads to a striking dimensional prediction:

• for $d>2$ the beta function changes sign → a metal–insulator transition exists
• for $d\le 2$ 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.