L-8

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 bindind model)

We consider disordered non-interacting particles hopping between nearest neighbors sites on a lattice. The hamiltonian reads:

$H=-t\sum _{<i,j>}(c_{i}^{\dagger }c_{j}+c_{j}^{\dagger }c_{i})\sum _{i}\epsilon _{i}c_{i}^{\dagger }c_{i}$

The single particle hamiltonian in 1d reads

$H={\begin{bmatrix}V_{1}&-t&0&0&0&0\\-t&V_{2}&-t&0&0&0\\0&-t&V_{3}&-t&0&0\\0&0&-t&\ldots &-t&0\\0&0&0&-t&\ldots &-t\\0&0&0&0&-t&V_{L}\end{bmatrix}}$

For simplicity we set the hopping $t=1$. The disorder are iid random variables drawn, uniformly from the box $(-{\frac {W}{2}},{\frac {W}{2}})$.

The final goal is to study the statistical properties of eigensystem

$H\psi =\epsilon \psi ,\quad {\text{with}}\sum _{n}|\psi _{n}|^{2}=1$

Density of states (DOS)

In 1d and in absence of disorder, the dispersion relation is $\epsilon (k)=-2\cos k,\quad k\in (-\pi ,\pi ),-2<\epsilon (k)<2$. From the dispersion relation, we compute the density of states (DOS) :

$\rho (\epsilon )=\int _{-\pi }^{\pi }{\frac {dk}{2\pi }}\delta (\epsilon -\epsilon (k))={\frac {1}{\pi }}{\frac {1}{\sqrt {4-\epsilon ^{2}}}}\quad {\text{for }}\epsilon \in (-2,2)$

In presence of disorder the DOS becomes larger, and display sample to sample fluctuations. One can consider its mean value, avergaed over disorder realization.

Eigenstates

In absence of disorder the eigenstates are plane waves delocalized along all the system. In presence of disorder, three situations can occur and to distinguish them it is useful to introduce the inverse participation ratio, IPR

$IPR(q)=\sum _{n}|\psi _{n}|^{2q}\sim L^{-\tau _{q}}$

The normalization imposes $\tau _{1}=0$ and $\tau _{0}=-d$.

• Delocalized eigenstates In this case, $|\psi _{n}|^{2}\approx L^{-d}$. Hence, we expect

$IPR(q)=L^{d}(1-q)\quad \tau _{q}=d(1-q)$

• Localized eigenstates In this case, $|\psi _{n}|^{2}\approx 1/\xi _{\text{loc}}^{1/d}$ for $\xi _{\text{loc}}^{d}$ sites and zero elsewhere. Hence, we expect

$IPR(q)=O(1)\quad \tau _{q}=0$

• Multifractal eigenstates At the transition, namely at the mobility edge, an anomalous scaling is observed elsewhere. Hence, we expect

$IPR(q)=L^{d}(1-q)\quad \tau _{q}=D_{q}(1-q)$

Here $D_{q}$ is q-dependent fractal dimension, smaller than $d$.

Transfer matrices and Lyapunov exponents

Product of random variables and Central limit theorem

Consider a set of positive iid random variables $x_{1},x_{2},\ldots x_{N}$ with finite mean and variance and compute their product

$\Pi _{N}=\prod _{n=1}^{N}x_{i},\quad {\text{or}}\;\ln \Pi _{N}=\sum _{n=1}^{N}\ln x_{i}$

For large N, the Central Limit Theorem predicts:

$\log \Pi _{N}=\gamma N+\gamma _{2}{\sqrt {N}}\chi$

• $\chi$ is a Gaussian number of zero mean and unit variance
• $\gamma ,\gamma _{2}$ are N indepent and can be written as

$\gamma ={\overline {\ln x}}<\ln {\overline {x}},\quad \gamma _{2}={\sqrt {{\overline {(\ln x)^{2}}}-({\overline {\ln x}})^{2}}}$

Log-normal distribution

The distribution of $\Pi _{N}$ is log-normal

$P(\Pi _{N})={\frac {1}{\gamma _{2}^{2}{\sqrt {2\pi N}}\Pi _{N}}}\exp \left[-{\frac {(\ln(\Pi _{N})-\gamma N)^{2}}{2\gamma _{2}^{2}N}}\right]$

Quenched and Annealed averages

For the log-normal distribution the mean ${\overline {\Pi _{N}}}=\exp[(\gamma -\gamma _{2}^{2})N]$ is larger than the median value $\Pi _{N}^{\text{median}}=\exp(\gamma N)$ (which is larger than the mode). Hence, $\Pi _{N}$ is not self averaging, while $\ln \Pi _{N}$ is self averaging. This is the reason why in the following we will take quenched averages.

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

${\begin{bmatrix}\psi _{n+1}\\\psi _{n}\end{bmatrix}}={\begin{bmatrix}V_{n}-\epsilon &-1\\1&0\end{bmatrix}}{\begin{bmatrix}\psi _{n}\\\psi _{n-1}\end{bmatrix}}$

We can continue the recursion

${\begin{bmatrix}\psi _{n+1}\\\psi _{n}\end{bmatrix}}={\begin{bmatrix}V_{n}-\epsilon &-1\\1&0\end{bmatrix}}{\begin{bmatrix}V_{n-1}-\epsilon &-1\\1&0\end{bmatrix}}{\begin{bmatrix}\psi _{n-1}\\\psi _{n-2}\end{bmatrix}}$

It is useful to introduce the transfer matrix

$T_{n}={\begin{bmatrix}V_{n}-\epsilon &-1\\1&0\end{bmatrix}}$

and the product of random matrices

$M_{n}=T_{n}\cdot T_{n-1}\cdot \ldots T_{1}$