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 binding model)

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

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

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

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

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.

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

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

Quenched and Annealed averages

To compute the moments of the log-normal distribution, it is convenient to introduce the variable

$X\equiv \ln(\Pi _{N})$

which is Gaussian distributed:

$p(X)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\exp \left[-{\frac {(X-\mu )^{2}}{2\sigma ^{2}}}\right]$

with $\mu =\gamma N$ and $\sigma ^{2}=\gamma _{2}^{2}N$.

The moments of $\Pi _{N}$ can be easily computed:

${\overline {\Pi _{N}^{n}}}=\int dX\,e^{nX}p(X)=\exp \left[\mu n+\sigma ^{2}{\frac {n^{2}}{2}}\right]=\exp \left[(\gamma n+\gamma _{2}^{2}{\frac {n^{2}}{2}})N\right]$

The variable $\Pi _{N}$ is therefore not self-averaging (see Valentina's lecture 1) since its fluctuations grow with $N$ faster than its mean:

${\frac {\overline {\Pi _{N}^{2}}}{({\overline {\Pi _{N}}})^{2}}}=\exp \left[\gamma _{2}^{2}N\right]$

Hence, $\Pi _{N}$ is not self-averaging, while $\ln \Pi _{N}$ is self-averaging.

In particular, the mean ${\overline {\Pi _{N}}}=\exp[(\gamma +\gamma _{2}^{2}/2)N]$ grows much faster than the typical value $\Pi _{N}^{\text{typ}}\equiv \exp(\gamma N)$.

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 and their product

$T_{n}={\begin{bmatrix}V_{n}-\epsilon &-1\\1&0\end{bmatrix}},\quad \Pi _{n}=T_{n}\cdot T_{n-1}\cdot \ldots T_{1}$

The Schrodinger equation can be written as

${\begin{bmatrix}\psi _{n+1}\\\psi _{n}\end{bmatrix}}=\Pi _{n}{\begin{bmatrix}\psi _{1}\\\psi _{0}\end{bmatrix}}={\begin{bmatrix}\pi _{11}&\pi _{12}\\\pi _{21}&\pi _{22}\end{bmatrix}}{\begin{bmatrix}\psi _{1}\\\psi _{0}\end{bmatrix}}$

Fustenberg Theorem

We define the norm of a 2x2 matrix:

$\|\Pi _{n}\|^{2}={\frac {\pi _{11}^{2}+\pi _{21}^{2}+\pi _{12}^{2}+\pi _{22}^{2}}{2}}$

For large N, the Fustenberg theorem ensures the existence of the non-negative Lyapunov exponent, namely

$\lim _{n\to \infty }{\frac {\ln \|\Pi _{n}\|}{n}}=\gamma \geq 0$

In absence of disorder $\gamma =0$ for $\epsilon \in (-2,2)$. Generically the Lyapunov is positive, $\gamma >0$, and depends on $\epsilon$ and on the distribution of $V_{i}$.

Consequences

Localization length

Together with the norm, also $|\psi _{n}|^{2}$ grows exponentially with n. We can write

$\ln |\psi _{n}|\sim \gamma n+\gamma _{2}\chi {\sqrt {n}}$

which means that $\ln |\psi _{n}|$ is performing a random walk with a drift.

However, our initial goal is a properly normalized eigenstate at energy $\epsilon$. 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

$\xi _{\text{loc}}(\epsilon )\equiv \gamma ^{-1}(\epsilon )$

Fluctuations

We expect strong fluctuations on quantites like $|\psi _{n}|,\|\Pi _{n}\|,G,\ldots$, while their logarithm is self averaging.