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

Central limit theorem and log-normal distribution

Consider a set of positive iid random variables $x_{1},x_{2},\ldots x_{N}$ with finite mean and variance. Consider the multiplication of random variables

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

In the large N limit, the Central Limit Theorem applies, we write it in the following form:

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

Here, $\chi$ is a Gaussian number of zero mean and unit variance, $\gamma ,\gamma _{2}$ are constant that we can determine. Show that

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

Product of random matrices

Instead of imposing Dirichelet boundary condition on the extremeties of the box we can impose 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 recusion

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

It is useful to intruduce the transfer matrix

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

and the product of random matrices

Failed to parse (unknown function "\codt"): {\displaystyle M_n= T_n \cdot T_{n-1} \codt\ldots T_1 }