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: In 1d the dispersion relation is $\epsilon (k)=-2\cos _{k},k\in (-\pi ,\pi ),-2<\epsilon (k)<2$ and the density of states is

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