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:

$H=-t\sum _{<i,j>}(c_i^{†}{c}_j+c_j^{†}{c}_i)\sum _i{ϵ}_i{c}_i^{†}{c}_i$

The single particle hamiltonian in 1d reads

$H=\left[\begin{array}{cccccc}{V}_1& -t& 0& 0& 0& 0\\ -t& V_2& -t& 0& 0& 0\\ 0& -t& V_3& -t& 0& 0\\ 0& 0& -t& \dots & -t& 0\\ 0& 0& 0& -t& \dots & -t\\ 0& 0& 0& 0& -t& V_L\end{array}\right]$

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

$H\psi =ϵ\psi ,\text{with}\sum _n|{\psi }_n{|}^2=1$

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

$\rho (ϵ)=\underset{-\pi }{\overset{\pi }{\int }}\frac{dk}{2\pi }\delta (ϵ-ϵ(k))=\frac{1}{\pi }\frac{1}{\sqrt{4-{ϵ}^2}}forϵ\in (-2,2)$