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 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 it is simple to 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 for\epsilon \in (-2,2)

In presence of disorder the DOS becomes larger, and display sample to sample fluctuations. One can consider the 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 IPR