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 = [\begin{matrix} 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{matrix}]

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 ψ = ϵ ψ, with \sum_{n} | ψ_{n} |^{2} = 1

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Anderson model (tight bindind model)

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Anderson model (tight bindind model)

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