1d Anderson model

Numerical experiments » 1d Anderson model

The one-dimensional Anderson model is a one-dimensional lattice with on-site random energy and hopping only to the nearest neigbor. The Hamiltonian is given by:

H = sum_{n=-infty}^{+infty} {w_n |nranglelangle n| + (t |nrangle langle n+1| + H.c.)

where w_n are independent random numbers uniformly distributed in [-W/2,W/2]. It is customary to take the hopping amplitude t as unit of energy, i.e. t=1.

The time-independent Schroedinger equation for the wavefunction psi at energy E is thus:

psi_{n-1} + psi_{n+1} + (w_n-E)psi_n = 0

 This equation can be simply recast as a linear relation between the two-components vectors (psi_{n+1},psi_n) and (psi_n,psi_{n-1}) (see 3rd displayed equation on the left side), allowing to use the usual transfer matrix method for 1d systems.

Here, we will use three different tools to test localization properties of the 1d Anderson model:

  1. Exact diagonalization of the Hamiltonian for a finite lattice.
  2. Computation of the localization length using a variant of the transfer matrix method.
  3. Time propagation of an initially localized wavepacket.