Anderson localisation from a generalised master equation

The authors consider here a tight-binding model for the motion of a single electron and an energetically disordered lattice. They show that the Anderson transition in this model can be studied using a generalised master equation with a nearest-neighbour memory function. It is first demonstrated that because the generalised master equation with a nearest-neighbour memory function is isomorphic to a classical bond-percolation problem, an exact expansion can be constructed for the diffusion constant using the bond flux method of Kundu, Parris and Phillips (1987). The authors show that at the effective medium level, the diffusion constant vanishes for any non-zero value of the disorder. They then calculate the probability distribution of the self-energy for the bond flux Green function on a Cayley tree of connectivity K. The authors approach predicts an Anderson transition at W/V=13 for K=2 and W/V=20 for K=3. W is the width of the distribution for the site energies and V, the nearest-neighbour matrix element. These results are in good agreement with the exact values of W/V=17 K (K=2) and W/V=29 (K=3). Further applications of the generalised master equation to disordered systems and the prospect of constructing the exact memory function for Anderson localisation are discussed.