A nearly exact method of solving certain localisation problems

Abstract

The old theories of localisation of Anderson (1958) and Abou-Chacra et al (1973) are reexamined. It is argued that (a) the convergence properties of the renormalised perturbation series for the self-energy are predominantly governed by its first term; and (b) the localisation problem in a real lattice can be mapped on to the localisation problem in a Cayley tree lattice in which the non-contributing branches are trimmed off. The connectivity constant for the trimmed Cayley tree, which can be evaluated exactly, should be used in the Abou-Chacra et al method (1973-1974) to obtain results for a real lattice. Calculations for two-dimensional lattices show partial agreement with the well known result that all states should be localised at any disorder-the triangular lattice (coordination number C=6) appears to show complete localisation only above a critical value of disorder, the honeycomb lattice (C=3) shows complete localisation always, and the square lattice (C=4) is found to be the marginal case.