Electron localization and statistical mechanics of a zero-component classical field

Abstract

The formal similarity between the exact analytical expressions for the configuration-averaged propagator for an electron in a random lattice with a Gaussian site energy distribution, and the statistical-mechanical two point correlation function for a zero component classical field with a quartic self-interaction is investigated. This enables one to identify the critical point with the mobility edge and thus to obtain the latter as a function of disorder from the recursion relations of the renormalization group. Two distinct situations corresponding to the random potential being real, or imaginary arise naturally in the treatment. The latter case describes the possibility of random absorption or emission of the particle, e.g., by recombination-generation processes. Explicit calculations for the case of an imaginary potential show that the mobility edge moves inward from the ordered band edge and that an Anderson-like transition occurs when the disorder exceeds a certain value. When the random site potential is real, the mobility edge is not displayed by the treatment, which leads instead to an unphysical fixed point.