On the Mott formula for the a.c. conductivity and binary correlators in the strong localization regime of disordered systems

TL;DR

This paper develops a method to analyze disordered systems in the strong localization regime, deriving the Mott formula for low frequency conductivity and asymptotic correlators based on deep random wells.

Contribution

It introduces a novel approach assuming deep random wells with small density to derive key characteristics in strong localization, linking the density of states to the well density.

Findings

Derived the Mott formula for low frequency conductivity.

Obtained asymptotic formulas for two-point correlators.

Showed the density of states matches the density of wells in leading order.

Abstract

We present a method that allows us to find asymptotic form of various characteristics of disordered systems in the strong localization regime, i.e., when either the random potential is big enough or the energy is close enough to the spectrum edges. The method is based on the hypothesis that relevant realizations of the random potential in the strong localization regime have the form of deep random wells that are uniformly and chaotically distributed in the space with a sufficiently small density. Assuming this and using the density expansion, we show first that the density of wells coincides in the leading order with the density of states. Thus the density of states is in fact the small parameter of the theory in the strong localization regime. Then we derive the Mott formula for the low frequency conductivity and the asymptotic formulas for certain two-point correlators when the difference of respective energies is small.