Quantum interference effects for strongly localized electrons

We examine the role of quantum interference phenomena for strongly localized electrons. The probability distribution for tunneling between two sites separated at a distance t is computed numerically and analytically by summing all forward scattering paths. We find a universal probability distribution that is approximately log normal; its mean proportional to t, its variance growing as ${\mathit{t}}^{2\mathrm{\omega }}$, with ω depending on the dimension d. Since the mean and variance of the distribution are independent, two parameters are necessary to describe the tunneling probability. High moments of the distribution are, however, nonuniversal, and dominated by exceptionally good samples. We also study the response of the system to a magnetic field B, with and without spin-orbit (SO) scattering. Without SO a magnetic field leads to a small (nonuniversal) increase in the localization length ξ scaling as ${\mathit{B}}^{1/2}$. With SO there is still a positive magnetoconductance (initially scaling as ${\mathit{B}}^2$${\mathit{t}}^3$, although there is no change in the localization length). The above results, obtained from extensive numerical simulations, can be analytically explained by a replica analysis of moments. They follow from properties of a bound state between replicas: the attraction factor is related to the symmetries of the underlying Hamiltonian. Our results are compared and contrasted with current literature of the subject of strong localization.