Dynamics of Anderson localization in disordered wires

TL;DR

This paper derives exact expressions for electron return probabilities in disordered wires, revealing how Anderson localization and topologically protected channels influence electron dynamics over time.

Contribution

It introduces a novel exact mapping between one-dimensional sigma models and zero-dimensional random matrix ensembles for analyzing disordered wire dynamics.

Findings

Return probability approaches a nonzero limit without protected channels.

Presence of protected channels causes return probability to decay as a power law.

The derived formulas apply across different symmetry classes and include topological effects.

Abstract

We consider the dynamics of an electron in an infinite disordered metallic wire. We derive exact expressions for the probability of diffusive return to the starting point in a given time. The result is valid for wires with or without time-reversal symmetry and allows for the possibility of topologically protected conducting channels. In the absence of protected channels, Anderson localization leads to a nonzero limiting value of the return probability at long times, which is approached as a negative power of time with an exponent depending on the symmetry class. When topologically protected channels are present (in a wire of either unitary or symplectic symmetry), the probability of return decays to zero at long time as a power law whose exponent depends on the number of protected channels. Technically, we describe the electron dynamics by the one-dimensional supersymmetric non-linear sigma model. We derive an exact identity that relates any local dynamical correlation function in a disordered wire of unitary, orthogonal, or symplectic symmetry to a certain expectation value in the random matrix ensemble of class AIII, CI, or DIII, respectively. The established exact mapping from one- to zero-dimensional sigma model is very general and can be used to compute any local observable in a disordered wire.