A mean-field model for the electron glass dynamics

TL;DR

This paper introduces a mean-field model for electron glass dynamics, revealing a universal logarithmic decay in return to equilibrium due to broad eigenvalue distributions of the rate matrix.

Contribution

It develops a microscopic mean-field framework linking eigenvalue distributions to glassy relaxation, suggesting universality across different glassy systems.

Findings

Decay is non-exponential and lacks a characteristic time scale.

Eigenvalues of the rate matrix follow a 1/|λ| distribution.

Logarithmic relaxation emerges naturally from the eigenvalue spectrum.

Abstract

We study a microscopic mean-field model for the dynamics of the electron glass, near a local equilibrium state. Phonon-induced tunneling processes are responsible for generating transitions between localized electronic sites, which eventually lead to the thermalization of the system. We find that the decay of an excited state to a locally stable state is far from being exponential in time, and does not have a characteristic time scale. Working in a mean-field approximation, we write rate equations for the average occupation numbers, and describe the return to the locally stable state using the eigenvalues of a rate matrix, A, describing the linearized time-evolution of the occupation numbers. Analyzing the probability distribution of the eigenvalues of A we find that, under certain physically reasonable assumptions, it takes the form $P(\lambda) \sim \frac{1}{|\lambda|}$, leading naturally to a logarithmic decay in time. While our derivation of the matrix A is specific for the chosen model, we expect that other glassy systems, with different microscopic characteristics, will be described by random rate matrices belonging to the same universality class of A. Namely, the rate matrix has elements with a very broad distribution, i.e., exponentials of a variable with nearly uniform distribution.