Slow relaxation and equilibrium dynamics in a two-dimensional Coulomb glass: Demonstration of stretched exponential energy correlations

Abstract

We have simulated energy relaxation and equilibrium dynamics in Coulomb glasses using the random energy lattice model. We show that in a temperature range where the Coulomb gap is already well developed $(T=0.03--0.1)$, the system still relaxes to an equilibrium behavior within the simulation time scale. For all temperatures $T$, the relaxation is slower than exponential. Analyzing the energy correlations of the system at equilibrium $C(\ensuremath{\tau})$, we find a stretched exponential behavior, $C(\ensuremath{\tau})={e}^{\ensuremath{-}{(\ensuremath{\tau}/{\ensuremath{\tau}}_{0})}^{\ensuremath{\gamma}}}$. We study the temperature dependence of ${\ensuremath{\tau}}_{0}$ and $\ensuremath{\gamma}$. ${\ensuremath{\tau}}_{0}$ is shown to increase faster than exponentially with decreasing $T$. $\ensuremath{\gamma}$ is proportional to $T$ at low temperature and approaches unity for high temperature. We define a time ${\ensuremath{\tau}}_{\ensuremath{\gamma}}$ from these stretched exponential correlations and show that this time corresponds well with the time required to reach equilibrium. From our data it is not possible to determine whether ${\ensuremath{\tau}}_{\ensuremath{\gamma}}$ diverges at any finite temperature, indicating a glass transition, or whether this divergence happens at zero temperature. While the time dependence of the system energy can be well fitted by a random walker in a harmonic potential for high temperatures $(T=10)$, this simple model fails to describe the long time scales observed at lower temperatures. Instead we present an interpretation of the configuration space as a structure with fractal properties and the time evolution as a random walk on this fractal-like structure.