Mean-field model for electron-glass dynamics

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 $⟨n_i⟩$ and describe the return to the locally stable state by using the eigenvalues of a rate matrix $A$ describing the linearized time evolution of the occupation numbers. By analyzing the probability distribution $P\left(\lambda \right)$ of the eigenvalues of $A$, we find that, under certain physically reasonable assumptions, it takes the form $P\left(\lambda \right)\sim \frac{1}{\mid \lambda \mid }$, 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, as in the case of exponentials of a variable with nearly uniform distribution.

Figure 1

(Color online) Histogram of the site energies in two dimensions, for $N=10000$ and $\mu =0$ (half filling), obtained by solving the self-consistent Eq. (5). The sites were uniformly distributed in a square with $\frac{e^2}{r_{nn}T}=20$, where $r_{nn}$ denotes the average nearest-neighbor distance and $T$ is the temperature. The energies ${ϵ}_i$ were uniformly distributed in the interval $[-\frac{W}{2},\frac{W}{2}]$ with $W=1$. The $y$ axis denotes the probability density of the energies $E_i$. The graph is the average over 300 instances. Notice the finite value of the density at the minimum due to the finite temperature.

Figure 2

(Color online) Demonstration of correlations between the distance and energy difference of two sites chosen at random. The parameters were as for Fig. 1 with $N=1000$. The graph shows the energy difference of all the pairs of sites as a function of their distance. The parameterless fit clearly shows that at small distances, $\delta E\sim \frac{e^2}{r}$.

Figure 3

(Color online) Comparison of the eigenvalue magnitudes of the full linearized matrix (stars) and the ones obtained after neglecting the “Coulomb” term (crosses), the second one in Eq. (8). The difference is mostly seen for the large magnitude eigenvalues, which are not relevant for the behavior at long time scales. $N=1000$, $\frac{\xi }{r_{nn}}=0.1$, and $\frac{e^2}{r_{nn}T}=10$. The energies ${ϵ}_i$ were uniformly distributed in the interval $[-\frac{W}{2},\frac{W}{2}]$ with $\frac{W}{T}=10$. The inset shows the relative error in replacing the full matrix with the approximated one, which is defined as the difference between the approximation and exact diagonalization divided by the exact value.

Figure 4

(Color online) Reciprocal of the distribution of the real part of the eigenvalues, obtained numerically, averaged over 1000 realizations. The parameters were as for Fig. 3. The fit to the reciprocal distribution is linear.

Figure 5

(Color online) Distribution of the eigenvalues in two dimensions for a low density $(\frac{\xi }{r_{nn}}=0.05)$. $N=1000$, and the results were obtained after averaging over 1000 instances. The points were uniformly distributed in the unit square, and the matrix element $A_{ij}=e^{-d_{ij}∕\xi }$ with $\xi =0.0016$. The dashed curve is a plot of Eq. (15), while the solid curve is a plot of Eq. (21).

Figure 6

(Color online) The effect of interactions on the distribution of the eigenvalues of $A$. The graph shows $P\left(\lambda \right)$ for various interaction strengths, each denoted by a different color and mark (dots are the case with strongest interaction, while stars are the weakest). The temperature, disorder, density, and localization length were as for Fig. 3, and $\frac{e^2}{r_{nn}T}$ was varied from 1 to 10. It is observed that $P\left(\lambda \right)$ follow the same $\sim \frac{1}{\lambda }$ dependence with the same prefactor, but as the interaction strength grows, the lower and upper cutoffs are pushed to lower eigenvalues.