We use a local mean-field (Hartree-like) approach to study the conductance of a strongly localized electron system in two dimensions. We find a crossover between a regime where Coulomb interactions modify the conductance significantly to a regime where they are negligible. We show that under rather general conditions the conduction obeys a scaling relation which we verify using numerical simulations. The use of a local mean-field approach gives a clear physical picture, and removes the ambiguity of the use of single-particle tunneling density of states (DOS) in the calculation of the conductance. Furthermore, the theory contains interaction-induced correlations between the on site energy of the localized states and distances, as well as finite temperature corrections of the DOS.
We study the dynamics of a simple model for quantum decay, where a single state is coupled to a set of discrete states, the pseudocontinuum, each coupled to a real continuum of states. We find that for constant matrix elements between the single state and the pseudocontinuum the decay occurs via one state in a certain region of the parameters, involving the Dicke and quantum Zeno effects. When the matrix elements are random, several cases are identified. For a pseudocontinuum with small bandwidth there are weakly damped oscillations in the probability to be in the initial single state. For intermediate bandwidth one finds mesoscopic fluctuations in the probability with amplitude inversely proportional to the square root of the volume of the pseudocontinuum space. They last for a long time compared to the nonrandom case.
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 ⟨ni⟩ 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(λ) of the eigenvalues of A, we find that, under certain physically reasonable assumptions, it takes the form P(λ)∼1/∣λ∣, 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.