Shallow Coulomb gap and weak level-level correlations in a deeply insulating electron system with interactions

We consider a system of two-dimensional electrons strongly localized by disorder. Interactions create a gap in the average tunneling density of states nu (E) at energies E close to the Fermi level. We derive a system of self-consistent equations for the correlators of local densities of occupied and empty states. When either the interactions are screened by the gate or the temperature T is high enough, so that the Coulomb gap is shallow, the perturbative solution of the system yields analytical expressions for the interaction-induced correction delta nu (E) and the level-level correlator. We show that even with short-range interactions, delta nu (E) exhibits a singular energy dependence at T=0. We also demonstrate that at high T this energy dependence is a universal function of the the ratio E/T. Regarding the level correlations, we trace how the correlator falls off as a function of spatial and energy separation between the levels. We also trace how the correlations vanish with increasing T. Our most noticeable observation is that for two close energies the correlator changes sign from positive (attraction) at small distances to negative (repulsion) at large distances.