A Renormalization-Group approach to the Coulomb Gap

TL;DR

This paper applies a renormalization-group approach using diagrammatic perturbation theory to analyze the Coulomb Gap, deriving the density of states behavior across different dimensions and disorder strengths.

Contribution

It introduces a renormalization-group method for the Coulomb Gap problem, extending analysis to various dimensions and disorder regimes using diagrammatic techniques.

Findings

Derived the standard density of states result in d=1

Extended the method to all disorder levels

Predicted crossover behavior in weak disorder regimes

Abstract

The free energy of the Coulomb Gap problem is expanded as a set of Feynman diagrams, using the standard diagrammatic methods of perturbation theory. The gap in the one-particle density of states due to long-ranged interactions corresponds to a renormalization of the two-point vertex function. By collecting the leading order logarithmic corrections we have derived the standard result for the density of states in the critical dimension, d=1. This method, which is shown to be identical to the approach of Thouless, Anderson and Palmer to spin glasses, allows us to derive the strong-disorder behaviour of the density of states. The use of the renormalization group allows this derivation to be extended to all disorders, and the use of an epsilon-expansion allows the method to be extended to d=2 and d=3. We speculate that the renormalization group equations can also be derived diagrammatically, allowing a simple derivation of the crossover behaviour observed in the case of weak disorder.