The $\theta$ vacuum reveals itself as the fundamental theory of the quantum Hall effect II. The Coulomb interaction

TL;DR

This paper explores how the $ heta$ vacuum concept underpins the quantum Hall effect, extending topological quantization to include Coulomb interactions and connecting to variable range hopping theory.

Contribution

It introduces a topological framework that incorporates Coulomb interactions into the quantum Hall effect within a non-linear sigma model.

Findings

Topological arguments extend to Coulomb-interacting systems.

The framework links quantum Hall physics with variable range hopping.

Insights into dynamical scaling in interacting electron gases.

Abstract

Within the Grassmannian $U(2N) / U(N) \times U(N)$ non-linear $\sigma $ model representation of localization one can study the low energy dynamics of both the free and interacting electron gas. We study the cross-over between these two fundamentally different physical problems. We show how the topological arguments for the exact quantization of the Hall conductance are extended to include the Coulomb interaction problem. We discuss dynamical scaling and make contact with the theory of variable range hopping.