QED$_{2+1}$ with Nonzero Fermion Density and Quantum Hall Effect

TL;DR

This paper derives a general expression for conductivity in 2+1 dimensional QED with nonzero fermion density, showing it depends on the Chern-Simons coefficient and exhibits step-function behavior with respect to chemical potential and magnetic field.

Contribution

It provides a new analytical formula linking conductivity to the Chern-Simons coefficient in QED$_{2+1}$ under magnetic fields and finite fermion density.

Findings

Conductivity is determined solely by the Chern-Simons coefficient.

Conductivity exhibits step-function dependence on chemical potential and magnetic field.

The derived expression applies to uniform magnetic fields in QED$_{2+1}$.

Abstract

A general expression for the conductivity in the QED$_{2+1}$ with nonzero fermion density in the uniform magnetic field is derived. It is shown that the conductivity is entirely determined by the Chern-Simons coefficient: $\sigma_{ij}=\varepsilon_{ij}~{\cal C}$ and is a step-function of the chemical potential and the magnetic field.