Massive Dirac fermions and the zero field quantum Hall effect

TL;DR

This paper calculates the quantized Hall conductivity for massive Dirac fermions in 2+1 dimensions, showing it equals half-integer values when certain symmetries are broken, linking it to the zero-field quantum Hall effect.

Contribution

It explicitly demonstrates the half-integer quantization of the filling factor for massive Dirac fermions in QED3 and discusses the role of Chern-Simons terms in the Lagrangian.

Findings

Filling factor is half when time reversal and parity are broken.

Quantized electrical conductivity corresponds to a zero-field quantum Hall effect.

The analysis includes both irreducible and reducible Dirac representations.

Abstract

Through an explicit calculation for a Lagrangian in quantum electrodynamics in (2+1)-space--time dimensions (QED$_3$), making use of the relativistic Kubo formula, we demonstrate that the filling factor accompanying the quantized electrical conductivity for massive Dirac fermions of a single species in two spatial dimensions is a half (in natural units) when time reversal and parity symmetries of the Lagrangian are explicitly broken by the fermion mass term. We then discuss the most general form of the QED$_3$ Lagrangian, both for irreducible and reducible representations of the Dirac matrices in the plane, with emphasis on the appearance of a Chern-Simons term. We also identify the value of the filling factor with a zero field quantum Hall effect (QHE).