Quantized Hall conductivity of Bloch electrons: topology and the Dirac fermion

TL;DR

This paper investigates the quantized Hall conductivity of two-dimensional Bloch electrons under rational magnetic flux, revealing that changes in conductivity at gap closures are governed by Dirac fermion physics and that multiple band gaps close simultaneously.

Contribution

It challenges the naive Dirac fermion interpretation for individual bands and shows that the Hall conductivity change occurs at multiple gap closings, with a q-fold degeneracy in the dispersion relation.

Findings

Hall conductivity change at gap closing is described by Dirac fermion theory.

Multiple band gaps close simultaneously at q points.

Dispersion relation exhibits q-fold degeneracy in the magnetic Brillouin zone.

Abstract

We consider the Hall conductivity of two-dimensional non-interacting Bloch electrons when the magnetic flux per unit cell is a rational number $p/q$ where $p$ and $q$ are mutually coprime. We present a counter-example for the naive expectation that the Hall conductivity carried by a band is given by treating gap minima as Dirac fermions. Instead of the above expectation, we show that the {\em change\/} of the Hall conductivity at a gap-closing phenomenon is given by the Dirac fermion argument. Comparing with the Diophantine equation, our result implies that a band-gap closes at $q$ points simultaneously. Furthermore, we show that the dispersion relation is $q$-fold degenerate in the magnetic Brillouin zone.