Absence of Parity Anomaly in Massive Dirac Fermions on a Lattice

TL;DR

This paper demonstrates that the half-quantized Hall conductivity attributed to the parity anomaly does not occur for massive Dirac fermions on a lattice when proper regularization and symmetry considerations are applied.

Contribution

It clarifies that the parity anomaly is absent in massive Dirac fermions on a lattice, contrasting with previous assumptions in condensed matter physics.

Findings

Massive Dirac fermions on a lattice always yield integer quantized Hall conductivity.

Half-quantized Hall conductivity appears only in the unphysical infinite momentum cutoff limit.

Half-quantized Hall conductivity is associated with massless Dirac cones, not massive ones.

Abstract

The parity anomaly for Dirac fermions in two spatial dimensions has shaped perspectives in quantum field theory and condensed matter physics. In condensed matter it has evolved as a mechanism for half-quantized Hall responses in systems described by massive Dirac fermions. Here we reexamine the issue on a lattice and show that the half-quantized Hall conductivity is absent for massive Dirac fermions when lattice regularization is properly implemented and the translational invariant symmetry is taken into account. We realize that a single massive Dirac cone on a lattice always leads to an integer quantized Hall conductivity and to the half-quantized Hall conductivity only in the unphysical limit of infinite momentum cut-off. The half-quantized Hall conductivity appears with nonzero longitudinal conductance as a signature of a single massless Dirac cone on a lattice. Consequently, the parity anomaly is a property of massless Dirac fermions in a semimetal/metal, not of massive Dirac fermions in an insulator on a lattice.