Absence of Parity Anomaly in Massive Dirac Fermions on a Lattice

TL;DR

This work reexamines the parity anomaly for two-dimensional Dirac fermions on a lattice and shows that a single massive Dirac cone yields an integer Hall conductivity as a multiple of $e^2/h$ when lattice regularization and translational symmetry are properly enforced; the half-quantized value $\sigma_H = \frac{e^2}{2h}$ is absent. A lattice realization with a single massive Dirac cone is achieved by a momentum-dependent mass $m(k)=mv^2-Bk^2$ (or an equivalent Wilson mass) in a lattice tight-binding model, which yields a Chern number $n_c=1$, $-1$, or $0$ and hence $\sigma_H = n_c \frac{e^2}{h}$ for insulating gaps. The half-quantized response can only appear in the massless limit at the Dirac point as $\mu_F\to 0$, giving $\sigma_H = \frac{e^2}{2h}\mathrm{sgn}(B)$, whereas for finite mass the parity anomaly is not realized on the lattice. The work discusses implications for quantum valley Hall effect, gapped surface states of topological insulators, and axion insulators, showing that, due to the TKNN theorem, the total Hall conductance remains quantized as an integer, and the parity anomaly is a lattice-regularized feature of massless Dirac fermions rather than 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.