Absence of parity anomaly in massive Dirac fermions on a lattice

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.