Reinterpretation of chiral anomaly on a lattice

Chiral anomalies are a quantum mechanical effect for massless Dirac fermions in both particle physics and condensed matter physics. Here we present a set of effective models for single massless Dirac fermions in one and three dimensions in the whole Brillouin zone from higher-dimensional Chern insulators, which uniquely capture both the chiral fermion behavior near the Dirac point and high-energy states at Brillouin zone boundaries. In the presence of electromagnetic fields, the chiral coefficient ${C}_{5}$ is found to be chemical potential dependent in general but quantized precisely at the Fermi surface where chiral symmetry is preserved. This result provides an alternative interpretation of a chiral anomaly on a lattice: The anomaly is caused by symmetry-broken states far below the Fermi surface and protected by the local chiral symmetry. Our analysis here might provide a potential theoretical foundation for applying the concept of chiral anomalies in condensed matter physics.