Axionic field theory of (3+1)-dimensional Weyl semi-metals

TL;DR

This paper develops an axionic field theory to describe the electromagnetic response of (3+1)-dimensional Weyl semi-metals, revealing a non-quantized coefficient linked to Weyl node separation and exploring boundary effects and thermal responses.

Contribution

It introduces an axionic field theory with a non-quantized coefficient for Weyl semi-metals, connecting electromagnetic response, boundary anomalies, and thermal Hall effects.

Findings

The coefficient is proportional to Weyl node separation.

Boundary gauge invariance is restored by chiral surface states.

Thermal Hall conductivity is established but not from gravitational anomaly.

Abstract

From a direct calculation of the anomalous Hall conductivity and an effective electromagnetic action obtained via Fujikawa's chiral rotation technique, we conclude that an axionic field theory with a non-quantized coefficient describes the electromagnetic response of the $(3+1)$-dimensional Weyl semi-metal. The coefficient is proportional to the momentum space separation of the Weyl nodes. Akin to the Chern-Simons field theory of quantum Hall effect, the axion field theory violates gauge invariance in the presence of the boundary, which is cured by the chiral anomaly of the surface states via the Callan-Harvey mechanism. This provides a unique solution for the radiatively induced CPT-odd term in the electromagnetic polarization tensor of the Lorentz violating spinor electrodynamics, where the source of the Lorentz violation is a constant axial four vector term for the Dirac fermion. A direct linear response calculation also establishes anomalous thermal Hall effect and a Wiedemann-Franz law, but thermal Hall conductivity does not directly follow from the well known formula for the gravitational chiral anomaly.