Torsional Anomalies and Bulk-Dislocation Correspondence in Weyl Systems

TL;DR

This paper investigates torsional and gauge anomalies in Weyl systems with complex geometries, revealing how these anomalies relate to dislocations, disclinations, and the position of Weyl nodes, with implications for materials like Weyl semimetals.

Contribution

It provides a systematic analysis of anomalies in Weyl systems with torsion and curvature, highlighting the role of Weyl node positions and the insensitivity of certain anomalies to ultra-violet physics.

Findings

Torsional anomalies depend on Weyl node positions in energy-momentum space.

Anomalies originate from particles pumped through Weyl nodes and are UV insensitive.

Zero modes in dislocations lead to current and energy-momentum non-conservation.

Abstract

Based on the supersymmetric quantum mechanical approach, we have systematically studied both the $U\left(1\right)$ gauge anomaly and the diffeomorphism anomaly in Weyl systems with torsion, curvature and external electromagnetic fields. These anomalies relate to the chiral current (or current) non-conservation and chiral energy-momentum (or energy-momentum) non-conservation, respectively, which can be applied to the $^{3}\text{He-A}$ phase, the chiral superconductors and the Weyl semimetals with dislocations and disclinations. In sharp difference with other anomalies, there exist torsional anomalies depending on the position of Weyl nodes in the energy-momentum space. These anomalies originate from particles pumped up through the Weyl nodes and they are thus insensitive to the ultra-violet physics, while the Nieh-Yan anomaly is from the particle inflow through the ultra-violet cut-off. The current non-conservation as well as the energy-momentum non-conservation are found, which stem from the zero modes trapped in the dislocations and they can be understood from the Callan-Harvey mechanism. Finally, by comparing our results with the well-established momentum anomaly in the $^{3}\text{He-A}$ phase, the Nieh-Yan term as well as other cut-off dependent terms are shown to be negligible, because the ratio between the Lorentz symmetry breaking scale and the chemical potential is of order $10^{-5}$.