Chiral anomaly in inhomogeneous systems with nontrivial momentum space topology

TL;DR

This paper extends the understanding of chiral anomalies in 4D systems with inhomogeneous Dirac operators, linking topological invariants in phase space to physical phenomena like Fermi surface stability and conductivity.

Contribution

It introduces a method to compute the chiral anomaly for inhomogeneous Dirac operators using covariant Wigner-Weyl calculus and relates the anomaly to phase space topological invariants.

Findings

The anomaly factorizes into gauge field and phase space topological invariants.

The phase space invariant $N_3$ ensures the topological stability of Fermi points.

The results connect topological invariants to the chiral separation effect conductivity.

Abstract

We consider the chiral anomaly for systems with a wide class of Hermitian Dirac operators ${Q}$ in 4D Euclidean spacetime. We suppose that $ Q$ is not necessarily linear in derivatives and also that it contains a coordinate inhomogeneity unrelated to that of the external gauge field. We use the covariant Wigner-Weyl calculus (in which the Wigner transformed two point Greens function belongs to the two-index tensor representation of the gauge group) and point splitting regularization to calculate the global expression for the anomaly. The Atiyah-Singer theorem can be applied to relate the anomaly to the topological index of $ Q$. We show that the topological index factorizes (under certain assumptions) into the topological invariant $\frac{1}{8\pi^2}\int \text{tr}(F\wedge F)$ (composed of the gauge field strength) multiplied by a topological invariant $N_3$ in phase space. The latter is responsible for the topological stability of Fermi points/Fermi surfaces and is related to the conductivity of the chiral separation effect.