Chiral anomaly in inhomogeneous systems with nontrivial momentum space topology
Praveen D. Xavier, M. A. Zubkov
TL;DR
The paper extends the chiral anomaly analysis to systems governed by a broad class of Hermitian Dirac operators $Q$ in $D=4$ with coordinate inhomogeneity, using the covariant Wigner-Weyl calculus and point-splitting regularization. It shows that the global anomaly factorizes into the product of the external gauge-field density $\propto \int \mathrm{tr}(F \wedge F)$ and a momentum-space topological invariant $N_3$, yielding $\mathscr A = -2 i N_3 \frac{1}{16\pi^2} \int d^4x \mathrm{tr}(F F^\star)$; $N_3$ is defined by a 3-form integral over a momentum-space surface and is robust under smooth deformations of $Q_W$. An illustrative example demonstrates $N_3=n$, giving $\mathscr A = -n \frac{i}{4\pi^2} \int \mathrm{tr}(F \wedge F)$, generalizing the standard Dirac result ($n=1$). The work highlights a topological factorization linking phase-space topology to gauge-field topology, with implications for Fermi-surface stability and the chiral separation effect, and suggests extensions to interactions and non-Abelian gauge fields in condensed-matter analogs and beyond.
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 $\frac{1}{8π^2}\int \text{tr}(F\wedge F)$ (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.