Chiral anomaly in inhomogeneous systems with nontrivial momentum space topology

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.