Wigner transformation, Momentum space topology, and anomalous transport: anomalous quantum Hall effect, (the absence of) chiral magnetic effect, chiral separation effect

We use derivative expansion applied to the Wigner transform of the two-point Green function in the lattice models of solid state physics and in the lattice regularized relativistic quantum field theory. This approach allows to analyze the anomalous quantum Hall effect (AQHE), the chiral magnetic effect (CME), and the chiral separation effect (CSE) taking into account properly the ultraviolet effects. We find for the first time that the corresponding currents are proportional to the momentum space topological invariants. We reproduce the conventional expression for the Hall conductivity in the ideal tight – binding models of $(2+1) D$ topological insulators and in the $3+1 D$ Weyl semimetals. At the same time we predict the appearance of the AQHE in $3+1 D$ topological insulators of certain types. Using the same method we prove that in the equilibrium $(3+1) D$ theory the CME is absent both is solids and in the properly regularized relativistic quantum field theory. We demonstrate that the chiral separation effect appears in the continuum limit of the lattice theory with the current given by its conventional expression.

References

1. M. A. Zubkov, “Absence of equilibrium chiral magnetic effect”, Phys. Rev. D, 93:10 (2016), 105036, arXiv: 1605.08724      
2. M. A. Zubkov, “Wigner transformation, momentum space topology, and anomalous transport”, Annals Phys., 373 (2016), 298–324, arXiv: 1603.03665      
3. Z. V. Khaidukov, M. A. Zubkov, “Chiral Separation effect in lattice regularization”, Phys. Rev. D, 95 (2017), 074502