Wigner transformation, momentum space topology, and anomalous transport

Abstract

Using derivative expansion applied to the Wigner transform of the two - point Green function we analyse the anomalous quantum Hall effect (AQHE), and the chiral magnetic effect (CME). The corresponding currents are proportional to the momentum space topological invariants. We reproduce the conventional expression for the Hall conductivity in $2+1$ D. In $3+1$ D our analysis allows to explain systematically the AQHE in topological insulators and Weyl semimetals. At the same time using this method it may be proved, that the equilibrium CME is absent in the wide class of solids, as well as in the properly regularized relativistic quantum field theory.Comment: Latex, 26 page

Precise Wigner-Weyl calculus for lattice models

“…The meaning of Q(p) for the lattice models of electrons in crystals is the inverse propagator of Bloch electron. Introduction of an external gauge field A(x) defined as a function of coordinates effectively leads to the Peierls substitution (see, for example, [38,78,79]):…”

“…The previous definitions of the Wigner-Weyl formalism (used earlier for the description of the quantum Hall effect), [78,79], given for completeness in Appendix B and Appendix C, are sufficient for the description of the systems in the presence of slowly varying fields. This assumes, in particular the requirement for the real crystals that the magnetic field is much smaller than about 10000 Tesla.…”

“…One has in this approximation In a more general case of the homogeneous system withQ = f (p), where f (p) is the function defined in momentum space with periodic boundary conditions, we have Q C (p, x) = f (p). An alternative version of Wigner-Weyl formalism has been proposed in [78,79], although still giving only approximate results for lattice models.…”

Feynman rules in terms of the Wigner transformed Green functions

“…It satisfies the Groenewold equation Q W (R, p) ⋆ G W (R, p) = 1 [13,38], where Q W is the Weyl symbol of operatorQ,…”

Hall conductivity as topological invariant in phase space

“…In the works of one of the authors of the present paper the Wigner-Weyl formalism has been applied to the study of the nondissipative transport phenomena [47][48][49][50][51][52]. In particular, it was shown that the response of nondissipative currents to the external field strength is expressed through the topological invariants that are robust to the smooth deformations of the system.…”