Origin of Bardeen-Zumino current in lattice models of Weyl semimetals

For a generic lattice Hamiltonian of the electron states in Weyl semimetals, we calculate the electric charge and current densities in the first order in background electromagnetic and strain-induced pseudoelectromagnetic fields. We show that the resulting expressions for the densities contain contributions of two types. The contributions of the first type coincide with those in the chiral kinetic theory. The contributions of the second type contain the information about the whole Brillouin zone and cannot be reproduced in the chiral kinetic theory. Remarkably, the latter coincide exactly with the Bardeen-Zumino terms that are usually introduced in relativistic quantum field theory in order to define the consistent anomaly. We demonstrate the topological origin of the Bardeen-Zumino (or, equivalently, Chern-Simons) corrections by expressing them in terms of the winding number in the lattice Hamiltonian model.