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.

Figure 1

Left panel: The energy spectrum of the lattice model (1) given by Eq. (6) for $d_0\ne 0$ (red solid lines) and $d_0=0$ (blue dashed lines) as a function of $k_z$ at $k_x=k_y=0$. Right panel: The energy spectrum (6) for several different values of parameter $t_1$. The complete set of model parameters is given in Appendix pp1.

Figure 2

The dependence of the zero-temperature Hall conductivity ${\sigma }_{12}$ in a Weyl semimetal (red solid line), a double-Weyl semimetal (blue dashed line), and a triple-Weyl semimetal (green dotted line) on the chemical potential. Here ${\epsilon }_0$ is the value of energy (6) at $\mathbf{k}=\mathbf{0}$. The numerical results are calculated for the model parameters in Appendix pp1.

Figure 3

The electric current density $J_z$ in the direction of the pseudomagnetic field as a function of ${\epsilon }_0$ (left panel), $t_1$ (middle panel), and $\mu /{\epsilon }_0$ (right panel). In the left and middle panels, the chemical potential is $\mu =43.45\text{meV}$. In the middle and right panels, we used ${\epsilon }_0=86.9\text{meV}$, which corresponds to the “dome” energy at $k=0$ in the model defined in Appendix pp1.

Figure 4

The dependence of the conductivity tensor component ${\sigma }_{03}^{\left(5\right)}$ on the chemical potential for three difference values of the quasiparticle decay width: ${\mathrm{\Gamma }}_0=0.1{\epsilon }_0$ (red solid line), ${\mathrm{\Gamma }}_0=0.15{\epsilon }_0$ (blue dashed line), and ${\mathrm{\Gamma }}_0=0.2{\epsilon }_0$ (green dotted line), respectively. The numerical values of other parameters are defined in Appendix pp1.