Chiral response in lattice models of Weyl materials

For a generic lattice Hamiltonian of the electron states in Weyl materials, we calculate analytically the chiral (or, equivalently, valley) charge and current densities in the first order in background electromagnetic and strain-induced pseudoelectromagnetic fields. We find that the chiral response induced by the pseudoelectromagnetic fields is not topologically protected. Although our calculations reproduce qualitatively the anomalous chiral Hall effect, the actual result for the conductivity depends on the definition of the chirality as well as on the parameters of the lattice model. In addition, while for the well-separated Fermi surfaces surrounding the individual Weyl nodes the current induced by the magnetic field coincides almost exactly with the current of the chiral separation effect in linearized models, there are clear deviations when the Fermi surfaces undergo the Lifshitz transition. In general, we find that all chiral response coefficients vanish at large chemical potential.

Figure 1

Left: the chiral analog of the Chern number ${\mathcal{C}}_{\chi }\left(k_z\right)$ as a function of $k_z/b_z$. The results are shown for the two definitions of the chirality: $\chi \left(\mathbf{k}\right)={\chi }_1\left(\mathbf{k}\right)$ (red solid line) and $\chi \left(\mathbf{k}\right)={\chi }_2\left(\mathbf{k}\right)$ (dashed blue line). Right: the chiral current density $J_z^5$ measured in units of $J_{z,\text{CSE}}^5$ [see Eq. (15)] as a function of $\mu /{\epsilon }_0$. The results in both panels are plotted for the numerical values of parameters defined in Appendix pp1.

Figure 2

The dependence of the generalized chiral conductivity ${\sigma }_{03}^5$ on $\mu /{\epsilon }_0$. The red solid, blue dashed, and green dotted lines correspond to ${\mathrm{\Gamma }}_0=0.1{\epsilon }_0,0.15{\epsilon }_0$, and $0.2{\epsilon }_0$, respectively. The results are plotted for the model parameters given in Appendix pp1.

Figure 3

The dependence of the relative difference of the chiral charge density $\mathrm{\Delta }{\rho }^5/{\rho }_{\text{BZCS}}^5=({\rho }_{\text{BZCS}}^5-{\rho }_0^5)/{\rho }_{\text{BZCS}}^5$ on ${\epsilon }_0$ for the two definitions of chirality: $\chi \left(\mathbf{k}\right)={\chi }_1\left(\mathbf{k}\right)$ (red solid line) and $\chi \left(\mathbf{k}\right)={\chi }_2\left(\mathbf{k}\right)$ (dashed blue line). The results are plotted for the model parameters given in Appendix pp1 and the vanishing chemical potential $\mu =0$.

Figure 4

The dependence of the diagonal components of the generalized chiral conductivity tensor on the chemical potential: ${\sigma }_{11}^{5,5}$ (left panel), and ${\sigma }_{33}^{5,5}$ (right panel). The red solid, blue dashed, and green dotted lines correspond to ${\mathrm{\Gamma }}_0=0.1{\epsilon }_0,0.15{\epsilon }_0$, and $0.2{\epsilon }_0$, respectively. The results are plotted for the model parameters given in Appendix pp1.

Figure 5

The dependence of the off-diagonal component of the generalized chiral conductivity tensor ${\tilde{\sigma }}_{12}^{5,5}$ on the chemical potential, normalized by ${\sigma }_{5,\text{AHE}}=-e^2{b}_z/\left(6{\pi }^2\right)$. The results are plotted for the model parameters given in Appendix pp1.

Figure 6

The energy spectrum (A9) at $d_0=0$ for several different values of $t_1$ at $k_x=k_y=0$.