Anomalous thermoelectric phenomena in lattice models of multi-Weyl semimetals

The thermoelectric transport coefficients are calculated in a generic lattice model of multi-Weyl semimetals with a broken time-reversal symmetry by using the Kubo's linear response theory. The contributions connected with the Berry curvature-induced electromagnetic orbital and heat magnetizations are systematically taken into account. It is shown that the thermoelectric transport is profoundly affected by the nontrivial topology of multi-Weyl semimetals. In particular, the calculation reveals a number of thermal coefficients of the topological origin which describe the anomalous Nernst and thermal Hall effects in the absence of background magnetic fields. Similarly to the anomalous Hall effect, all anomalous thermoelectric coefficients are proportional to the integer topological charge of the Weyl nodes. The dependence of the thermoelectric coefficients on the chemical potential and temperature is also studied.

Figure 1

The low-energy part of the quasiparticle spectrum in the lattice model (1) describing multi-Weyl semimetals with the topological charges of Weyl nodes (a) $n=1$, (b) $n=2$, and (c) $n=3$. For simplicity, we set $d_0=0$ and plotted the energy as a function of $k_x$ and $k_z$ at fixed $k_y=0$. We also used a characteristic energy scale set by the size of the “dome” between the Weyl nodes ${\epsilon }_0\equiv {\left|\mathbf{d}\right|}_{\mathbf{k}=\mathbf{0}}$. Black points label the positions of the Weyl nodes. The complete set of model parameters is given in Appendix pp1.

Figure 2

The dependence of the transport coefficients $L_{xx}^{11},L_{zz}^{11}$, and $L_{xy}^{11}$ on the chemical potential in a Weyl semimetal (red solid line), a double-Weyl semimetal (blue dashed line), and a triple-Weyl semimetal (green dotted line) at fixed $T=0.1{\epsilon }_0$. In panels (a) and (b), the quasiparticle transport width is modeled by $\mathrm{\Gamma }={\mathrm{\Gamma }}_0\left(1+{\omega }^2/{\epsilon }_0^2\right)$ with ${\mathrm{\Gamma }}_0=0.1{\epsilon }_0$. In panel (c), the results are plotted in the clean limit, $\mathrm{\Gamma }=0$. The numerical values of other model parameters are defined in Appendix pp1.

Figure 3

The dependence of the transport coefficients $L_{xx}^{11},L_{zz}^{11}$, and $L_{xy}^{11}$ on the temperature in a Weyl semimetal (red solid line), a double-Weyl semimetal (blue dashed line), and a triple-Weyl semimetal (green dotted line) at fixed $\mu =0.1{\epsilon }_0$. In panels (a) and (b), the quasiparticle transport width is modeled by $\mathrm{\Gamma }={\mathrm{\Gamma }}_0\left(1+{\omega }^2/{\epsilon }_0^2\right)$ with ${\mathrm{\Gamma }}_0=0.1{\epsilon }_0$. In panel (c), the results are plotted in the clean limit, $\mathrm{\Gamma }=0$. The numerical values of other model parameters are defined in Appendix pp1.

Figure 4

The dependence of the transport coefficients $L_{xx}^{21},L_{zz}^{21}$, and $L_{xy}^{21}$ on the chemical potential in a Weyl semimetal (red solid line), a double-Weyl semimetal (blue dashed line), and a triple-Weyl semimetal (green dotted line) at fixed $T=0.1{\epsilon }_0$. In panels (a) and (b), the quasiparticle transport width is modeled by $\mathrm{\Gamma }={\mathrm{\Gamma }}_0\left(1+{\omega }^2/{\epsilon }_0^2\right)$ with ${\mathrm{\Gamma }}_0=0.1{\epsilon }_0$. In panel (c), the results are plotted in the clean limit, $\mathrm{\Gamma }=0$. The numerical values of other model parameters are defined in Appendix pp1.

Figure 5

The dependence of the transport coefficients $L_{xx}^{21},L_{zz}^{21}$, and $L_{xy}^{21}$ on temperature in a Weyl semimetal (red solid line), a double-Weyl semimetal (blue dashed line), and a triple-Weyl semimetal (green dotted line) at fixed $\mu =0.1{\epsilon }_0$. In panels (a) and (b), the quasiparticle transport width is modeled by $\mathrm{\Gamma }={\mathrm{\Gamma }}_0\left(1+{\omega }^2/{\epsilon }_0^2\right)$ with ${\mathrm{\Gamma }}_0=0.1{\epsilon }_0$. In panel (c), the results are plotted in the clean limit, $\mathrm{\Gamma }=0$. The numerical values of other model parameters are defined in Appendix pp1.

Figure 6

The dependence of the transport coefficients $L_{xx}^{22},L_{zz}^{22}$, and $L_{xy}^{22}$ on the chemical potential in a Weyl semimetal (red solid line), a double-Weyl semimetal (blue dashed line), and a triple-Weyl semimetal (green dotted line) at fixed $T=0.1{\epsilon }_0$. In panels (a) and (b), the quasiparticle transport width is modeled by $\mathrm{\Gamma }={\mathrm{\Gamma }}_0\left(1+{\omega }^2/{\epsilon }_0^2\right)$ with ${\mathrm{\Gamma }}_0=0.1{\epsilon }_0$. In panel (c), the results are plotted in the clean limit, $\mathrm{\Gamma }=0$. The numerical values of other model parameters are defined in Appendix pp1.

Figure 7

The dependence of the transport coefficients $L_{xx}^{22},L_{zz}^{22}$, and $L_{xy}^{22}$ on temperature in a Weyl semimetal (red solid line), a double-Weyl semimetal (blue dashed line), and a triple-Weyl semimetal (green dotted line) at fixed $\mu =0.1{\epsilon }_0$. In panels (a) and (b), the quasiparticle transport width is modeled by $\mathrm{\Gamma }={\mathrm{\Gamma }}_0\left(1+{\omega }^2/{\epsilon }_0^2\right)$ with ${\mathrm{\Gamma }}_0=0.1{\epsilon }_0$. In panel (c), the results are plotted in the clean limit, $\mathrm{\Gamma }=0$. The numerical values of other model parameters are defined in Appendix pp1.

Figure 8

The dependence of the thermal conductivity ${\kappa }_{xx},{\kappa }_{zz}$, and ${\kappa }_{xy}$ on the chemical potential in a Weyl semimetal (red solid line), a double-Weyl semimetal (blue dashed line), a triple-Weyl semimetal (green dotted line) at fixed $T=0.1{\epsilon }_0$. In panels (a) and (b), the quasiparticle transport width is modeled by $\mathrm{\Gamma }={\mathrm{\Gamma }}_0\left(1+{\omega }^2/{\epsilon }_0^2\right)$ with ${\mathrm{\Gamma }}_0=0.1{\epsilon }_0$. In panel (c), the results are plotted in the clean limit, $\mathrm{\Gamma }=0$. The numerical values of other model parameters are defined in Appendix pp1.

Figure 9

The dependence of the thermal conductivity ${\kappa }_{xx},{\kappa }_{zz}$, and ${\kappa }_{xy}$ on temperature in a Weyl semimetal (red solid line), a double-Weyl semimetal (blue dashed line), a triple-Weyl semimetal (green dotted line) at fixed $\mu =0.1{\epsilon }_0$. In panels (a) and (b), the quasiparticle transport width is modeled by $\mathrm{\Gamma }={\mathrm{\Gamma }}_0\left(1+{\omega }^2/{\epsilon }_0^2\right)$ with ${\mathrm{\Gamma }}_0=0.1{\epsilon }_0$. In panel (c), the results are plotted in the clean limit, $\mathrm{\Gamma }=0$. The numerical values of other model parameters are defined in Appendix pp1.

Figure 10

The dependence of the thermopower $S_{xx},S_{zz}$, and $S_{xy}$ on the chemical potential in a Weyl semimetal (red solid line), a double-Weyl semimetal (blue dashed line), a triple-Weyl semimetal (green dotted line) at fixed $T=0.1{\epsilon }_0$. In panels (a) and (b), the quasiparticle transport width is modeled by $\mathrm{\Gamma }={\mathrm{\Gamma }}_0\left(1+{\omega }^2/{\epsilon }_0^2\right)$ with ${\mathrm{\Gamma }}_0=0.1{\epsilon }_0$. In panel (c), the results are plotted in the clean limit, $\mathrm{\Gamma }=0$. The numerical values of other model parameters are defined in Appendix pp1.

Figure 11

The dependence of the thermopower $S_{xx},S_{zz}$, and $S_{xy}$ on temperature in a Weyl semimetal (red solid line), a double-Weyl semimetal (blue dashed line), a triple-Weyl semimetal (green dotted line) at fixed $\mu =0.1{\epsilon }_0$. In panels (a) and (b), the quasiparticle transport width is modeled by $\mathrm{\Gamma }={\mathrm{\Gamma }}_0\left(1+{\omega }^2/{\epsilon }_0^2\right)$ with ${\mathrm{\Gamma }}_0=0.1{\epsilon }_0$. In panel (c), the results are plotted in the clean limit, $\mathrm{\Gamma }=0$. The numerical values of other model parameters are defined in Appendix pp1.

Figure 12

The dependence of the relative Lorenz number $L_{xx}/L_0={\kappa }_{xx}/\left(e^2{L}_{0}T{L}_{xx}^{11}\right),L_{zz}/L_0={\kappa }_{zz}/\left(e^2{L}_{0}T{L}_{zz}^{11}\right)$, and $L_{xy}/L_0={\kappa }_{xy}/\left(e^2{L}_{0}T{L}_{xy}^{11}\right)$ on the chemical potential in a Weyl semimetal (red solid line), a double-Weyl semimetal (blue dashed line), a triple-Weyl semimetal (green dotted line) at fixed $T=0.1{\epsilon }_0$. In panels (a) and (b), the quasiparticle transport width is modeled by $\mathrm{\Gamma }={\mathrm{\Gamma }}_0\left(1+{\omega }^2/{\epsilon }_0^2\right)$ with ${\mathrm{\Gamma }}_0=0.1{\epsilon }_0$. In panel (c), the results are plotted in the clean limit, $\mathrm{\Gamma }=0$. The numerical values of other model parameters are defined in Appendix pp1.

Figure 13

The dependence of the relative Lorenz number $L_{xx}/L_0={\kappa }_{xx}/\left(e^2{L}_{0}T{L}_{xx}^{11}\right),L_{zz}/L_0={\kappa }_{zz}/\left(e^2{L}_{0}T{L}_{zz}^{11}\right)$, and $L_{xy}/L_0={\kappa }_{xy}/\left(e^2{L}_{0}T{L}_{xy}^{11}\right)$ on the chemical potential in a Weyl semimetal (red solid line), a double-Weyl semimetal (blue dashed line), a triple-Weyl semimetal (green dotted line) at fixed $\mu =0.1{\epsilon }_0$. In panels (a) and (b), the quasiparticle transport width is modeled by $\mathrm{\Gamma }={\mathrm{\Gamma }}_0\left(1+{\omega }^2/{\epsilon }_0^2\right)$ with ${\mathrm{\Gamma }}_0=0.1{\epsilon }_0$. In panel (c), the results are plotted in the clean limit, $\mathrm{\Gamma }=0$. The numerical values of other model parameters are defined in Appendix pp1.