Effect of the chiral anomaly on circular dichroism and Hall angle in doped and tilted Weyl semimetals

From the Kubo formula for transport in a tilted Weyl semimetal, we calculate the absorptive part of the dynamic conductivity for both right- and left-handed circular polarized light. These depend on the real part of the longitudinal conductivity and the imaginary part of the transverse (Hall) conductivity. We include the effect of the chiral anomaly, which pumps charge from negative to positive chirality node when the usual $\mathit{E}·\mathit{B}$ term is included in the electrodynamics, and obtain analytic expressions. To calculate the Hall angle, we further provide expressions for the imaginary part of longitudinal and real part of the transverse conductivity and compare results with and without the pumping term. We also consider the case of a noncentrosymmetric Weyl semimetal in which the chiral nodes are displaced in energy by an amount $\pm {\mathcal{Q}}_0$. This leads to modification in dichroism and Hall angle which parallel the pumping case.

Figure 1

(a) Schematic of two tilted Weyl cones with nonzero chiral pumping. The positive chirality node and the negative chirality node are tilted anticlockwise and clockwise, respectively, and we include chiral pumping; both affect the Hall conductivity. Application of $\mathit{E}·\mathit{B}$ term results in a net transfer of charge carriers from one node to the other (blue arrow), thereby modifying the chemical potential of the two nodes. The chemical potential of the negative and the positive chirality nodes are denoted as ${\mu }_{-}$ and ${\mu }_{+}$, respectively. (b) The total longitudinal $\left({\sigma }_{xx}\right)$ and Hall $\left({\sigma }_{xy}\right)$ conductivity is a sum of the contribution from each chirality node. The upward and downward arrows refer to the positive and the negative contributions, respectively. We consider a scenario where ${\mu }_{+}$ is always positive, whereas the sign of ${\mu }_{-}$ depends on the chiral pumping [Eq. (5)]. Therefore, the contribution coming from positive chirality is always positive. The length of the arrows signifies the different doping levels in each of the two Weyl nodes. (c) The charge imbalance makes a nonvanishing contribution to longitudinal and Hall conductivity which leads to changes in the circular dichroism, i.e., the absorption of the right (red) and the left (green) circularly polarized light is modified by the chiral pumping.

Figure 2

(a) Schematic for various transitions involved in the presence of charge imbalance. (b) Contributions coming from different frequency intervals in the positive and negative chirality nodes for two cases, namely, $\left(1+\mathcal{C}\right){\mu }_{-}>\left(1-\mathcal{C}\right){\mu }_{+}$ [case I] and $\left(1+\mathcal{C}\right){\mu }_{-}<\left(1-\mathcal{C}\right){\mu }_{+}$ [case II] have been illustrated with distinct colors shading. For example, below the frequency value ${\mathrm{\Omega }}_L^{{\mu }_{-}}\left[=2{\mu }_{-}/\left(\mathcal{C}+1\right)\right]$, we do not expect the interband optical conductivity to be finite due to complete Pauli blocking. In case I, each node contributes partially in the frequency regime $2{\mu }_{-}/\left(\mathcal{C}+1\right)<\mathrm{\Omega }<2{\mu }_{-}/\left(1-\mathcal{C}\right)$. For frequencies greater than $2{\mu }_{+}/\left(1-\mathcal{C}\right)$, Pauli blocking is completely removed and we expect full contribution from both the nodes. This is indeed the case as can be seen in (c) and (d), where we have shown the real part of the longitudinal conductivity and the imaginary part of Hall conductivity as a function of frequency for ${\mu }_p=0.8{\mu }_0$, which falls under case I and ${\mu }_p=0.99{\mu }_0$ for the other case, respectively. Here we have considered ${\mathcal{C}}_{-}\left(=-{\mathcal{C}}_{+}\right)=\mathcal{C}=0.5$ and ${\mu }_0$ is chosen as the normalization parameter.

Figure 3

Optical conductivity, (${\sigma }_{\pm }$) as seen by left and right circularly polarized light as a function of frequency for (a) ${\mu }_p=0$, (b) ${\mu }_p=0.8{\mu }_0$, (c) ${\mu }_p=0.99{\mu }_0$, and (d) ${\mu }_p={\mu }_0$. For ${\mu }_p\to {\mu }_0$, ${\mu }_{-}\to 0$, implying that for the negative node, optical transitions start at zero frequency. Increasing frequency beyond ${\mathrm{\Omega }}_U^{{\mu }_{+}}$, we see that both the nodes contribute as expected. Color-coded numbers identify the values of the critical frequencies defining the various photon ranges. Other parameters are the same as those of Fig. 2.

Figure 4

(a) The imaginary part of longitudinal and (b) real part of the Hall conductivity is shown as a function of frequency. The bubbled line in (a) refers to the total sum of the contribution from the interband (red) and the intraband (blue) conductivity, black dashed line is for ${\mu }_p=0$ and green circles for ${\mu }_p=0.8{\mu }_0$. In (b), the dc value of Hall conductivity is consistent with Eq. (26). Other parameters are the same as those of Fig. 2.

Figure 5

Bulk Hall angle as a function of frequency for (a) the nonvanishing chiral pumping case, where the red curve is the no pumping case (${\mu }_p=0$) and is for comparison, and (b) the broken inversion symmetry case. Other parameters are the same as those of Fig. 2.

Figure 6

(a) Schematic for various transitions involved when inversion symmetry is broken. (b)Absorptive part of the conductivity for right and left circularly polarized light as a function of frequency for ${\mathcal{Q}}_0=0.5{\mu }_0$, ${\mathcal{Q}}_0=1.5{\mu }_0$, ${\mathcal{Q}}_0=2{\mu }_0$, and ${\mathcal{Q}}_0=2.5{\mu }_0$. This case is analogous to the chiral pumping case shown in Fig. 3, except that the red and green curves are less distorted here. Other parameters are the same as those of Fig. 2.

Figure 7

(a) The imaginary part of longitudinal and (b) real part of the Hall conductivity is shown as a function of frequency for broken inversion symmetry case. The bubbled line in (a) refers to the total sum of the contribution from the inter- and the intraband conductivity for ${\mathcal{Q}}_0=0.5{\mu }_0$. In panel (b), $\mathrm{Re}\left[{\sigma }_{xy}\left(\mathrm{\Omega }\right)\right]$ is shown for the same value of ${\mathcal{Q}}_0$. Note that its dc value is consistent with Eq. (26). Other parameters are the same as those of Fig. 2.