Electromagnetic Response of Weyl Semimetals

It has been suggested recently, based on subtle field-theoretical considerations, that the electromagnetic response of Weyl semimetals and the closely related Weyl insulators can be characterized by an axion term $\theta \mathit{E}·\mathit{B}$ with space and time dependent axion angle $\theta (\mathit{r},t)$. Here we construct a minimal lattice model of the Weyl medium and study its electromagnetic response by a combination of analytical and numerical techniques. We confirm the existence of the anomalous Hall effect expected on the basis of the field theory treatment. We find, contrary to the latter, that chiral magnetic effect (that is, ground state charge current induced by the applied magnetic field) is absent in both the semimetal and the insulator phase. We elucidate the reasons for this discrepancy.

Figure 1

Low energy spectra in Dirac and Weyl semimetals. (a) Doubly degenerate massless Dirac cone at the transition from a TI to a band insulator. Weyl semimetals with the individual cones shifted in (b) momenta and (c) energy. Panel (d) illustrates the Weyl insulator which can arise when the excitonic instability gaps out the spectrum indicated in (c). In all panels, two components of the 3D crystal momentum $\mathit{k}$ are shown.

Figure 2

The band structure of the Weyl semimetal lattice model, displayed along the path $\mathit{k}$: $(\pi ,0,\pi )\to (0,0,0)\to (0,0,\pi )\to (\pi ,0,\pi )$. (a) Doubly degenerate 3D Dirac point when $H_1=0$ and $ϵ=6t$. (b) Momentum-shifted Weyl point for $b=0.9$ and $b_0=0$. (c) Energy-shifted Weyl points for $b_z=0$ and $b_0=0.7$. (d) Weyl insulator with $b_z=0$ and $b_0=0.7$ and the exciton gap modeled by taking $ϵ=5.9t$. In all panels, we take $\lambda ={\lambda }_z=1.0$, $t=0.5$, and the energy is measured in units of $\lambda$. Red circles mark the location of the Dirac (Weyl) points.

Figure 3

(a) Charge density $\delta \rho (x,y)$ accumulated in the vicinity of the flux tubes $\Phi =0.01{\Phi }_0$ in the Weyl semimetal. (b) Total accumulated charge per layer $\delta Q$ near one of the flux tubes, in units of $e/2\pi$ for indicated values of $b_z$. Dashed lines represent the expectation based on Eq. (9). We use $\lambda ={\lambda }_z=t=0.5$, $ϵ=3.0$, $L=14$, and $L_z=160$ independent values of $k_z$. Panels (c), (d) show the charge accumulations as a function of $b_z$ in the presence of nonzero Dirac mass and $b_0$. Parameters as above except $b_0=0.1$, 0.2, 0.3 in (c) and $ϵ=3.0$, 2.9, 2.8, 2.7 for the curves in (d) from left to right.

Figure 4

(a) Chiral current $J_z$ as a function of energy offset $b_0$ for various values of the momentum cutoff $\Lambda$. The dashed line indicates the field theory prediction Eq. (5). (b) The slope $d{J}_z/d{b}_0$ in units of $e\eta /2\pi$ as a function of cutoff $\Lambda$. Slope 1.0 is expected on the basis of Eq. (5).