Magneto-optical conductivity of Weyl semimetals

Weyl semimetals are a topological phase of matter that have drawn recent interest and have been suggested as a possible phase of the pyrochlore iridates, among other materials. Here we compute the magneto-optical response of an isolated Weyl point. We find that the conductivity is a series of asymmetric peaks lying on top of a linear background. We find additional features in the conductivity that are explained by the underlying Landau level structure. We show that weak disorder tends to blur out the peaks, but they still show up as measurable oscillations on a linear background.

Figure 1

Dispersive Landau level structure for a Weyl semimetal at finite chemical potential. In this figure the chemical potential lies between the fifth and sixth Landau levels. The arrow at $k_z=0$ shows an interband transition from $n=5$ to $n=6$. These interband transitions at $k_z=0$ give the main set of peaks shown in Fig. 2. The two transitions at finite $k_z$ occur between the $n=0$ and $n=1$ Landau levels. They occur when $\mu$ lies between two Landau levels and are responsible for the additional bumps seen in Fig. 4.

Figure 2

The real part of the optical conductivity (in units of $e^2/8\pi l_B$) for $\mu l_B=0$, 1.4, and 2 as a function of $\omega l_B$. In this figure we took $T{l}_B=0.01$. The conductivity is a series of peaks which sit on a linear background. These peaks occur at the $k_z=0$ value of the Landau levels shown in Fig. 1. The inset shows the $\mu =0$ conductivity over a larger frequency range, demonstrating its linearity. The background conductivity should be the result for free fermions, which is a straight line of slope $1/3$ in our units.

Figure 3

The real part of the optical conductivity (in units of $e^2/8\pi l_B$) for graphene in the clean limit. Here $T{l}_B=0.01$. For $\mu =0$ there is a series of peaks at $\overline{\omega }=\sqrt{2(n+1)}+\sqrt{2n}$. As $\mu$ is increased these peaks disappear and are transferred into the intraband peak at low frequency.

Figure 4

The real part of the conductivity (in units of $e^2/8\pi l_B$) as a function of frequency for several values of the chemical potential. The top panel illustrates how the conductivity changes as the chemical potential sweeps between adjacent Landau levels. The bottom panel illustrates the change in the optical conductivity as a Landau level is crossed.

Figure 5

The real part of the optical conductivity (in units of $e^2/8\pi l_B$) for left handed circularly polarized light for $\overline{\mu }=2.0$ and $\overline{T}=0.01$. In the low temperature limit, the imaginary part of the Hall conductivity exactly cancels the real part of the diagonal conductivity for $\overline{\omega }\lesssim 2\overline{\mu }$.

Figure 6

The real part of the optical conductivity (in units of $e^2/8\pi l_B$) for $\overline{\mu }=0.0$ and $\overline{T}=0.01$ for several values of scattering rate $\overline{\Gamma }$. The red line shows the conductivity in the absence of a magnetic field. Increasing disorder tends to blur out the peaks towards the free limit.

Figure 7

The real part of the optical conductivity (in units of $e^2/8\pi l_B$) in the semiclassical limit. This occurs for large values of the chemical potential. All the spectral weight of the now forbidden interband transitions piles up into a single resonance at low frequency.