Dynamical conductivity of the Fermi arc and the Volkov-Pankratov states on the surface of Weyl semimetals

Weyl semimetals are known to host massless surface states called Fermi arcs. These Fermi arcs are the manifestation of the bulk-boundary correspondence in topological matter and thus are analogous to the topological chiral surface states of topological insulators. It has been shown that the latter, depending on the smoothness of the surface, host massive Volkov-Pankratov states that coexist with the chiral ones. Here, we investigate these VP states in the framework of Weyl semimetals, namely their density of states and magneto-optical response. We find the selection rules corresponding to optical transitions which lead to anisotropic responses to external fields. In the presence of a magnetic field parallel to the interface, the selection rules and hence the poles of the response functions are mixed.

Figure 1

(a) Sketch of the smooth topological heterojunction. Inside the Weyl semimetal ($x<0$) the Weyl nodes are separated in the momentum space along the $k_z$ direction by $k_0=2\sqrt{2m\mathrm{\Delta }}$. Inside the junction $(0<x<\ell )$, the Weyl nodes merge and gap out to finally form an insulating phase ($x>\ell$). (b) Sketch of the surface band structure. The Weyl nodes are split along the $k_z$ direction. The chiral Fermi arc (shown in pink) and the first VP state (shown in yellow) extend up to the bulk bands (shown in blue) along the $k_z$ axis.

Figure 2

The states on the surface reciprocal space for a semi-infinite slab geometry for different energies. For low energies ($E=0.1\mathrm{\Delta }$), only the Fermi arc (shown in dashed red line) is visible. As the energy is increased beyond ${\epsilon }_0(\approx 0.447\mathrm{\Delta }$ for the chosen set of parameter values), the first VP band (in green lines) appears. For larger $E$, the second VP state appears. It is interesting to note that the boundary of the VP states along the $k_z$ axis do not change for changing $E$. The parameters used are $v_F=0.5$ eV $\text{Å}$, $\mathrm{\Delta }=0.5$ eV, ${\mathrm{\Delta }}^{\prime }=\mathrm{\Delta }$, $m=0.25{\mathrm{eV}}^{-1}{\text{Å}}^{-2}$, $\ell =20\text{Å}$, and $\mathbf{B}=0$.

Figure 3

Comparison of density of states of the VP states with that of the bulk. The bulk density of states is multiplied by $v_F/\mathrm{\Delta }$ to make the two quantities dimensionally consistent. The dotted vertical line denotes the $E=\mathrm{\Delta }$ point, beyond which no new VP state can appear. The parameters used are the same as in Fig. 2. For these parameters, only four VP states can be seen.

Figure 4

Longitudinal optical conductivities (in units of $e^2$) for $\mu =0$ along the $\text{(a)}x,z$ and $\text{(b)}y$ directions. All possible transitions involving the VP states are shown. In (a), the poles are located at $\omega ={\epsilon }_0(\sqrt{n+1}+\sqrt{n})$ and in (b), they are located at $\omega =2\sqrt{n}{\epsilon }_0$. The parameters used are the same as in Fig. 2.

Figure 5

$\text{(a)}{\sigma }_{zz}$ and $\text{(b)}{\sigma }_{yy}$ (in units of $e^2$) for different values of $\mu$. In (a), as the chemical potential is increased, the absorption edge at $\omega ={\epsilon }_0$ splits into two. For larger chemical potential there is an overall redistribution of spectral weight and one can see a large accumulation for low frequencies (at $\omega \sim 0.5{\epsilon }_0$). In (b), the plot remains unchanged until $\mu ={\epsilon }_0$ beyond which the conductivity vanishes for $\omega \le \mu +{\epsilon }_0$ as the $n=1$ VP band starts getting occupied.

Figure 6

Constant energy contours for $E=0.5\mathrm{\Delta }$ with two different values of magnetic field. The continuous lines denote the contours for ${\ell }_S/{\ell }_B=0.2$ and the dashed lines are for ${\ell }_S/{\ell }_B=0.05$. The two red lines on the left denotes the $n=0$ Fermi arc states for the different values of the magnetic field whereas the green lines depict the $n=1$ VP states. Interestingly, the change in the length of the VP states along the $k_z$ axis is very small for small magnetic fields and is neglected in the optical conductivity calculations.

Figure 7

Longitudinal optical conductivities (in units of $e^2$) for $\mu =0$, ${\ell }_S/{\ell }_B=0.2\text{(a)}$ along $x$ and $z$ and (b) along $y$. In (a), the real part of ${\sigma }_{zz}$ (red dashed line) has poles at both $\omega ={\epsilon }_1(\sqrt{n+1}+\sqrt{n})$ and $\omega =2\sqrt{n}{\epsilon }_1$. The real part of ${\sigma }_{xx}$ (blue continuous line) remains unchanged apart from an overall scaling along the $x$ axis due to the change of the effective magnetic field. In panel (b), new poles at $\omega ={\epsilon }_1(\sqrt{n+1}+\sqrt{n})$, though very small, can be seen. Also, the effect of the absorption edge due to transitions from the Fermi arc can be seen below $\omega =2{\epsilon }_1$. This was prohibited in the $\mathbf{B}=0$ limit.

Figure 8

Optical conductivities (in units of $e^2$) by changing $\mu$ for ${\ell }_S/{\ell }_B=0.2$. We have used $\mu =0.25{\epsilon }_1$ in panels (a) and (b) and $\mu =1.2{\epsilon }_1$ in panels (c) and (d).

Figure 9

Comparison of the poles of ${\sigma }_{yy}$ with and without the magnetic field.