Optical conductivity of multi-Weyl semimetals

Multi-Weyl semimetals are new types of Weyl semimetals which have anisotropic nonlinear energy dispersion and a topological charge larger than one, thus exhibiting a unique quantum response. Using a unified lattice model, we calculate the optical conductivity numerically in the multi-Weyl semimetal phase and in its neighboring gapped states, and obtain the characteristic frequency dependence of each phase analytically using a low-energy continuum model. The frequency dependence of longitudinal and transverse optical conductivities obeys scaling relations that are derived from the winding number of the parent multi-Weyl semimetal phase and can be used to distinguish these electronic states of matter.

Figure 1

(a) Phase diagrams of $J=2$ lattice models on the $t_z/m_0$ and $m_z/m_0$ plane and (b) evolution of the energy band structure from the 3D quantum anomalous Hall (QAH) phase to the normal insulator (NI) phase. Here, we use several values of $m_z/m_0$ corresponding to different phases, indicated by circled numbers in the phase diagram. $\mathrm{QAH}|\mathrm{WSM}$ and $\mathrm{WSM}|\mathrm{NI}$ denote the transition phase between 3D QAH and WSM, and WSM and NI, respectively. The phase diagram for $J=1$ has a similar shape, but has a different phase boundary between the WSM and 3D QAH represented by the dashed line [12].

Figure 2

Real part of (a)–(d) longitudinal and (e), (f) transverse optical conductivities in units of ${\sigma }_0=\frac{e^2}{\hslash a}$ for the lattice (blue solid line) and continuum (black dotted line) models in the WSM phase. The arrows in the insets indicate interband transitions corresponding to kink structures in ${\sigma }_{xx}\left(\omega \right)$ and ${\sigma }_{xy}\left(\omega \right)$. Here, $m_z/m_0=0,b=0.5\pi /a$, and $k_{\mathrm{c}}=\pi /a$ are used for calculation.

Figure 3

Real part of (a)–(d) longitudinal and (e), (f) transverse optical conductivities in the 3D QAH phase for the lattice model (blue solid line), the continuum model (red dashed line), and the analytic results (black dotted line). For the longitudinal (transverse) conductivities, the analytic results are obtained for $\gamma =0$ $\left(\gamma =\frac{m_0{a}^2}{2}\right)$. Solid (dashed) lines in the inset to (a) represent the energy dispersion for $J=1$ along the $k_z$ direction with $k_x=0$ $\left(k_x=\frac{\pi }{a}\right)$ and $k_y=0$. The left inset to (b) represents the energy dispersion for $J=2$ along the $k_x$ direction with $k_z=\frac{\pi }{a}$ and $k_y=0$, and the right inset to (b) shows an enlarged view in ${\sigma }_{xx}\left(\omega \right)$ near the interband transition. Arrows in the insets indicate interband transitions corresponding to the kink structures appearing in ${\sigma }_{xx}$ and ${\sigma }_{xy}$. Here, $m_z/m_0=-0.8$ and $k_{\mathrm{c}}=\pi /a$ are used for the calculation.

Figure 4

Real part of (a)–(d) longitudinal and (e), (f) transverse optical conductivities at the transition between the 3D QAH and WSM phases for the lattice model (blue solid line), the continuum model (red dashed line), and the analytic results (black dotted line). For the longitudinal (transverse) conductivities, the analytic results are obtained for $\gamma =0$ $\left(\gamma =\frac{m_0{a}^2}{2}\right)$. Here, $m_z/m_0=-0.5$ and $k_{\mathrm{c}}=\pi /a$ are used for calculation.