Magnetic-Field-Induced Relativistic Properties in Type-I and Type-II Weyl Semimetals

We investigate Weyl semimetals with tilted conical bands in a magnetic field. Even when the cones are overtilted (type-II Weyl semimetal), Landau-level quantization can be possible as long as the magnetic field is oriented close to the tilt direction. Most saliently, the tilt can be described within the relativistic framework of Lorentz transformations that give rise to a rich spectrum, displaying new transitions beyond the usual dipolar ones in the optical conductivity. We identify particular features in the latter that allow one to distinguish between semimetals of different types.

Figure 1

Graphical representation of the tilt-parameter vector $\mathbf{t}$. The unit sphere $|\mathbf{t}|=1$ represents the boundary between type-I (blue arrow) and type-II (red arrows) WSM. In the presence of a magnetic field along the $z$ axis, the projection of a tilt-parameter vector on the $xy$ plane identifies the regime, the magnetic regime (i.e., inside the green disc) or in the electric regime (outside the disc). The vectors ${\mathbf{t}}_{m1}$, ${\mathbf{t}}_{m2}$, and ${\mathbf{t}}_{e2}$ indicate a type-I WSM, a type-II WSM in the magnetic regime, and a type-II WSM in the electric regime, respectively.

Figure 2

LL spectrum of a tilted Weyl cone for $\beta =0.75$, in units of $v_{\perp }/l_B\simeq 26\mathrm{m}\mathrm{eV}\times v_{\perp }[10^6\mathrm{m}/\mathrm{s}]/\sqrt{B[T]}$. The green lines are $n>0$ LLs and the blue (red) line represents the $n=0$ state for $\mathrm{sgn}(v_x{v}_y{v}_{z}B)=1$ [$\mathrm{sgn}(v_x{v}_y{v}_{z}B)=-1$]. The left figure corresponds to the case of a type-I Weyl semimetal where the $n=0$ LL group velocity changes sign with chirality. On the contrary, the right figure is for a type-II Weyl semimetal, where the $n=0$ slope representing the LL group velocity direction is independent of chirality.

Figure 3

Optical conductivity for the ${\mathbf{e}}_{+}$-elliptic polarization of undoped type-I [panel (a)] and type-II WSM [panel (b)], for different values of $\beta$ ($=0$, 0.1, 0.2, 0.4, and 0.6). The curves have been displaced up for increasing values of $\beta$. In the case $\beta =0$ one observes the usual Landau-level spectrum with $n\to (n\pm 1)$ transitions while for higher values of the in-plane tilt, more transitions are allowed. Insets are a reminder of the LL structure and the arrows indicate inter-LL transitions visible in the optical conductivity.