Realizing type-II Weyl points in an optical lattice

The recent discovery of the Lorentz symmetry-violating “type-II” Weyl semimetal phase has renewed interest in the study of Weyl physics in condensed-matter systems. However, tuning the exceptional properties of this novel state has remained a challenge. Optical lattices, created using standing laser beams, provide a convenient platform to tune tunneling parameters continuously in time. In this paper, we propose a generalized two level system exhibiting type-II Weyl points that can be realized using ultracold atoms in an optical lattice. The system is engineered using a three-dimensional lattice with complex $\pi$ phase tunneling amplitudes. Various unique properties of the type-II Weyl semimetal such as open Fermi surface, anomalous chirality, and topological Fermi arcs can be probed using the proposed optical lattice scheme.

Figure 1

Structure of 3D lattice proposed to realize type-II Weyl points. Tunneling amplitude along the $x$ direction alternately carries a complex phase of $\pi$ (depicted using dashed lines) and 0 (solid). The lattice can be constructed using two dissimilar sites depicted in red and blue. Tunneling along the $z$ direction has a constant phase of 0 ($\pi$) along the red (blue) sites and is shifted from 0 by a constant value $J_z$. Inset: A part of the lattice depicting the complex tunneling amplitudes in the $x$, $y$, and $z$ directions.

Figure 2

(a) Band structure of the lattice in the $k_x=0$ plane. The electron (orange) and hole (blue) bands touch to form type-II WPs at $(k_y,k_z)=(\pm \pi /2a,\pm \pi /2a)$. Each node, labeled $W1–W4$, is tilted in the $k_z$ direction. (b) Fermi surface ($E=0$) is depicted as a function of the crystal momentum $\mathit{k}$. The electron-hole pockets in the first Brillouin zone (shaded in gray) touch at WPs ($W1–W4$). (c) Berry curvature in the $k_x=0$ plane. Arrows represent the direction of the vector field. Type-II WPs $W2,W3$ act as sources and $W1,W4$ act as sinks of Berry curvature. The color scale, indicating the magnitude of the vector field, has been logarithmically scaled. In all calculations the magnitude of tunneling amplitudes is normalized to 1 and $\delta J_z$ is assumed to be $1/2$.

Figure 3

Energy spectra in the vicinity of WPs in the presence of a magnetic field ($l_B=10$) directed along the $\mathit{z}$ axis. First 50 Landau levels are plotted for type-II WPs ($\delta J_z=0.5$ and $J_x=J_y=J_z=1$) located at $(k_{x}a,k_{y}a,k_{z}a)=(0,\pi /2,\pi /2)$ in (a), $(0,\pi /2,-\pi /2)$ in (c), $(0,-\pi /2,\pi /2)$ in (e), and $(0,-\pi /2,-\pi /2)$ in (g). For comparison, spectra of corresponding type-I WPs ($\delta J_z=J_x=J_y=1$ and $J_z=0$) are plotted in (b), (d), (f), and (h). The group velocities of the chiral Landau level in type I WPs (b),(f) have opposite slopes reflecting their opposite chirality. However, chiral Landau levels in type-II WPs (a),(e) have positive group velocities in spite of their opposite chiralities; same for (c),(g) in contrast to (d),(h).

Figure 4

Energy spectra for an optical lattice consisting of 25 unit cells in the $\mathit{x}-\mathit{y}$ direction and infinite along the $\mathit{x}+\mathit{y}$ and $\mathit{z}$ directions (coordinates in Fig. 1). (a) Spectra plotted as a function of $k_z$ and $k_p=(k_x+k_y)/\sqrt{2}$. Surface states have been depicted in orange. Fermi arcs connecting type-II WPs $W1$ ($W3$) and $W2$ ($W4$) are highlighted using black lines. (b) Left: spectra plotted as a function of $k_p$ for $k_z=\pi /2a$. Fermi arc connecting $W1$ and $W2$ can be clearly seen. Right: spectra plotted as a function of $k_z$ for $k_p=\pi /8a$. Two surface states corresponding to propagation along the two edges of the lattice are evident.