Tunable Weyl semimetal and its possible realization in optical lattices

Weyl semimetal (WSM) is an exotic topological state in condensed-matter physics. In this paper, based on a two-band cubic lattice model, we study WSMs with a pair of tunable Weyl nodes. We point out that each Weyl node can be tuned freely, leading to three types of WSMs with different tilt strengths: types I, II, and 1.5. We study chiral modes, surface Fermi arcs, and quantum anomalous Hall effect to show the properties of a combination of independent nodes. In addition, we present an experimental setup to realize the different types of WSMs based on a timely technique.

Figure 1

Different WSMs: (a) Phase diagram of parameters $v_{oy}$ and $v_{sy}$, (b) WSM-I without tilts of nodes, (c) WSM-1.5 with one type-I and one type-II node, parameters are set as ${\mathbf{n}}_{+}=-(0,1,0)$, ${\alpha }_{+}=1.2$, and ${\alpha }_{-}=0$, (d) WSM-II with the same tilts perpendicular to the node separation, parameters are set as ${\mathbf{n}}_{+}={\mathbf{n}}_{-}=(0,1,0)$ and ${\alpha }_{+}={\alpha }_{-}=1.2$.

Figure 2

Surface Fermi arcs of WSMs with tilts along the $\widehat{x}$ direction. The edge of the red region (in the center) is Fermi surface $E=0$. (a) The dispersions of two lowest bands (surface states) at ${\mathbf{n}}_{+}=-{\mathbf{n}}_{-}=(1,0,0)$, ${\alpha }_{\pm }=0.7$. (b) Fermi arcs for WSM-I with ${\alpha }_{\pm }=0$. (c) Fermi arcs for WSM-1.5 with ${\mathbf{n}}_{-}=(1,0,0)$, ${\alpha }_{-}=1.2$ and ${\alpha }_{+}=0$. (d) Fermi arcs for WSM-II with ${\mathbf{n}}_{\pm }=(\pm 1,0,0)$, ${\alpha }_{\pm }=1.2$. These results show that the tilting effect along the $\widehat{x}$ direction will change the chemical potential of surface states so that will change the shape of the Fermi arcs.

Figure 3

Surface Fermi arcs when tilts are along the $\widehat{y}$ direction. Owing to the tilting of surface states, the electron and hole pockets appear with additional Fermi surface. (a) The dispersions of two lowest bands (surface states) at ${\mathbf{n}}_{+}={\mathbf{n}}_{-}=(0,1,0)$, ${\alpha }_{\pm }=0.5$. (b) Fermi arcs for WSM-I with ${\alpha }_{\pm }=0$. (c) Fermi arcs for WSM-1.5 with ${\mathbf{n}}_{-}=(0,1,0)$, ${\alpha }_{-}=1.2$, and ${\alpha }_{+}=0$. (d) Fermi arcs for WSM-II with ${\mathbf{n}}_{\pm }=(0,1,0)$, ${\alpha }_{\pm }=1.2$.

Figure 4

(a) Dispersion of WSM-I in a magnetic field along $\widehat{x}$ direction; there are two chiral modes near Fermi surface $E=0$. (b) Dispersion of WSM-II with ${\mathbf{n}}_{+}={\mathbf{n}}_{-}=(0,1,0)$ and ${\alpha }_{+}={\alpha }_{-}=1.2$ in a magnetic field along $\widehat{x}$ direction; there are no chiral modes and the spectrum is gapped. (c) Dispersion of WSM-1.5 with ${\alpha }_{-}=0$, ${\mathbf{n}}_{+}=(0,1,0)$ and ${\alpha }_{+}=1.2$ in a magnetic field along $\widehat{x}$ direction; there exists one chiral mode.

Figure 5

(a) QAHC in WSMs with a tilting along $\widehat{x}$ direction ${\mathbf{n}}_{\pm }=(\pm 1,0,0)$, the green solid line is ${\alpha }_{+}={\alpha }_{-}$ and the red dashed line is ${\alpha }_{-}=0$. The red dashed line decreases at the beginning due to the energy shift in Eq. (12). QAHC turns to zero for this case. (b) Dispersion of type-II WSM with ${\mathbf{n}}_{+}=-{\mathbf{n}}_{-}=(1,0,0)$ and ${\alpha }_{+}={\alpha }_{-}=4$, the purple flat plane is Fermi energy $E_F=0$. (c) Dispersion of type-II WSM with ${\mathbf{n}}_{+}=-{\mathbf{n}}_{-}=(1,0,0)$ and ${\alpha }_{+}={\alpha }_{-}=100$. The purple flat plane is Fermi energy $E_F=0$. For the states in the topological region $-k_0<k_x<k_0$, there exists an energy gap. (d) The energy dispersion for the yellow circle in (c).

Figure 6

(a) QAHC in WSMs with a tilting along the $\widehat{y}$ direction ${\mathbf{n}}_{\pm }=(0,1,0)$, the blue solid line is ${\alpha }_{+}={\alpha }_{-}$ and the black dashed line is ${\alpha }_{-}=0$. QAHC turns to a constant. (b) Dispersion of type-II WSM with ${\mathbf{n}}_{+}={\mathbf{n}}_{-}=(0,1,0)$ and ${\alpha }_{+}={\alpha }_{-}=1.2$, the purple flat plane is Fermi energy $E_F=0$. (c) Dispersion of type-II WSM with ${\mathbf{n}}_{+}={\mathbf{n}}_{-}=(0,1,0)$ and ${\alpha }_{+}={\alpha }_{-}=1000$. The purple flat plane is Fermi energy $E_F=0$. In the topological region $-k_0<k_x<k_0,$ the states are partly filled. (d) The dispersion for yellow circle in (c).

Figure 7

(a) Our purpose to realize WSMs. Blue balls (at the corners of cube) represent minima sites and green (on the bonds) and yellow (on center of faces) balls represent second minima sites. Double-sided arrows denote the Raman couplings. Red arrows (outside the cube with one helix arrow around it) denote Raman lasers for a Raman coupling with $\mathrm{\Delta }{m}_F=-1$, of which a circle arrow marks a unit of angular momentum carried by a $\sigma$ laser. (b) The lattice potential for all hyperfine states without gradient potential. M denotes minima sites, S denotes second minima sites, and C center second minima sites. (c) Scheme of realizing spin-flipping terms in the presence of a gradient potential.

Figure 8

QAHC vs chemical potential in a (left) 2D normal insulator, which is a slice of WSMs with a trivial topological property (QAHC becomes zero when the lower band is filled and the upper band is empty), and in a (right) 2D topological insulator, which is a slice of WSM with a nontrivial topological property.