Exotic topological states with Raman-induced spin-orbit coupling

We propose a simple experimental scheme to realize simultaneously the one-dimensional spin-orbit coupling and the staggered spin flip in ultracold pseudospin-$1/2$ atomic Fermi gases trapped in optical lattices. In the absence of interspecies interactions, the system supports type-I and II Weyl semimetals in three-dimensional and gapped Chern insulators and gapless topological semimetal states in two-dimensional lattices. By turning on the $s$-wave interactions, a rich variety of gapped and gapless inhomogeneous topological superfluids can emerge. In particular, a gapped topological Fulde-Ferrell superfluid, which supports the chiral edge states at opposite boundaries with the same chirality, is predicted.

Figure 1

(a) Level diagram of a four-level atomic system. (b) Proposed scheme for generating Raman-assisted hopping in the $xy$ lattice plane. The two lines with the same color linked to $M_{x,y}$ denote the process of staggered spin-flip Raman.

Figure 2

Energy spectra of (a) type-I WSM with $\vartheta =\pi /4$ and (b) type-II WSM with $\vartheta =0$ for $k_{x}a=0, t_0=t_z=t$, and $\delta /t=-2$. The surface states are denoted by the red solid lines.

Figure 3

(a) Phases of the Hamiltonian (3) on the ${\varphi }_x\delta$ plane. Equations (5) are plotted as solid lines, and the numbers represent the Chern numbers of the different phases. Dashed lines denote the boundaries between CI and tSM phases for $t_0/t=0.71$, with shaded regions marking the tSM phase. (b) and (c) show the band spectra for ${\varphi }_x/\pi =0.2$ (CI phase) and 0.3 (tSM phase), respectively, for the parameters $t_0/t=0.71$ and $\delta /t=4$.

Figure 4

(a) Phase diagram in the $U_0{\varphi }_x$ parameter plane. (b) $|U_0|$ dependence of ${\mathrm{\Delta }}_{Q_x}$ (solid line) and $Q_x$ (dashed line). (c) $|U_0|$ dependence of $\mathcal{C}$ (solid line) and $E_g^{\left(i\right)}$ (dashed line).

Figure 5

(a)–(d) Quasiparticle spectra for $|U_0|/t=6.67$, 8.17, 8.5, and 10 with ${\varphi }_x/\pi =0.25$, respectively. The corresponding phase is denoted. The black dots mark the TR-invariant points.

Figure 6

Spatial distribution of the synthetic magnetic ${\mathbf{B}}_{\mathrm{GF}}$ (in T) for $\delta =0.1E_L$ and $\vartheta =\pi /2$. For optical lattices generated by the red-detuned (blue-detuned) lasers, atoms are trapped close to the red (blue) sites.