Weyl points and topological nodal superfluids in a face-centered-cubic optical lattice

We point out that a face-centered-cubic (fcc) optical lattice, which can be realized by a simple scheme using three lasers, provides one a highly controllable platform for creating Weyl points and topological nodal superfluids in ultracold atoms. In noninteracting systems, Weyl points automatically arise in the Floquet band structure when shaking such fcc lattices, and sophisticated design of the tunneling is not required. More interestingly, in the presence of attractive interaction between two hyperfine spin states, which experience the same shaken fcc lattice, a three-dimensional topological nodal superfluid emerges, and Weyl points show up as the gapless points in the quasiparticle spectrum. One could either create a double Weyl point of charge 2, or split it into two Weyl points of charge 1, which can be moved in the momentum space by tuning the interactions. Correspondingly, the Fermi arcs at the surface may be linked with each other or separated as individual ones.

Figure 1

(a) Schematic of the fcc lattice with $A$ and $B$ sublattices. (b) The first BZ (red truncated octahedron) of the fcc lattice folded from that (black cubic) of the sc lattice. (c) The lowest two bands (blue solid), ${\epsilon }_{A\mathbf{k}}^0$ and ${\epsilon }_{B\mathbf{k}}^0$, of the fcc lattice. Red dashed line represents the subband of ${\epsilon }_{A\mathbf{k}}^0$ after absorbing one photon via shaking. The parameters are $V=8E_R,\hslash \omega =0.7E_R,\alpha =0.06,f=0.16d_0$. (d) One laser setup to realize the fcc lattice. Red (purple) arrows represent the lasers with wavelength $\lambda$ (${\lambda }^{\prime }$) and frequency $\omega$ (${\omega }^{\prime }$). Details can be referred to in the Supplemental Material [29].

Figure 2

(a)–(c) Fermi arcs with (100) surface of (a) noninteracting single particles for either spin up or spin down, and (b), (c) Bogoliubov quasiparticles for half-filling with $(U_A,U_B)=(0.15,0)E_R$ and $(0.15,0.08)E_R$, respectively. ${\tilde{k}}_{y,z}=(k_z\pm k_y)/\sqrt{2}$. (d)–(f) Schematics of nodal points in the bulk with no interaction, only one interaction, and both interactions, respectively. Pink (green) and red (blue) dots represent the nodal points with $+(-)1$ and $+(-)2$ charges, respectively. Spheres in (e) and (f) enclosing the nodal points have the same total Chern number, $-2$, of the lowest two bands. Parameters for single particles are the same as those in Fig. 1.