Chiral $f$-wave topological superfluid in triangular optical lattices

We demonstrate that an exotically chiral $f$-wave topological superfluid can be induced in cold-fermionic-atom triangular optical lattices through a laser-field-generated effective non-Abelian gauge field, controllable Zeeman fields, and $s$-wave Feshbach resonance. We find that the chiral $f$-wave topological superfluid is characterized by three gapless Majorana edge states located on the boundary of the system. More interestingly, these Majorana edge states degenerate into one Majorana fermion bound to each vortex in the superfluid. Our proposal enlarges the topological superfluid family and specifies a unique experimentally controllable system to study Majorana fermion physics.

Figure 1

(a) Uniform triangular lattices are formed from the maxima of the potential given by $V\left(\mathbf{r}\right)=V_L{\sum }_{i=1}^3\cos ({\mathbf{k}}_i·\mathbf{r})$. The defined lattice vectors ${\mathbf{s}}_i$ and ${\mathbf{d}}_i$ are shown. The hexagonal zone encircled by the black dashed lines is the unit cell of the triangular lattice. (b) The Brillouin zone (BZ) of triangular lattices, with the high-symmetry points marked.

Figure 2

Illustration of the light-atom interaction for generation of effective non-Abelian gauge fields and effective Zeeman fields. (a) The configuration of the hyperfine levels of the ultracold atom and three sets of laser beams characterized by the Rabi frequencies ${\Omega }_{a,i}$, ${\Omega }_{b,i}$, and ${\Omega }_{c,j}$ with $i=1,2,3$ and $j=1,2$. The atom and the laser fields have interaction through Raman-type coupling with a large single-photon detuning ${\delta }_{a/b/c}$. The laser beam configuration for ${\Omega }_a$, ${\Omega }_b$, and ${\Omega }_c$ are shown in (b), (c), and (d). (e) The relative energy levels modulated by the atom-laser couplings.

Figure 3

(a) The band structures of $E_k={\varepsilon }_k\pm {\Theta }_k$ with $\cos \phi =-0.101$ and $\mu =-2.59$. Here, we set $k_x\in [-4\pi /3,4\pi /3]$ and $k_y\in [-2\pi /\sqrt{3},2\pi /\sqrt{3}]$. (b) The band structures along high-symmetry lines [Fig. 2b]. The dashed blue lines are ${\varepsilon }_k\pm \left|g_k\right|$ and the solid red lines correspond to (a). Four different fillings with chemical potential ${\mu }_{1/2/3/4}=-2.59t$,$-3.436t$,$-3.89t$,$0.45t$ are shown. (c), (d), (e) are the SF quasiparticle spectra corresponding to ${\mu }_1$, ${\mu }_2$, and ${\mu }_3$. (f) The phase diagram $\cos \phi$-$\mu$. Two different phases NSF and TSF are identified. ${\Delta }_0=0.5t$ and $\hslash {\Omega }_p=3{\Delta }_0$.

Figure 4

The spectrums of Hamiltonian (19) with edges at x direction $i_x$ $\in (1,31)$. (a), (b), and (c) correspond to ${\mu }_1$, ${\mu }_2$, and ${\mu }_3$ cases in Fig. 3(b). The dashed black lines indicate the contributions from the first BZ. (d) The edge states [the states crossing the gap and denoted with the solid red lines in (a)] transport along two edges.