Majorana Doublets, Flat Bands, and Dirac Nodes in $s$-Wave Superfluids

Topological superfluids protected by mirror and time-reversal symmetries are exotic states of matter possessing Majorana Kramers pairs (MKPs), yet their realizations have long been hindered by the requirement of unconventional pairing. We propose to realize such a topological superfluid by utilizing $s$-wave pairing and emergent mirror and time-reversal symmetries in two coupled 1D ultracold atomic Fermi gases with spin-orbit coupling. By stacking such systems into 2D, we discover topological and Dirac-nodal superfluids hosting distinct MKP flat bands. We show that the emergent symmetries make the MKPs and their flat bands stable against pairing fluctuations that otherwise annihilate Majorana pairs. Exploiting new experimental developments, our scheme provides a unique platform for exploring MKPs and their applications in quantum computation.

Figure 1

Schematics of proposed experimental setups. (a) 1D SOC generated by two counterpropagating Raman lasers along ${\mathit{e}}_{\mathit{x}}$, i.e., one ${\mathrm{HG}}_{01}$ beam (red arrow) polarized along ${\mathit{e}}_{\mathit{z}}$ with frequency ${\omega }_1$ and one Gaussian beam (blue arrow) polarized along ${\mathit{e}}_{\mathit{y}}$ with frequency ${\omega }_2$. The green line shows the resulting Zeeman field along ${\mathit{e}}_{\mathit{y}}$. (b) Two-photon process induced by the two Raman lasers in (a) with a detuning $\delta$.

Figure 2

(a) Phase diagram in the $\mathrm{\Omega }\text{−}\mu$ plane, symmetric with respect to $\mu =0$ and $\mathrm{\Omega }=0$. The contour plot shows the site-averaged pairing $⟨\mathrm{\Delta }⟩$ in the normal superfluid (N), topological superfluid (T), metal with SOC (M), polarized insulator (I), and trivial vacuum (V). The dotted red lines are the phase boundaries determined by Eq. (5). (b) Phase transitions along the white dotted line in (a). The black solid (red dotted) lines denote the first (second) quasiparticle excitation states (ES) in the spectrum, both of which are twofold degenerate. (c) Probability distributions of the left (L) and right (R) MKPs at the red cross in (a). $\sum _i{P}_L(i)=\sum _i{P}_R(i)=2$ are the hallmarks of MKPs. $\alpha =1$ and $t_{\perp }=0.5$ are used in (a)–(c).

Figure 3

(a) Phase diagram in the $\mathrm{\Omega }\text{−}\mathrm{\Delta }$ plane for the 2D model [Eq. (8)]. The red, green, and blue regions denote the normal (N), topological (T), and Dirac-nodal (D) superfluids, respectively. (b) Bulk quasiparticle spectrum for the Dirac superfluid labeled by the red star in (a). Each Dirac point is indexed by a winding number ${\gamma }_t=\pm 1$. (c) and (d) Quasiparticle spectrum with MKP edge flat bands under open boundary condition for the Dirac and topological superfluids labeled in (a). $t_1=t_2=0.5$, $\alpha =1$, and $\mu =-2$ are used in (a)–(d).

Figure 4

(a) Self-consistent quasiparticle spectrum for the $100\times 8$ lattice model. The red lines denote the eight lowest quasiparticle excitation states. (b) Vector plot of the local pairing fields ${\mathrm{\Delta }}_j{e}^{i{\varphi }_j}$ for $\mathrm{\Omega }=1.1$. The length (direction) of each arrow denotes the strength (phase) of the local pairing field. $t_1=t_2=0.5$, $\alpha =1$, and $\mu =-2$ are used in (a) and (b).