Fulde-Ferrell-Larkin-Ovchinnikov or Majorana superfluids: The fate of fermionic cold atoms in spin-orbit-coupled optical lattices

The recent experimental realization of spin-orbit (SO) coupling for ultracold atoms opens a completely new avenue for exploring new quantum matters. In experiments, the SO coupling is implemented simultaneously with a Zeeman field. Such SO-coupled Fermi gases are predicted to support Majorana fermions with non-Abelian exchange statistics in one dimension (1D). However, as shown in recent theory and experiments for 1D spin-imbalanced Fermi gases, the Zeeman field can lead to the long-sought Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluids with nonzero center-of-mass momentum Cooper pairings, in contrast to the zero center-of-mass momentum pairings in Majorana superfluids. Therefore a natural question to ask is which phase, FFLO or Majorana superfluids, will survive in SO-coupled Fermi gases in the presence of a large out-of-plane Zeeman field. In this paper, we address this question by studying the mean-field quantum phases of 1D (quasi-1D) SO-coupled fermionic cold-atom optical lattices.

Figure 1

Phase diagram of 1D SO-coupled optical lattices as a function of Zeeman field $h$ and (a) and (c) filling factor $n$ or (b) and (d) chemical potential $\mu$. (top) SO coupling $\alpha =0.5t$; (bottom) $\alpha =1.0t$. Five different phases are identified in each phase diagram: normal BCS superfluid (NS), topological superfluid (TS), FFLO, normal gas (NG), and insulator phase (Ins).

Figure 2

Representative density of states for (a) the insulator phase, (b) normal BCS superfluid phase, (c) topological superfluid phase, and (d) FFLO phase. The insets show the order parameters in each phase. The corresponding phase points are marked by the plus signs in Fig. 1. Note that there is a zero-energy peak in the DOS in the topological superfluid phase as shown in (c).

Figure 3

(a) Plot of the order parameter and the two lowest eigenenergies $E_1$ and $E_2$ as a function of Zeeman field for the transition from normal BCS superfluids to topological superfluids. $\alpha =1.0t,\mu =-2.0t$. (b) and (c) The Majorana zero-energy-state wave function.

Figure 4

Single-particle band structure of the SO-coupled optical lattices. $\alpha =1.0t$, $h=1.7t$ [corresponding to the dashed line in Fig. 1]. The corresponding chemical-potential regions for different phases are identified. (b) Plot of $\mu$ as a function of Zeeman field for fixed filling factor $n=0.25$, $0.5$, $0.75$, $1.0$. The regions between the two circles on each line label the topological superfluid phase (for $n=0.25,0.5$) or the FFLO phase (for $n=0.75,1.0$).

Figure 5

Phase diagram in the presence of a harmonic trap. (a) Order parameter profiles and (b) atom density distributions for different trapping frequencies: ${\omega }_x=0.0$ (red dotted line), $0.05$ (orange dashed line), $0.15$ (green solid line), and $0.2$ (blue dash-dotted line). In (a) and (b), we fix the Zeeman field $h=1.0t$ and the average filling factor $n=0.475$. (c) and (d) are phase diagrams as a function of Zeeman field $h$ and filling factor $n$ or chemical potential $\mu$. In all the plots, $\alpha =1.0t$.

Figure 6

Phase diagram with the inclusion of the Hartree shift. $\alpha =0.5t$. All other parameters and notations are the same as those in Fig. 1.

Figure 7

Topological and FFLO phases in quasi-1D ($100\times 3\times 3$) lattices. The order parameters in (a) topological superfluid and (b) FFLO phases. Red solid lines: quasi-1D; green dashed lines: 1D for the same parameters. (c) The BdG excitation spectrum in the topological superfluid phase. The inset shows the nine zero-energy states (18 zero energies total shown due to the intrinsic particle-hole symmetry of the BdG equation). (d) The corresponding edge-state wave function along nine tubes (denoted by different colors) of the quasi-1D lattice for one of the zero-energy states. Because of the weak transversal tunneling, each of the nine zero-energy states expands but is distributed unevenly over the nine tubes. Other parameters are $h=1.2t$, $\mu =-2t$ for the topological superfluid phase and $h=1.8t$, $\mu =0$ for the FFLO phase.