Gapless Topological Fulde-Ferrell Superfluidity in Spin-Orbit Coupled Fermi Gases

Topological superfluids usually refer to a superfluid state which is gapped in the bulk but metallic at the boundary. Here we report that a gapless, topologically nontrivial superfluid with an inhomogeneous Fulde-Ferrell pairing order parameter can emerge in a two-dimensional spin-orbit coupled Fermi gas, in the presence of both in-plane and out-of-plane Zeeman fields. The Fulde-Ferrell pairing—induced by the spin-orbit coupling and in-plane Zeeman field—is responsible for this gapless feature. This exotic superfluid has a significant Berezinskii-Kosterlitz-Thouless transition temperature and has robust Majorana edge modes against disorder owing to its topological nature.

Figure 1

Low-temperature phase diagram of a Rashba spin-orbit coupled 2D Fermi gas at $E_b=0.2E_F$, $\lambda =E_F/k_F$, and $T=0.05T_F$. By tuning the out-of-plane and in-plane Zeeman fields, ${\mathrm{\Omega }}_R$ and $\delta$, the system may evolve from a gapped FF (gFF) to a nodal FF (nFF), then a gapless topological FF (tnFF), and, finally, to a gapped topological FF (tgFF). Above the dot-dashed line, the order parameter $\mathrm{\Delta }$ becomes less than $0.001E_F$ (therefore indicated as a normal phase $N$). The dashed line indicates the boundary where the critical BKT temperature is reached. In between, the pseudogap regime is highlighted by the shadowed area. The color map shows the phase stiffness $\pi \mathcal{J}/2$, in units of $E_F$.

Figure 2

(a) Evolution of the minimum excitation gap $E_g$ and the Chern number with increasing ${\mathrm{\Omega }}_R$ at a fixed $\delta =0.4E_F$. The inset shows the evolution of the FF pairing momentum $Q$ and pairing gap $\mathrm{\Delta }$. (b) Dispersion relation of the lower helicity branch $E_{\eta =2}^{\nu }(k_x,k_y=0)$ across the topological transition at ${\mathrm{\Omega }}_R/E_F=0.24$ (green thick line), 0.325 (red dot-dashed line), and 0.38 (blue thin line). (c) The corresponding contours of zero-energy energy spectrum (nodal points) on the $k_x\text{−}{k}_y$ plane.

Figure 3

Contour plots of the Berry curvature ${\mathrm{\Gamma }}^{\nu =-}\equiv {\gamma }_1^{\nu =-}+{\gamma }_2^{\nu =-}$ before [(a), ${\mathrm{\Omega }}_R=0.24E_F$] and after [(b), ${\mathrm{\Omega }}_R=0.38E_F$] the topological phase transition at an in-plane field $\delta =0.4E_F$.

Figure 4

Energy spectrum of a 2D strip formed by imposing open boundary condition (i.e., hard wall confinement) in the $y$ direction. The edge states due to the confinement are highlighted by the red lines. The parameters are $\delta =0.4E_F$ and ${\mathrm{\Omega }}_R=0.24E_F$ (a) or ${\mathrm{\Omega }}_R=0.38E_F$ (b), the same as in Fig. 3.

Figure 5

Finite-temperature phase diagram as a function of ${\mathrm{\Omega }}_R$ at $\delta =0.4E_F$. The solid line shows the BKT critical temperature and the dashed lines give the boundaries between different phases (see the lower panel for the dependence over the entire temperature scale).