Superfluid density and Berezinskii-Kosterlitz-Thouless transition of a spin-orbit-coupled Fulde-Ferrell superfluid

We theoretically investigate the superfluid density and Berezinskii-Kosterlitz-Thouless (BKT) transition of a two-dimensional Rashba spin-orbit-coupled atomic Fermi gas with both in-plane and out-of-plane Zeeman fields. It was recently predicted that, by tuning the two Zeeman fields, the system may exhibit different exotic Fulde-Ferrell (FF) superfluid phases, including the gapped FF, gapless FF, gapless topological FF, and gapped topological FF states. Due to the FF paring, we show that the superfluid density (tensor) of the system becomes anisotropic. When an in-plane Zeeman field is applied along the $x$ direction, the tensor component along the $y$ direction $n_{s,yy}$ is generally larger than $n_{s,xx}$ in most parameter space. At zero temperature, there is always a discontinuity jump in $n_{s,xx}$ as the system evolves from a gapped FF into a gapless FF state. With increasing temperature, such a jump is gradually washed out. The critical BKT temperature has been calculated as functions of the spin-orbit-coupling strength, interatomic interaction strength, and in-plane and out-of-plane Zeeman fields. We predict that the novel FF superfluid phases have a significant critical BKT temperature, typically at the order of $0.1T_F$, where $T_F$ is the Fermi degenerate temperature. Therefore, their observation is within the reach of current experimental techniques in cold-atom laboratories.

Figure 1

Phase diagrams of a 2D spin-orbit-coupled atomic Fermi gas at a broad Feshbach resonance and at a typical low temperature $0.05T_F$ with (a) $h_z=0.1E_F$ or (b) $h_x=0.4E_F$. The strength of spin-orbit coupling is $\lambda =E_F/k_F$. There are four superfluid phases: gFF, nFF, tnFF, and tgFF (whose phase stiffness $\pi \mathcal{J}/2$—in units of $E_F$—is illustrated in color), as well as a pseudogap phase (gray area). We treat the system as a normal gas (shown in white) when the pairing gap $\Delta <{10}^{-3}$. In the gapless topological phase, the notations ${\mathrm{tnFF}}_1,{\mathrm{tnFF}}_2$, and ${\mathrm{tnFF}}_3$ distinguish different zero-energy contours in energy spectrum. For details, see the contour plots in Fig. 2.

Figure 2

Dispersion relation of the lower branch $E_{\eta =2}^{\nu }(k_x,k_y=0)$ (left panel, red curves for particle excitations $\nu =+$ and blue curves for hole excitations $\nu =-$) and the corresponding contour of zero-energy nodes (right panel). (a) and (b) correspond to the red point in Fig. 1 for the ${\mathrm{tnFF}}_1$ phase, (c) and (d) the yellow point for the ${\mathrm{tnFF}}_2$ phase, and (c) and (f) the magenta point for the ${\mathrm{tnFF}}_3$ phase.

Figure 3

Diagonal elements of the superfluid density tensor as a function of $h_x$ and $h_z$ at zero temperature (left panel) and at a finite temperature $T=0.05T_F$ (right panel). The superfluid density is measured in units of the total density $n=k_F^2/\left(2\pi \right)$. In (a) and (b), the out-of-plane Zeeman field strength $h_z=0.1E_F$. In (c) and (d), the in-plane Zeeman field strength $h_x=0.4E_F$. Other parameters are $E_b=0.2E_F$ and $\lambda =E_F/k_F$.

Figure 4

Temperature dependence of the diagonal elements of the superfluid density tensor, at the six points shown in Fig. 1: (a) $E_b=0.21E_F$ and $h_x=0.58E_F$, the ${\mathrm{tnFF}}_1$ phase; (b) $E_b=0.21E_F$ and $h_x=0.71E_F$, the ${\mathrm{tnFF}}_2$ phase; (c) $E_b=0.33E_F$ and $h_x=0.8E_F$, the ${\mathrm{tnFF}}_3$ phase; (d) $E_b=0.4E_F$ and $h_x=0.789E_F$, the tgFF phase; (e) $E_b=0.21E_F$ and $h_x=0.2E_F$, the gFF phase; and (f) $E_b=0.1E_F$ and $h_x=0.33E_F$, the nFF phase. The superfluid density is measured in units of the total density $n=k_F^2/\left(2\pi \right)$. Other parameters are $h_z=0.1E_F$ and $\lambda =E_F/k_F$.

Figure 5

Critical BKT transition temperature as a function of the binding energy $E_b$ at different in-plane Zeeman fields (a) or out-of-plane Zeeman fields (b). Here and in the next two figures, the colors green, red, blue, and yellow in the curves denote the superfluid phase gFF, nFF, tnFF, and tgFF, respectively.

Figure 6

Critical BKT transition temperature as a function of the out-of-plane Zeeman field $h_z$ (a) or the in-plane Zeeman field $h_x$ (b).

Figure 7

Critical BKT transition temperature as a function of the spin-orbit-coupling strength $\lambda$ at different binding energies (a), in-plane Zeeman fields (b), and out-of-plane Zeeman fields (c).

Figure 8

Spin imbalance as a function of the spin-orbit-coupling strength $\lambda$. The inset shows the FF moment $Q$ and the order parameter $\Delta$ at the same setting of parameters.