Fulde-Ferrell states and Berezinskii-Kosterlitz-Thouless phase transition in two-dimensional imbalanced Fermi gases

We study the superfluid properties of two-dimensional spin-population-imbalanced Fermi gases to explore the interplay between the Berezinskii-Kosterlitz-Thouless (BKT) phase transition and the possible instability towards the Fulde-Ferrell (FF) state. By the mean-field approximation together with quantum fluctuations, we obtain phase diagrams as functions of temperature, chemical potential imbalance, and binding energy. We find that the fluctuations change the mean-field phase diagram significantly. We also address possible effects of the phase separation and/or the anisotropic FF phase to the BKT mechanism. The superfluid density tensor of the FF state is obtained, and its transverse component is found always vanishing. This causes divergent fluctuations and possibly precludes the existence of the FF state at any nonzero temperature.

Figure 1

Framework of the paper, where the yellow oblate indicates a use of an ansatz, the green rectangles indicate approximations, while orange diamonds imply different applications of the theory. The related sections and important equations and figures are indicated by underlined bold font.

Figure 2

Mean-field phase diagrams as functions of $E_b$ and $T$ without the FF ansatz at $h=0.5$ and $h=1$, respectively. The phase boundaries plotted as solid curves correspond to first-order phase transitions, while the dashed curves correspond to continuous phase transitions. The tricritical point is indicated by a brown dot where three different phases meet, i.e., the normal phase (NP) (white), the phase separation (PS) region (between the red and the blue curves), and the superfluid (SF) phase. The contours show the values of superfluid density (in the units of total density, $n=1/2\pi$) in the PS region and the SF phase, which is positive definite and approaches $n/2$ as $T\to 0$ in the SF phase, since then all particles are fully paired. In the PS region it can be larger than $n/2$ because the superfluid only takes part of the spatial volume. We emphasize that the superfluid density shown here is a MF result and its nonzero value does not necessarily mean superfluidity. The phase boundaries agree with the results in Ref. [28].

Figure 3

The thermodynamic potential $\Omega$ (with ${\Omega }_{\mathrm{w}}$ included) as functions of ${\Delta }_0$ at $h=0.5$ and $T=0.1$, with $E_b=0.2$ (red) for the NP with only one minimum at ${\Delta }_0=0$; $E_b=0.24$ (orange) for the NP with an unstable local minimum at ${\Delta }_0\approx 0.68$; $E_b\approx 0.26$ (green) for the NP-PS boundary where two minima ${\Omega }_{\mathrm{N}}$ and $\Omega ({\Delta }_0\approx 0.73)$ are equal; $E_b\approx 0.29$ (blue) for the PS-SF boundary with two equal minima ${\Omega }_{\mathrm{N}}$ and $\Omega ({\Delta }_0\approx 0.74)$; $E_b=0.35$ (purple) for the SF phase with ${\Delta }_0\approx 0.82$ while ${\Omega }_{\mathrm{N}}$ becomes a local minimum; and $E_b=0.6$ (black) for the SF phase with ${\Delta }_0\approx 1.1$ as the only one minimum. Note that when all the particles are in the NP, $\mu$ is completely determined by $T$ and $h$ (here $\mu \approx 1.0$); consequently, ${\Omega }_{\mathrm{N}}$ is constant.

Figure 4

Phase diagrams including the fluctuations. The MF phase boundaries (thin curves) and the tricritical points are shown for comparison. The new PS-SF boundaries (thick blue) show the strong effect of fluctuations. The difference between thick and thin red curves shows the history dependence of the NP-PS boundary. The PS region does not end with a tricritical point but with a region where no solution can be found to satisfy the equilibrium condition, as indicated by pink dotted lines. Besides the solid curves corresponding to first-order phase transitions and the dashed curves to continuous phase transitions, the black dot-dashed curves correspond to the topological BKT phase transition with $T_{\mathrm{BKT}}$ obtained by Eq. (17). The curves of $T_{\mathrm{BKT}}$ bend in the PS region as the corresponding superfluid density increases in the superfluid portion. The colored region shows the values of nonzero order parameter ${\Delta }_0$, but only the part below $T_{\mathrm{BKT}}$ can be taken as a superfluid, while the remaining part is the pseudogap phase where no phase coherence exists and the superfluid density vanishes in accordance with the BKT mechanism. The first-order NP-SF phase boundaries are not quite smooth because of the numerical difficulty to find the exact locations of this phase transition.

Figure 5

Contour plots of ${\Omega }^{T0}$ at $h=0.8$ for various $E_b$ corresponding to different phases at $T=0$. (a) $E_b=0.3$ (NP); (b) $E_b=0.5$ (FF); (c) $E_b=0.6$ (PS${}_{\mathrm{F}}$); (d) $E_b=0.7$ (PS${}_{\mathrm{N}}$); (e) $E_b=0.9$ (SF). For the acronyms of the phases see the text. The global minima are indicated by red dots. Similar results have been shown in Ref. [72]. The pit close to the $Q$ axis in (a) indicates the emergence of a FF state in (b) at finite $Q$. The contour labels are the values of ${\Omega }^{T0}$ divided by $n=1/2\pi$, i.e., the thermodynamic potential per particle.

Figure 6

Phase diagram of a 2D imbalanced Fermi gas at $T=0$. Here NP is the normal phase, PS${}_{\mathrm{F}}$ is the phase separation region with the FF and the superfluid (SF) states coexisting, while in PS${}_{\mathrm{N}}$ the SF phase and the NP coexist. The phase boundaries plotted as solid curves correspond to first-order phase transitions, while the dashed curves correspond to continuous phase transitions. The colors in the FF-related regions show the values of $Q$ of the FF states.

Figure 7

Left: Mean-field phase diagrams as functions of $E_b$ and $T$ with the FF ansatz at $h=0.5$ and $h=0.8$, respectively. The new boundaries after including the FF ansatz are shown by thick curves, while the phase boundaries (thin curves) and the tricritical points in the corresponding non-FF case are shown for comparison. Since in the case with $h=1$ the FF state does not exist at zero temperature (cf. Fig. 6), the case with $h=0.8$ is used instead, where the PS${}_{\mathrm{N}}$ region exists at $T=0$. The phase boundaries plotted as solid curves correspond to first-order phase transitions, while the dashed curves correspond to continuous phase transitions. Middle and right: The density plots for the values of ${\Delta }_0$ and $Q$ of the FF states, zoomed in for the FF-related regions.

Figure 8

$\kappa$, ${\tilde{\rho }}_{xx}$, and ${\tilde{\rho }}_{zz}$ of the FF state as functions of $T$. The parameters are chosen from the FF-PS${}_{\mathrm{F}}$ boundary at $T=0$. $T$ ranges from 0 to where the FF state can still be found.

Figure 9

The same as Fig. 8 but with $Q$ determined by Eq. (21). The transverse superfluid density ${\tilde{\rho }}_{xx}$ is magnified 100 times for the sake of clarity.

Figure 10

By including the FF state, newly obtained PS${}_{\mathrm{F}}$-SF segments (thick green) for various $h$ are added to the phase diagrams of the corresponding non-FF cases. The non-FF results are plotted as in Fig. 4, namely with the red curves for the NP-PS boundaries and the blue curves for the PS-SF boundaries, in which the thin curves are the MF results and the thick ones include the fluctuations. Also, the pink dotted lines indicate the regions where no solution satisfies the phase equilibrium condition, and the black dot-dashed curves are $T_{\mathrm{BKT}}^{\mathrm{SF}}$ obtained in the non-FF case. Because the PS${}_{\mathrm{F}}$-SF segments are quite close to the PS-SF boundaries of the non-FF case, we only focus on the relevant part in each plot to demonstrate the difference clearly. In these plots $h$ ranges from $0.1$ to $0.6$ as the PS${}_{\mathrm{F}}$-SF boundary does not exist when $h\ge 0.7$.