Larkin-Ovchinnikov superfluidity in a two-dimensional imbalanced atomic Fermi gas

We present an extensive study of two-dimensional Larkin-Ovchinnikov (LO) superfluidity in a spin-imbalanced two-component atomic Fermi gas. In the context of Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase, we explore a wide and generic class of pairing gap functions with explicit spatial dependency. The mean-field theory of such phases is applied through the Bogoliubov-de Gennes equations in which the pairing gap can be determined self-consistently. To systematically explore the configuration space we consider both the canonical and grand canonical ensembles where we control the polarization or chemical potentials of the system, respectively. The mean-field calculations enable us to understand the nature of the phase transitions in the fully paired Bardeen-Cooper-Schrieffer (BCS) state, exotic LO phase, and partially polarized free Fermi gas ($N_{\text{PP}}$). The order of the phase transitions has been examined and, in particular, we find a weak first-order phase transition between the exotic inhomogeneous LO phase and the BCS phase. In comparison to the three-dimensional case, where the phase diagram is dominated by a generic separation phase, we predict a broader parameter space for the spatially inhomogeneous LO phase. By computing the superfluid density of the LO phase at different polarization, we show how the superfluidity of the system is suppressed with increasing spin polarization.

Figure 1

Thermodynamic potential per particle for the FF Ansatz in two dimensions. The $z$ axis is rescaled by using a logarithmic function $f\left(z\right)=ln(1+\alpha z/\left(2\pi {\varepsilon }_F\right))$, with $\alpha =9.916$.

Figure 2

The energy difference $\delta F=F_{{\text{LO}}_{\text{g}}}-F_{\text{free}}$, for the Helmholtz free energy, $F_{{\text{LO}}_{\text{g}}}$, and the free Fermi gas, $F_{\text{free}}$, at temperature $T/T_F=0$, and interaction strength ${\varepsilon }_B=0.25{\varepsilon }_F$. Rescaled for graphic purposes through the constant $F_0=2\pi \times {10}^{-2}N{\varepsilon }_F$. The transition polarization $p_{c1}\simeq 0.02$, from BCS to ${\text{LO}}_{\text{g}}$, and $p_{c2}=0.58$, from ${\text{LO}}_{\text{g}}$ to $N_{\text{PP}}$, are denoted by bold black vertical lines. The region $p>0.4$ denotes the ${\text{LO}}_{\text{g}}$ phase that is indistinguishable from the ${\mathrm{LO}}_1$ phase.

Figure 3

The inhomogeneous order parameter $\mathrm{\Delta }\left(x\right)/{\varepsilon }_F$ as a function of the $x$ direction (solid line) and the local spin polarization $p\left(x\right)$ (dashed line) calculated via the self-consistent method at temperature $T/T_F=0$ and interaction strength ${\varepsilon }_B=0.25{\varepsilon }_F$, for polarizations (a) $p=0.08$, (b) $p=0.10$, and (c) $p=0.28$.

Figure 4

(a) The weight ${\mathrm{\Delta }}_n$ (in Fermi energy units) of each Fourier cosine mode $n$ of the ${\text{LO}}_{\text{g}}$ phase, calculated according to Eq. (53) at zero temperature and ${\varepsilon }_B=0.25{\varepsilon }_F$, at different spin polarizations $p$. (b) The weight of the main Fourier mode, shown in percentage with respect to the other nonvanishing modes, as a function of the spin polarization. The insets are a guide to show the behavior of the order parameter for a given polarization.

Figure 5

The Helmholtz free-energy difference between the ${\text{LO}}_{\text{g}}$ (for $Q\ne 0$ and BCS at $Q=0$) and the free Fermi gas free energy rescaled by $F_0=2\pi \times {10}^{-2}N{\varepsilon }_F$: (a) at zero temperature varying the polarization and (b) at fixed polarization $p=0.07$ increasing the temperature. The semitransparent bullets indicate the absolute minimum position $Q_{\text{min}}$ for each configuration. The dashed horizontal line in panel (a) is the free Fermi gas free energy: every time the $Q$ is too high the model mimics the free Fermi gas destroying the order parameter. Here, the binding energy is ${\varepsilon }_B=0.25{\varepsilon }_F$.

Figure 6

The behavior of $\mu$ and $\delta \mu$ with respect to their expected behavior as free Fermi gas thermodynamic variables, as a function of the polarization, at zero temperature, density $n=k_F^2/\left(2\pi \right)$ and binding energy ${\varepsilon }_B=0.25{\varepsilon }_F$. The phase transition in the canonical ensemble are denoted by the background color as found in Fig. 2.

Figure 7

The critical phase-transition values of $\delta \mu$ as a function of the binding energy ${\varepsilon }_B$ at zero temperature, $\delta {\mu }_1$ (stars) from the BCS superfluid to ${\text{LO}}_{\text{g}}$ and $\delta {\mu }_2$ (crosses) from ${\text{LO}}_{\text{g}}$ to $N_{\text{PP}}$.

Figure 8

The zero-temperature minima of FF, BCS, and ${\text{LO}}_{\text{g}}$ thermodynamic potentials. The upper panel (a) is for interaction strength ${\varepsilon }_B=0.10{\varepsilon }_F$ and panel (b) is for interaction ${\varepsilon }_B=0.25{\varepsilon }_F$. The critical chemical-potential imbalance, $\delta {\mu }_2$ denotes the ${\text{LO}}_{\text{g}}$ to $N_{\text{PP}}$ transition and ${\mathrm{\Delta }}_0$ is the BCS energy gap at zero temperature. The thermodynamic potentials are defined with respect to the free Fermi gas thermodynamic potential, ${\mathrm{\Omega }}_{\mathrm{free}}$, and scaled by the constant ${\mathrm{\Omega }}_0=2\pi \times {10}^{-3}N{\varepsilon }_F$. In panel (a), the labels from (1) to (4) refer to Fig. 9 and denote the configurations for which we studied the shape of the thermodynamic potential itself.

Figure 9

The thermodynamic potential as a function of $Q$ for interaction strength ${\varepsilon }_B=0.1{\varepsilon }_F$ and chemical-potential difference (a) $\delta \mu =0.3$, (b) $\delta \mu =0.31$, (c) $\delta \mu =0.33$, and (d) $\delta \mu =0.36$. The plots (a)–(d) correspond to points (1)–(4), respectively, in Fig. 8. (e) BCS-${\text{LO}}_{\text{g}}$ transition at higher binding energy ${\varepsilon }_B=0.35{\varepsilon }_F$ to emphasize the first-order nature of the transition. The thermodynamic potential is defined with respect to the free Fermi gas through $\delta \mathrm{\Omega }=\mathrm{\Omega }-{\mathrm{\Omega }}_{\text{free}}$, and scaled by a constant, ${\mathrm{\Omega }}_0=2\pi \times {10}^{-3}N{\varepsilon }_F$.

Figure 10

Critical temperature of the second-order BCS-$N_{\text{PP}}$ (solid) transition, the CC limit (dashed), weak first-order BCS-${\text{LO}}_{\text{g}}$ transition through $\delta {\mu }_1\left(T\right)$ (crosses), and second-order ${\text{LO}}_{\text{g}} N_{\text{PP}}$ through $\delta {\mu }_2\left(T\right)$ (stars). The graph is scaled with respect to the BCS critical temperature $T_c^{\text{BCS}}$ at $\delta \mu =0$ and the BCS pairing gap at zero temperature ${\mathrm{\Delta }}_0$. The inset shows the full phase diagram and the black dashed line is the BKT transition temperature.

Figure 11

Effect of a phase twist imposed on the pairing order parameter. In the parallel direction (red broken line), superfluid current can hardly be produced, while in the perpendicular direction (blue line), superfluid current is allowed to flow, except the regions where the order parameter vanishes and the spin imbalance is high.

Figure 12

The perpendicular Helmholtz free-energy difference $\delta F_{\perp }=F\left({\mathbf{Q}}_{\perp }\right)-F\left(\mathbf{0}\right)$ as a function of the phase twist, for various polarizations. The free-energy variation is calculated by using ${\mathbf{Q}}_{\perp }=Q_s\widehat{y}$ and is plotted in units of a rescaling constant $F_0=2\pi \times {10}^{-3}N{\varepsilon }_F$. The continuous lines are the data fit using a two-parameter polynomial curve $a{x}^2+b{x}^4$.

Figure 13

The superfluid fraction ${\rho }_s/n$ as a function of the polarization $p$ at ${\varepsilon }_B=0.25{\varepsilon }_F$ at zero temperature. The polarization is rescaled by the critical polarization $p_{c2}=0.58$, at which the ${\text{LO}}_{\text{g}}-N_{\text{PP}}$ transition occurs. The green and yellow regions correspond to the ${\text{LO}}_{\text{g}}\to {\mathrm{LO}}_1$ and BCS regions shown in Fig. 2, respectively. The solid line is a guide line.