Fulde-Ferrell-Larkin-Ovchinnikov state of two-dimensional imbalanced Fermi gases

The ground-state phase diagram of attractively interacting Fermi gases in two dimensions with a population imbalance is investigated. We find the regime of stability for the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase, in which pairing occurs at a finite wave vector, and determine the magnitude of the pairing amplitude $\mathrm{\Delta }$ and FFLO wave vector $q$ in the ordered phase, finding that $\mathrm{\Delta }$ can be of the order of the two-body binding energy. Our results rely on a careful analysis of the zero-temperature gap equation for the FFLO state, which possesses nonanalyticities as a function of $\mathrm{\Delta }$ and $q$, invalidating a Ginzburg-Landau expansion in small $\mathrm{\Delta }$.

Figure 1

Top: Phase diagram for 2D imbalanced Fermi gases at fixed total chemical potential $\mu$ and Zeeman field $h$ (normalized to the two-body binding energy ${\epsilon }_b)$. The two curves represent $h_{\mathrm{FFLO}}$ in Eq. (30) and $h_c$ in Eq. (16). Bottom: Phase diagram for fixed total particle number $N=N_{\uparrow }+N_{\downarrow }$ as a function of the binding energy ${\epsilon }_b$ normalized to the Fermi energy. The three curves represent $h_{\mathrm{FFLO}}$ in Eq. (32), $h_{c1}$ in Eq. (17), and $h_{c2}$ in Eq. (18).

Figure 2

Phase diagram at fixed total particle number and fixed polarization $P$ as a function of the binding energy normalized to the Fermi energy. The two curves represent $P_{\mathrm{FFLO}}$ in Eq. (33) and $P_{c2}$ in Eq. (19).

Figure 3

FFLO pairing amplitude $\mathrm{\Delta }$ (top) and wave vector $q$ (bottom) for the case of $\mu =3{\epsilon }_b$. For this case, $h_{\mathrm{FFLO}}=\sqrt{5}{\epsilon }_b\simeq 2.24{\epsilon }_b$ and $h_c\simeq 1.8{\epsilon }_b$. Filled (red) circles derive from a direct numerical minimization of Eq. (6), except for the points $\left(\mathrm{\Delta }=0,q=2\sqrt{{\epsilon }_b}\right)$ at $h=h_{\mathrm{FFLO}}$, following from the continuous nature of the transition. Dashed lines are the approximate formulas, Eq. (84).

Figure 4

Plots of the curves $p_{\pm }\left(\theta \right)$, Eq. (21), for $\mu =1.4$ and $h=0.4$ with $q=0.25$ (so that $q<q_c$; left) and $q=0.4$ (so that $q>q_c$; right). Physically, these correspond to the Fermi surfaces for spin-$\uparrow$ and spin-$\downarrow$ fermions, boosted by $\pm q/2$. The dashed (yellow) curve represents $0={\xi }_{\mathbf{p}-\frac{1}{2}\mathbf{q}\uparrow }+{\xi }_{\mathbf{p}+\frac{1}{2}\mathbf{q}\downarrow }$ (or $p^2=2\tilde{\mu })$, the curve along which the denominator of the first term in Eq. (20) vanishes. For an intermediate $q$ (at $q_c)$, it is clear that the circles will touch at one point; this is the case at the FFLO phase transition.

Figure 5

Top: $S(h,q,\mathrm{\Delta })$ at $\mathrm{\Delta }=0$ for $\mu =4,{\epsilon }_b=1$, and $h=3.5$ [long-dashed (green) curve], $h=3$ [short-dashed (blue) curve], and $h=\sqrt{7}\simeq 2.65$ [solid (red) curve], with the latter showing a finite $q$ pairing instability where $S(h,q,0)=0$. Bottom: $S(h,q,\mathrm{\Delta })$ as a function of $\mathrm{\Delta }$ for two sets of parameters (note that they overlap at small $\mathrm{\Delta }$ and at large $\mathrm{\Delta })$. The upper, black curve represents the case of $\tilde{\mu }=1.3,q=1.8$, and $h=0.28$; the lower (red) curve, the same $\tilde{\mu }$ and $q$ values but a larger $h$ $\left(h=0.5\right)$. These curves show that this function exhibits slope discontinuities at two values of $\mathrm{\Delta }$, which we denote ${\mathrm{\Delta }}_{c1}$ and ${\mathrm{\Delta }}_{c2}$. The equilibrium value of $\mathrm{\Delta }$ in the FFLO state, which satisfies the gap equation $S(h,q,\mathrm{\Delta })=0$, occurs for ${\mathrm{\Delta }}_{c1}<\mathrm{\Delta }<{\mathrm{\Delta }}_{c2}$. Curves in the top panel represent our analytical results; curves in the bottom panel, numerical results for ${\mathrm{\Delta }}_{c1}<\mathrm{\Delta }<{\mathrm{\Delta }}_{c1}$ and analytical results for $\mathrm{\Delta }<{\mathrm{\Delta }}_{c1}$, for $\mathrm{\Delta }>{\mathrm{\Delta }}_{c2}$, and given in Eq. (34) (although these analytical results agree with numerical evaluations).

Figure 6

Left: Where $E_{\mathbf{p}+}<0$ [left shaded (blue) region] and $E_{\mathbf{p}-}<0$ [right shaded (blue) region], with parameters chosen so that $\mathrm{\Delta }<{\mathrm{\Delta }}_{c1}$. Right: The same, but with larger $\mathrm{\Delta }$ such that $\mathrm{\Delta }<{\mathrm{\Delta }}_{c1}$. We chose $\mu =1.4,q=1$, and $h=0.3$ for each (in units such that ${\epsilon }_b=1)$. For these parameters, ${\mathrm{\Delta }}_{c1}\simeq 0.523$. $\mathrm{\Delta }=0.3$ (left); $\mathrm{\Delta }=0.5$ (right).

Figure 7

Plot of the quartic equation discriminant, ${\mathrm{\Delta }}_d\left(\theta \right)$, at angle $\theta =0$ (and multiplied by ${10}^{-6})$ for $\tilde{\mu }=1.3,h=0.6$, and $q=2$ and as a function of $\mathrm{\Delta }$. As shown in Sec. 5c, the qualitative behavior shown here, in which ${\mathrm{\Delta }}_d\left(0\right)$ vanishes at ${\mathrm{\Delta }}_{c1}$ and ${\mathrm{\Delta }}_{c2}$, always occurs in the FFLO regime. Therefore, we can determine ${\mathrm{\Delta }}_{c1}$ and ${\mathrm{\Delta }}_{c2}$ by solving ${\mathrm{\Delta }}_d\left(0\right)=0$ for the pairing amplitude $\mathrm{\Delta }$.

Figure 8

Top: The function $S(h,q,\mathrm{\Delta },T)$ for $\mu =1.3,h=0.5$, and $q=1.8$ (in units such that ${\epsilon }_b=1)$ and $k_{B}T=0.1\mu$. The solid black curve is an exact numerical integration and the dashed (red) curve is the quadratic-order approximation, Eq. (85), with the coefficients calculated numerically, showing a large discrepancy between the gap equation solutions [given by $S(h,q,\mathrm{\Delta },T)=0]$ within the two approaches. Bottom: The same curves, but for a higher temperature $\left(k_{B}T=0.25\mu \right)$, showing that the quadratic-order approximation holds at larger $T$.