Intermediate-Temperature Superfluidity in an Atomic Fermi Gas with Population Imbalance

We derive the underlying finite temperature theory which describes Fermi gas superfluidity with population imbalance in a homogeneous system. We compute the pair formation temperature, superfluid transition temperature $T_c$, and superfluid density in a manner consistent with the standard ground state equations and, thereby, present a complete phase diagram. Finite temperature stabilizes superfluidity, as manifested by two solutions for $T_c$ or by low $T$ instabilities. At unitarity, the polarized state is an “intermediate-temperature superfluid.”

Figure 1

Mean-field behavior of $T_c^{\mathrm{MF}}$ as a function of $1/k_{F}a$ for different $p$. Shown in the inset is the pairing gap $\Delta (T)$ at different $1/k_{F}a$ for $p=0.3$. Here $E_F\equiv k_B{T}_F\equiv {\hslash }^2{k}_F^2/2m$ is the noninteracting Fermi energy for $p=0$.

Figure 2

$T_c$ as a function of $1/k_{F}a$ for different $p$. The $T_c$ curve splits into two disconnected curves for $0.14\lesssim p\lesssim 0.185$. The $T_c$ solution inside the shaded area is unstable.

Figure 3

Phase diagram on the $p-1/k_{F}a$ plane with the nearly vertical line corresponding to $T_c=0$. Yellow region (shaded) corresponds to intermediate-temperature superfluidity. Region I, the system is normal; IIA, $M^{*}<0$ so no solution for $T_c$ exists; IIB, solution for $T_c$ exists but superfluid state is unstable; IIC, superfluid state exists at intermediate $T$ between upper and lower $T_c$’s but not $T=0$; IID, superfluid state exists at finite $T$ but becomes unstable at a low temperature $T_{\mathrm{unstable}}$ (shown in the inset for $1/k_{F}a=0.5$) where $n_s$ is finite; III, superfluid state exists for all $T\le T_c>0$. The chemical potential $\mu$ changes sign within regions IIB and IID (close to $1/k_{F}a\approx 0.6$ for all $p$), while $n_s(0)$ vanishes along a nearly vertical line (not shown) between $(p,1/k_{F}a)=(1,0.3)$ and $(0,0.6)$. The $p\equiv 0$ boundary is not continuously connected to the rest of the phase diagram.

Figure 4

Normalized superfluid density $n_s/n$ as a function of $T/T_F$ at $p=0.1$ for various $1/k_{F}a$ from BCS to BEC, corresponding to regions IIC ($1/k_{F}a=-0.5$ and 0), IID (0.2, 0.5, 1.0), and III (1.5), respectively. The inset plots $n_{s0}\equiv n_s(T=0)$ versus $1-p$ for different $1/k_{F}a$, indicating $n_{s0}\to 2n_{\downarrow }$ in the BEC limit. The dotted (segments of the) curves represent unstable solutions within region IIB (IID).