Strongly interacting Sarma superfluid near orbital Feshbach resonances

We investigate the nature of superfluid pairing in a strongly interacting Fermi gas near orbital Feshbach resonances with spin-population imbalance in three dimensions, which can be well described by a two-band or two-channel model. We show that a Sarma superfluid with gapless single-particle excitations is favored in the closed channel at large imbalance. It is thermodynamically stable against the formation of an inhomogeneous Fulde-Ferrell-Larkin-Ovchinnikov superfluid and features a well-defined Goldstone-Anderson-Bogoliubov phonon mode and a massive Leggett mode as collective excitations at low momentum. At large momentum, the Leggett mode disappears and the phonon mode becomes damped at zero temperature, due to the coupling to the particle-hole excitations. We discuss possible experimental observation of a strongly interacting Sarma superfluid with ultracold alkaline-earth-metal Fermi gases.

Figure 1

The landscape of thermodynamic potential (in units of $N{\varepsilon }_F$) in the ${\mathrm{\Delta }}_c-{\mathrm{\Delta }}_o$ plane, in a three-dimensional plot (a) or in a contour plot (b). Here, we take $1/k_F{a}_{s0}=-0.5$ and $1/k_F{a}_{s1}=-0.05,\mu =0.6{\varepsilon }_F$, $\delta \left(B\right)=0$, and $\delta \mu =0.5{\varepsilon }_F$. In the contour plot, the different competing phases corresponding to the local minima are indicated. The two channels are symmetric due to zero detuning. In (b), the in-phase BCS phase, where the two order parameters have the same sign, is an excited state with an energy much higher than other local minima. We note that, the FFLO pairing is not allowed with this set of interaction parameters (i.e., too strong interband coupling).

Figure 2

The contour plots of thermodynamic potential in the ${\mathrm{\Delta }}_c-{\mathrm{\Delta }}_o$ plane, with increasing detuning: (a) $\delta \left(B\right)=0.2{\varepsilon }_F$, (b) $\delta \left(B\right)=0.4{\varepsilon }_F$, and (c) $\delta \left(B\right)=0.6{\varepsilon }_F$. The color shows the magnitude of the thermodynamic potential ${\mathrm{\Omega }}_{\mathrm{MF}}$, from blue (small ${\mathrm{\Omega }}_{\mathrm{MF}}$) to red (large ${\mathrm{\Omega }}_{\mathrm{MF}}$). Various stable phases (i.e., local minima) are indicated. As the detuning increases, the pairing type in the closed channel changes from the BCS to the Sarma pairing. Here, we take $1/k_F{a}_{s0}=-0.5$ and $1/k_F{a}_{s1}=-0.05,\mu =0.6{\varepsilon }_F$ and $\delta \mu =0.5{\varepsilon }_F$.

Figure 3

The landscape of thermodynamic potential (in units of $N{\varepsilon }_F$) in the $Q_o-{\mathrm{\Delta }}_o$ plane, in a three-dimensional plot (a) or in a zoom-in contour plot to better view the extremely shallow FF solution (b). Here, we use $1/k_F{a}_{s0}=-0.5$ and $1/k_F{a}_{s1}=-5.0$, $\mu =0.8{\varepsilon }_F,\delta \left(B\right)=0.4{\varepsilon }_F$, and $\delta \mu =0.235{\varepsilon }_F$.

Figure 4

The landscape of thermodynamic potential in the ${\mathrm{\Delta }}_c-{\mathrm{\Delta }}_o$ plane, with (a) $Q_o=0$ and (b) $Q_o\simeq 0.326k_F$.

Figure 5

Thermodynamic potentials of different completing phases, relative to the ideal Fermi gas result ${\mathrm{\Omega }}_{IG}$ and in units of $N{\varepsilon }_F$, as a function of the chemical potential difference $\delta \mu$. The two arrows indicate the positions of the two phase transitions at $\delta {\mu }_{c1}\simeq 0.543{\mathrm{\Delta }}_o\left(0\right)$ and $\delta {\mu }_{c2}\simeq 0.662{\mathrm{\Delta }}_o\left(0\right)$, respectively. Here, we take $1/k_F{a}_{s0}=-0.5$ and $1/k_F{a}_{s1}=-5.0$, $\mu =0.8{\varepsilon }_F$, and $\delta \left(B\right)=0.4{\varepsilon }_F$. At $\delta \mu =0$, the pairing gap in the open channel of the low-energy out-of-phase solution is ${\mathrm{\Delta }}_o\left(0\right)=0.3898{\varepsilon }_F$.

Figure 6

Free energies of different completing phases (in units of $N{\varepsilon }_F$) as a function of the chemical potential difference $\delta \mu$, at a fixed total density $\rho =3{\pi }^2{k}_F^3$. The ${\left[\mathrm{BCS}\right]}_o$ ${\left[\mathrm{Sarma}\right]}_c$ state becomes favorable above the threshold $\delta {\mu }_c\simeq 0.486{\varepsilon }_F$. Here, we use $1/k_F{a}_{s0}=-0.5$ and $1/k_F{a}_{s1}=-0.05$, and $\delta \left(B\right)=0.4{\varepsilon }_F$.

Figure 7

(a)–(j) The spectral function of Cooper pairs $-\mathrm{Im}{\mathrm{\Gamma }}_{11}(q,\omega )$ at different transferred momenta, which increase from $0.1k_F$ to $1.0k_F$ with a step $0.1k_F$. Here, we consider a balanced BCS state with a fixed total density $\rho =3{\pi }^2{k}_F^3$ and $\delta \mu =0$. The detuning is $\delta \left(B\right)=0.4{\varepsilon }_F$. (k) The dispersion relation of the phonon (black circles) and Leggett modes (red squares). The shaded area in cyan color shows the two-particle excitation continuum. The error bar in red squares in the two-particle continuum indicates the damping width of the peak, due to the coupling to fermionic quasiparticles. The straight line shows $\omega =c_{s}q$, where $c_s\simeq 0.348v_F$ is the sound velocity.

Figure 8

(a)–(j) The spectral function of Cooper pairs $-\mathrm{Im}{\mathrm{\Gamma }}_{11}(q,\omega )$ at different transferred momenta, which increase from $0.1k_F$ to $1.0k_F$ with a step $0.1k_F$. Here, we take the parameters $\delta \left(B\right)=0.4{\varepsilon }_F$ and $\delta \mu =0.5{\varepsilon }_F$ and consider the case of a fixed total density $\rho =3{\pi }^2{k}_F^3$, which lead to a Sarma state in the closed channel with ${\mu }_c=\mu -\delta \left(B\right)/2\simeq 0.25{\varepsilon }_F$ and ${\mathrm{\Delta }}_c\simeq -0.08{\varepsilon }_F$. (k) The dispersion relation of the phonon (black circles) and Leggett modes (red squares). The shaded areas in cyan and yellow colors correspond to the two-particle [Eq. (A2)] and particle-hole excitation continua [Eq. (B2)], respectively. The error bar in symbols indicates the full width at half maximum of the peak in the spectral function. As the momentum increases, the Leggett peak shrinks gradually and disappears at $q\sim 0.4k_F$. The straight line shows $\omega =c_{s}q$, where $c_s\simeq 0.380v_F$ is the sound velocity.

Figure 9

The dispersion relation $E_{c\mathbf{k},+}=\sqrt{{\xi }_{c\mathbf{k}}^2+{\mathrm{\Delta }}_c^2}-\delta \mu$ of the lower branch of fermionic quasiparticles in the closed channel (black curve) and the square of the coherence factor ${\left(u_{c\mathbf{k}}{v}_{c\mathbf{k}}\right)}^2$ (red curve). Here, we take the parameters $\delta \left(B\right)=0.4{\varepsilon }_F$ and $\delta \mu =0.5{\varepsilon }_F$ and consider the case of a fixed total density $\rho =3{\pi }^2{k}_F^3$, which lead to a Sarma state in the closed channel with ${\mu }_c=\mu -\delta \left(B\right)/2\simeq 0.25{\varepsilon }_F$ and ${\mathrm{\Delta }}_c\simeq -0.08{\varepsilon }_F$. The dispersion relation has gapless particle-hole excitations near the wave vectors $\left|\mathbf{k}\right|=k_{\mathrm{ph}}\simeq 0.83k_F$. The factor ${\left(u_{c\mathbf{k}}{v}_{c\mathbf{k}}\right)}^2$ peaks near the minimum of the dispersion relation ($k_c\simeq 0.5k_F$).