Precursor of superfluidity in a strongly interacting Fermi gas with negative effective range

We investigate theoretically the effects of pairing fluctuations in an ultracold Fermi gas near a Feshbach resonance with a negative effective range. By employing a many-body $T$-matrix theory with a coupled fermion-boson model, we show that the single-particle density of states exhibits the so-called pseudogap phenomenon, which is a precursor of superfluidity induced by strong pairing fluctuations. We clarify the region where strong pairing fluctuations play a crucial role in single-particle properties, from the broad-resonance region to the narrow-resonance limit at the divergent two-body scattering length. We also extrapolate the effects of pairing fluctuations to the positive-effective-range region from our results near the narrow Feshbach resonance. Results shown in this paper are relevant to the connection between ultracold Fermi gases and low-density neutron matter from the viewpoint of finite-effective-range corrections.

Figure 1

Self-energy corrections of (a) Fermi atoms and (b) diatomic molecules. The single line ($G_0$) and double line ($D$) represent Green's functions of noninteracting atoms and dressed molecules, respectively. The shaded circle is the Feshbach coupling $g_{\mathrm{r}}$.

Figure 2

(a) Superfluid phase transition temperature $T_{\mathrm{c}}$ and (b) critical chemical potential ${\mu }_{\mathrm{c}}$ in the whole BCS-BEC crossover regime with the finite effective range $r_{\mathrm{e}}$. The quantity ${\tilde{g}}_{\mathrm{r}}=g_{\mathrm{r}}\sqrt{N}/{\varepsilon }_{\mathrm{F}}$ is the dimensionless Feshbach coupling.

Figure 3

Superfluid phase transition temperature $T_{\mathrm{c}}$ at various Feshbach couplings as a function of $1/k_{\mathrm{F}}{a}_{\mathrm{eff}}$, where $a_{\mathrm{eff}}$ is the effective scattering length defined by Eq. (16).

Figure 4

Effective range dependence of the inverse effective scattering length $1/k_{\mathrm{F}}{a}_{\mathrm{eff}}$ at $1/k_{\mathrm{F}}a=-0.5$ (dashed line), 0 (dotted line), and 0.5 (solid line) at $T=T_{\mathrm{c}}$. The inset shows $1/k_{\mathrm{F}}{a}_{\mathrm{eff}}$ as a function of the dimensionless Feshbach coupling ${\tilde{g}}_{\mathrm{r}}$. In each figure, we use the same line style at each scattering length.

Figure 5

Contributions to the particle number as a function of the dimensionless Feshbach coupling ${\tilde{g}}_{}$ at (a) $1/k_{\mathrm{F}}a=-0.5$, (b) $1/k_{\mathrm{F}}a=0$, and (c) $1/k_{\mathrm{F}}a=0.5$ at $T=T_{\mathrm{c}}$. The particle number of diatomic molecules $2N_{\mathrm{b}}/N$ (solid line) and the Fermi atomic number $2N_{\mathrm{f}}/N$ (dash-dotted line) are shown. We also compare the fluctuation corrections of the Fermi atomic number $2\delta N_{\mathrm{f}}/N$ (dashed line) and noninteracting contributions $2N_{\mathrm{f}}^0/N$ (dotted line). In each figure, we use the same line style.

Figure 6

Single-particle density of states $\rho \left(\omega \right)$ at $T=T_{\mathrm{c}}$ with various effective ranges: (a) $1/k_{\mathrm{F}}a=-0.5$, (b) $1/k_{\mathrm{F}}a=0$, and (c) $1/k_{\mathrm{F}}a=0.5$. Here ${\rho }_0\left(0\right)$ is the single-particle density of states at the Fermi level in an ideal Fermi gas at $T=0$.

Figure 7

Single-particle density of states $\rho \left(\omega \right)$ at $1/k_{\mathrm{F}}a=0$ at different temperatures. In each figure, the solid, dotted, and dashed lines represent the results of $T=T_{\mathrm{c}}, T_{\mathrm{pg}}$, and $0.5T_{\mathrm{F}}$, respectively. Here $T_{\mathrm{pg}}$ is the pseudogap temperature defined as the temperature where the dip structure in $\rho \left(\omega \right)$ disappears. The effective range is set to (a) $r_{\mathrm{e}}{k}_{\mathrm{F}}=-0.034$ (${\tilde{g}}_r=10$), (b) $r_{\mathrm{e}}{k}_{\mathrm{F}}=-3.40$ (${\tilde{g}}_r=1$), and (c) $r_{\mathrm{e}}{k}_{\mathrm{F}}=-16.3$ (${\tilde{g}}_r=0.5$).

Figure 8

Superfluid phase transition temperature $T_{\mathrm{c}}$ (solid line) and pseudogap temperature $T_{\mathrm{pg}}$ (dashed line) as a function of the effective range $r_{\mathrm{e}}$ at $1/k_{\mathrm{F}}a=0$. The dotted line shows the linear fitting of $T_{\mathrm{pg}}$ in the small negative effective range region ($|r_{\mathrm{e}}{k}_{\mathrm{F}}|\lesssim 0.05$). The inset shows $T_{\mathrm{c}}$ and $T_{\mathrm{pg}}$ versus the dimensionless Feshbach coupling ${\tilde{g}}_{\mathrm{r}}$.

Figure 9

The $\delta$ dependence of the single-particle density of states at $T=T_{\mathrm{c}}$. The inverse scattering length and the effective range are given by $1/k_{\mathrm{F}}a=-0.5$ and $r_{\mathrm{e}}{k}_{\mathrm{F}}=-0.034$ (where $k_{\mathrm{F}}$ is the Fermi momentum), respectively. Here ${\rho }_0\left(0\right)$ is the single-particle density of states at the Fermi level in an ideal Fermi gas at $T=0$.