Normal-state properties of a resonantly interacting $p$-wave Fermi gas

Motivated by the recent experimental progress in the study of $p$-wave resonant Fermi gases, we investigate the normal-state properties of such a gas. We calculate the universal equation of state and the two $p$-wave contacts that characterize the universal properties of the system, in good agreement with experiments. Furthermore our calculation obtains the superfluid transition temperature $T_c$ within the Nozières and Schmitt-Rink (NSR) scheme and determines the analytic expressions for $T_c$ in the weak-coupling limit. We show explicitly the nonperturbative nature of the effective range corrections.

Figure 1

(a) Diagrams that contribute to the thermodynamic potential in the Nozières and Schmitt-Rink scheme. The solid lines represent the Green's functions for fermions $G_0^{\mathrm{A}}(\mathbf{k},i{\omega }_n)={[i{\omega }_n-({\epsilon }_k-\mu )]}^{-1}$. The dashed lines represent the Green's function for molecules $G_0^{\mathrm{M}}(\mathbf{q},i{\nu }_n)$ of Eq. (7). The vertex is given by $g_m\left|{\mathbf{k}}_1\right|Y_{1m}\left({\widehat{k}}_1\right)$, indicating the $p$-wave scattering through channel $m$. (b) Schematic diagrams of an actual bound state ($E_b>0$) that is below the scattering threshold when $v_m>0$. (c) For $v_m<0$, there is a quasibound state in the continuum with energy $-E_b$ above the threshold. The scattering energy of two fermions extends to $2E_F$ in a degenerate Fermi gas.

Figure 2

Relative free energy $\mathrm{\Delta }{F}_{xy,z}=F_{xy,z}-F_{xy,z}^0$ as a function of $-E_b^m/E_F$ close to the $xy$ and $z$ resonances at $k_{\mathrm{B}}T=0.8E_F$. Here $F_{xy,z}^0$ is the corresponding free energy for noninteracting system. In our calculation, we have set $k_{F}R=0.04$, appropriate to the experiment. $F_{xy}$ (solid line) is always smaller than that of the $F_z$ (dashed line) due to multiple molecular bound states. Inset shows the chemical potential ${\mu }_{xy,z}$ as a function of $-E_b^m/E_F$ for the same set of parameters.

Figure 3

Contacts $C_v^{xy,z}$ (left) and $C_R^{xy,z}$ (right) as a function of $-E_b^m/E_F$ for $k_{\mathrm{B}}T=0.8E_F$ and $k_{F}R=0.04$. $C_v^{xy}$ (solid line) and $C_v^z$ (dashed line) decrease monotonically from the BEC to the BCS side with $C_v^z$ always smaller than $C_v^{xy}$. On the other hand, both $C_R^{xy}$ (solid line) and $C_R^z$ (dashed line) vanish at resonance $v=\pm \infty$ and depend on $-E_b^m/E_F$ nonmonotonically. The magnitude of $C_R^z$ is always smaller than $C_R^{xy}$.

Figure 4

Comparison with fermion-dimer mixture model shows that, in the resonance region, the prediction of NSR tracks closely that of the mixture model. Blue lines shows the NSR calculation and black dashed line indicates the prediction of mixture model, Eq. (14).

Figure 5

Critical temperature $T_c$ as a function of $-E_b^m/E_F$ for the $xy$ (solid line) and $z$ resonances (dashed line). The dotted line is the asymptotic $T_c$ in the BCS limit for the $z$ resonance given by Eq. (16). Inset shows the respective chemical potentials for the $xy$ and $z$ resonances.