Thermodynamics of the two-component Fermi gas with unequal masses at unitarity

We consider mass-imbalanced two-component Fermi gases for which the unequal-mass atoms interact via a zero-range model potential with a diverging $s$-wave scattering length $a_s$, that is, with $1/a_s=0$. The high-temperature thermodynamics of the harmonically trapped and homogeneous systems are examined using a virial expansion approach up to third order in the fugacity. We find that the universal part of the third-order virial coefficient associated with two light atoms and one heavy atom is negative, while that associated with two heavy and one light atom changes sign from negative to positive as the mass ratio $\kappa$ increases and diverges when Efimov physics sets in at $\kappa =13.61$. By examining the Helmholtz free energy, we find that the equilibrium polarization of the trapped and homogeneous systems is 0 for $\kappa =1$, but finite for $\kappa \ne 1$ (with a majority of heavy particles). Compared to the equilibrium polarization of the noninteracting system, the equilibrium polarization at unitarity is increased for the trapped system and decreased for the homogeneous system. We find that unequal-mass Fermi gases are stable for all polarizations.

Figure 1

Hyperangular eigenvalues $s_{l,\nu }$ as a function of $\overline{\kappa }$ for (a) $l=0$, (b) $l=1$, and (c) $l=2$ at unitarity. Note that $\overline{\kappa }$ is shown on a log scale. Solid lines show the $s_{l,\nu }$ obtained by solving Eq. (16); dashed lines show the limiting expressions, Eqs. (17)–(20), for $\overline{\kappa }\ll 1$; and thin dotted lines show the noninteracting $s_{l,\nu }^{\mathrm{NI}}$.

Figure 2

Angular momentum contributions $b_{2,1}\left(l\right)$ for $l=0–4$ for (a) $\overline{\kappa }=1$ and (b) $\overline{\kappa }=6.67$ as a function of $\tilde{\omega }$ at unitarity. Solid, dotted, dashed, dash-dotted, and dash-dot-dotted lines show $b_{2,1}\left(l\right)$ for $l=0$, 1, 2, 3, and 4, respectively.

Figure 3

Solid, dashed, dash-dotted, and dash-dot-dotted lines show the virial coefficient $b_{2,1}$ as a function of $\tilde{\omega }$ for $\overline{\kappa }=2$, 3, 4, and 6.67. The thin dotted line at 0 is a guide to emphasize the sign change of $b_{2,1}$ for $\overline{\kappa }=4$ around $\tilde{\omega }\approx 1.67$.

Figure 4

Circles show the universal parts $b_{1,2}^{\left(0\right)}$ ($\overline{\kappa }<1$) and $b_{2,1}^{\left(0\right)}$ ($\overline{\kappa }>1$) of the virial coefficients as a function of $\overline{\kappa }$. The inset (the axes use the same labels as the main panel) shows the first nonuniversal corrections $b_{1,2}^{\left(2\right)}$ and $b_{2,1}^{\left(2\right)}$ (squares) and the second nonuniversal corrections $b_{1,2}^{\left(4\right)}$ and $b_{2,1}^{\left(4\right)}$ (triangles) as a function of $\overline{\kappa }$. The lines are guides to the eye.

Figure 5

Trapped system at unitarity. Solid and dashed lines (dash-dotted and dotted lines) show the fugacities $z_1$ and $z_2$, respectively, for $\kappa =1$ ($\kappa =13$) as a function of the polarization $P$ for $N=5\times {10}^7$ and (a) $T/T_F=1$ and (b) $T/T_F=1/2$.

Figure 6

Free energy per particle $F/N$ ($N=5\times {10}^7$) as a function of the polarization $P$ for (a) $T/T_F=1$ and (b) $T/T_F=1/2$ for the trapped system at unitarity. Alternating solid and dotted curves, from top to bottom, show the free energy for $\kappa =1–13$ in units of 1. The dashed lines connect the $(F_{\mathrm{eq}},P_{\mathrm{eq}})$ points for different $\kappa$.

Figure 7

Thermodynamic observables $F/N$, $S/N$, and $U/N$ as a function of the polarization $P$ for $\kappa =6.67$ and $N=5\times {10}^7$ for the trapped system at unitarity. Alternating dotted and solid lines show (a) the free energy per particle $F/N$ for $T/T_F=1,1.2,...,1.8$ (from top to bottom), (b) the entropy per particle $S/N$ for $T/T_F=0.95,0.96,...,1.04$ (from bottom at $P\approx 0$ to top at $P\approx 0.5$), and (c) the energy per particle $U/N$ for $T/T_F=0.99,1,...,1.06$ (from bottom at $P\approx 0.5$ to top at $P\approx 0.1$). In panel (a), the dashed line connects $(F_{\mathrm{eq}},P_{\mathrm{eq}})$ values for different $T$. In panels (b) and (c), the dashed lines show $S/N$ for $U/N=2.91E_F$ and $U/N$ for $S/N=5.73k_B$, respectively. In all three panels, the vertical lines indicate the value of $P_{\mathrm{eq}}$ for $T/T_F=1$.

Figure 8

Percent difference between $P_{\mathrm{eq}}$ for $N=5\times {10}^7$ and $P_{\mathrm{eq}}$ for $N=500$ as a function of the mass ratio $\kappa$ for the trapped system at unitarity. Solid, dashed, and dotted lines show the percent difference for $T/T_F=1/2$, 1, and 2, respectively.

Figure 9

Fugacities $z_1$ and $z_2$ of the homogeneous system as a function of the polarization $P$ for (a) $\kappa =1$ and $T/T_F=6.67$, (b) $\kappa =6.67$ and $T/T_F=6.67$, (c) $\kappa =1$ and $T/T_F=2$, and (d) $\kappa =2$ and $T/T_F=2$. Dotted and dash-dotted lines show $z_1$ and $z_2$, respectively, for the unitary system obtained by solving Eqs. (44) and (45) self-consistently. For comparison, solid and dashed lines show $z_1$ and $z_2$, respectively, for the noninteracting system.

Figure 10

Free energy per particle $F/N$ of the homogeneous system as a function of the polarization $P$ for $T/T_F=10$ and mass ratios $\kappa =1$, 2, and 6.67. The curves are grouped in sets of two, each labeled by their corresponding mass ratio. The solid lines are calculated using the virial equation of state at unitarity. For comparison, the dotted lines show $F/N$ for the noninteracting system.

Figure 11

Equilibrium polarization $P_{\mathrm{eq}}$ of the homogeneous system at unitarity as a function of the mass ratio $\kappa$ for $T/T_F=10$. The dashed and solid lines show $P_{\mathrm{eq}}$ calculated using the virial equation of state up to second and third order, respectively. The dashed and solid lines are nearly indistinguishable on the scale shown. For comparison, the dotted line shows $P_{\mathrm{eq}}$ for the noninteracting system. The solid line in the inset shows the difference between the $P_{\mathrm{eq}}$'s calculated using the virial equation of state up to the third and second orders (the dashed line shows zero as a reference).

Figure 12

Equilibrium polarization $P_{\mathrm{eq}}$ of the homogeneous system at unitarity as a function of the temperature $T$ for $\kappa =6.67$. The dashed and solid lines show $P_{\mathrm{eq}}$ calculated using the virial equation of state up to second and third order, respectively. The dashed and solid lines are nearly indistinguishable on the scale shown. For comparison, the dotted line shows $P_{\mathrm{eq}}$ for the noninteracting system. The solid line in the inset shows the difference between the $P_{\mathrm{eq}}$'s calculated using the virial equation of state up to the third and second orders.

Figure 13

Virial coefficients $b_n$ of the single-component harmonically trapped Fermi gas as a function of $\tilde{\omega }$. Solid, dotted, dashed, dash-dotted, and dash-dot-dotted lines show $b_2$, $b_3$, $b_4$, $b_5$, and $b_6$, respectively.

Figure 14

Contour plot of the percent difference $100(z+2b_2{z}^2+3b_3{z}^3-N/Q_1)/(N/Q_1)$. The dashed line shows the percent difference of 1. The solid lines below the dashed line show the percent difference in steps of powers of 10 down to a value of 0.001, while solid lines above the dashed line show the percent difference in steps of 1 up to a value of 7. We consider the region where the percent difference is less than 1 to be well converged.