Natural and unnatural parity states of small trapped equal-mass two-component Fermi gases at unitarity and fourth-order virial coefficient

Equal-mass two-component Fermi gases under spherically symmetric external harmonic confinement with large $s$-wave scattering length are considered. Using the stochastic variational approach, we determine the lowest 286 and 164 relative eigenenergies of the $(2,2)$ and $(3,1)$ systems at unitarity as a function of the range $r_0$ of the underlying two-body potential and extrapolate to the $r_0\to 0$ limit. Our calculations include all states with vanishing and finite angular momentum $L$ (and natural and unnatural parity $\Pi$) with relative energy up to 10.5 $\hslash \Omega$, where $\Omega$ denotes the angular trapping frequency of the external confinement. Our extrapolated zero-range energies are estimated to have uncertainties of 0.1$\%$ or smaller. The $(2,2)$ and $(3,1)$ energies are used to determine the fourth-order virial coefficient of the trapped unitary two-component Fermi gas in the low-temperature regime. Our results are compared with recent predictions for the fourth-order virial coefficient of the homogeneous system. We also calculate small portions of the energy spectra of the $(3,2)$ and $(4,1)$ systems at unitarity.

Figure 1

Illustration of convergence for the $(3,1)$ system at unitarity with $1^{+}$ symmetry and $r_0=0.04a_{\mathrm{HO}}$. Solid and dashed lines show the quantity $\Delta {\epsilon }_{3,1}^{\mathrm{rel}}$, where $\Delta {\epsilon }_{3,1}^{\mathrm{rel}}=[E_{3,1}^{\mathrm{rel}}\left(N_b\right)-E_{3,1}^{\mathrm{rel}}(N_b\to \infty )]/E_{3,1}^{\mathrm{rel}}(N_b\to \infty )$, for states 1 and 12 as a function of $1/N_b$. Dotted lines show the extrapolation to the $N_b\to \infty$ limit. The inset shows an enlargement of the small $1/N_b$ region.

Figure 2

Illustration of finite-range dependence for the $(3,1)$ system with $1^{+}$ symmetry at unitarity. Squares show the relative eigenenergies $E_{3,1}^{\mathrm{rel}}\left(N_b\right)$ for various ranges $r_0$ of the underlying two-body interaction potential for (a) the ground state (state 1), (b) state 12, and (c) state 5; $N_b$ is the largest basis set considered. The energies provide variational upper bounds and the estimated basis set extrapolation error is indicated by error bars; in panels (a) and (b), the basis set extrapolation error is smaller than the symbol size and, thus, not visible. In panels (a) and (b), solid lines show linear fits to the energies $E_{3,1}^{\mathrm{rel}}\left(N_b\right)$ [the fit shown in panel (a) includes the energies for the five smallest $r_0$ values].

Figure 3

Illustration of finite-range dependence for the $(3,1)$ system with $3^{-}$ symmetry at unitarity. Circles and squares show the relative eigenenergies $E_{3,1}^{\mathrm{rel}}\left(N_b\right)$ for states 15 and 16, respectively, for three different ranges $r_0$ of the underlying two-body interaction potential; $N_b$ is the largest basis set considered (typically, $N_b$ increases with decreasing $r_0/a_{\mathrm{HO}}$). The energies provide variational upper bounds and the estimated basis set extrapolation error is indicated by error bars. Solid lines show linear fits to the energies $E_{3,1}^{\mathrm{rel}}\left(N_b\right)$.

Figure 4

Panels (a) and (b) show the density of states for the $(2,2)$ and $(3,1)$ systems at unitarity; only the relative degrees of freedom are accounted for. The histograms show the number of energies corresponding to shifted states per $\hslash \Omega /4$ while the crosses show the number of energies corresponding to unshifted states. The histograms and the crosses account for the $(2L+1)$ multiplicity of the energies.

Figure 5

Virial coefficient $b_2$ of the trapped two-component Fermi gas at unitarity as a function of the inverse temperature $\tilde{\omega }$. The solid line shows $b_2$, Eq. (17), while the dash-dot-dotted, dash-dotted, and dashed lines show $b_2$ obtained by setting $q_{\max }$ in Eq. (16) to 0, 1, and 10, respectively. The solid horizontal line shows the high-temperature limit $b_2^{\left(0\right)}$.

Figure 6

Virial coefficient $b_3$ of the trapped two-component Fermi gas at unitarity as a function of the inverse temperature $\tilde{\omega }$. The solid line shows $b_3$ with $L_{\max }$ and ${\nu }_{\max }$ set to very large values (see Ref. [29] for details) while the dash-dot-dotted, dash-dotted, and dashed lines show $b_3$ obtained by limiting $L_{\max }$ and ${\nu }_{\max }$ in Eq. (18) such that $s_{L,\nu }\le 11/2$, $\le 19/2$, and $\le 50$, respectively. The solid horizontal line shows the high-temperature limit $b_3^{\left(0\right)}$, Eq. (19).

Figure 7

Virial coefficient $b_4$ of the trapped two-component Fermi gas at unitarity as a function of the inverse temperature $\tilde{\omega }$. The dash-dot-dotted, dash-dotted, and dashed lines show $b_4$ obtained by limiting $s_{L,\nu }$ to be smaller than $11/2$, $15/2$, and $19/2$, respectively. The dotted line shows our attempt to extrapolate to the high-temperature limit; this extrapolation assumes that $b_4$ changes “predictably” from the low- to the medium- to the high-temperature regime. The inset shows the same data as the main figure. In addition, the solid horizontal line shows the high-temperature limit $b_4^{\left(0\right)}$ determined by applying the LDA to the fourth-order virial coefficient predicted for the homogeneous system [26].