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

Abstract

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 ${r}_{0}\ensuremath{\rightarrow}0$ limit. Our calculations include all states with vanishing and finite angular momentum $L$ (and natural and unnatural parity $\ensuremath{\Pi}$) with relative energy up to 10.5 $\ensuremath{\hbar}\ensuremath{\Omega}$, where $\ensuremath{\Omega}$ 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.