Unitary Fermi gas in a harmonic trap

We present an ab initio calculation of small numbers of trapped, strongly interacting fermions using the Green’s function Monte Carlo method. The ground-state energy, density profile, and pairing gap are calculated for particle numbers $N=2–22$ using the parameter-free “unitary” interaction. Trial wave functions are taken in the form of correlated pairs in a harmonic oscillator basis. We find that the lowest energies are obtained with a minimum explicit pair correlation beyond that needed to exploit the degeneracy of oscillator states. We find that the energies can be well fitted by the expression $a_{\mathit{TF}}{E}_{\mathit{TF}}+\Delta \mathrm{mod}(N,2)$ where $E_{\mathit{TF}}$ is the Thomas-Fermi energy of a noninteracting gas in the trap and $\Delta$ is the pairing gap. There is no evidence of a shell correction energy in the systematics, but the density distributions show pronounced shell effects. We find the value $\Delta =0.7\pm 0.2\omega$ for the pairing gap. This is smaller than the value found for the uniform gas at a density corresponding to the central density of the trapped gas.

Figure 1

(Color online) Energy systematics of the trapped unitary Fermi gas. Circles, GFMC results calculated with the HOSD trial function; triangles, GFMC results with the CL trial function; crosses, virial formula. The dotted line is the energy of the HOSD for free particles. The dashed line is the fit [Eq. (6)] to the CL calculated energies. The unit of energies is $\omega$.

Figure 2

(Color online) Radial densities for $N=2$, 8, 14, and 20. For $N=20$, the free-particle density distribution is also shown (dotted line). The unit on the radial axis is $1∕\sqrt{m\omega }$.