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 r0 of the underlying two-body potential and extrapolate to the r0 → 0 limit. Our calculations include all states with vanishing and finite angular momentum L (and natural and unnatural parity Π) with relative energy up to 10.5 Ω, where Ω 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.

Path integral Monte Carlo ground state approach: formalism, implementation, and applications

“…In the (3, 1) system, the angular momenta of each of the two unpaired atoms can couple to an angular momentum 0, 1, or 2, with the parity being even. The (1, +1) channel turns out to have the lowest energy [102]. Last, in the (4, 1) system, the angular momenta of each of the three unpaired atoms can couple to an angular momentum 0, 1, 2, or 3, with the parity being odd.…”

“…Nevertheless, the simple picture correctly suggests that the ground state of the (N − 1, 1) system does not have (L, Π) = (0, +1) symmetry. Rather, the ground state of the (2, 1), (3, 1) and (4, 1) systems has (L, Π) = (1, −1), (1, +1) and (0, −1) symmetry [99,101,102]. Roughly, this can be understood by realizing that the (2, 1), (3, 1) and (4, 1) systems contain one, two and three unpaired spin-up atoms, each of which carry one quantum of angular momentum (the p-shell is being filled).…”

“…The uncertainties for the ECG and DMC calculations are, according to Refs. [102,103,104] Figure 13. Symbols show the contact C of the (3, 1) system as a function of τ −1 .…”

Hyperspherical explicitly correlated Gaussian approach for few-body systems with finite angular momentum

“…. ,γ n−2 ) = sin 2 γ 2 × · · · × sin 2 γ n−2 (44) for n > 3. In the following, we suppress the dependence of H on the angles γ j (j 2) and rewrite f o ( s) such that the dependence on γ 1 is "isolated":…”

“…However, using scale invariance arguments [43], the s ZR 0,unit can be extracted from the energy spectrum of the harmonically trapped (2,2) system. The s ZR 0,unit values for the zero-range system at unitarity, obtained by analyzing the energy spectrum of the trapped system, are s ZR 0,unit = 4.5978 for L = 1 − symmetry and s ZR 0,unit = 4.0820 for L = 1 + symmetry [44]. Our values reported above are in very good agreement with these literature values, indicating that the HECG approach is capable of reliably describing strongly correlated few-body systems with finite angular momentum and positive and negative parity.…”