Abstract
We study the virial relations for ultracold trapped two-component Fermi gases in the case of short finite range interactions. Numerical verifications for such relations are reported through the Bardeen–Cooper–Schrieffer (BCS) Bose–Einstein-condensate (BEC) crossover. As an intermediate step, it is necessary to evaluate the partial derivatives of the many-body energy with respect to the inverse of the scattering length and with respect to the interaction range. Once the binding energy of the formed molecules in the BEC side is subtracted, the corresponding energy derivatives are found to have extreme values at the unitary limit. The value of the derivative with respect to the potential range in that limit is large enough to yield measurable differences between the total energy and twice the trapping energy unless the interacting system is described by extremely short potential ranges. The virial results are used to check the quality of the variational wavefunction involved in the calculations.
Highlights
We study the virial relations for ultracold trapped two-component Fermi gases in the case of short finite range interactions
We consider the case where at most one bound state is admitted by the potential and find the ground state of the many-body Schrödinger equation approximately, via a variational Monte Carlo calculation, for several scattering lengths a and short potential ranges rv rho ≡ √h /mω
We show a comparison between the trapping energy curve as predicted by the virial relation, equation (17), and specific values of that energy evaluated directly from the variational functions for a potential range kFrv = 0.0109
Summary
We study 2N fermionic atoms of mass m in two populated hyperfine states (N = N↑ = N↓ = 165) confined by an isotropic three-dimensional harmonic trap of frequency ω, and interacting through an attractive finite range potential V = −|V0|e−r/rv. The ground-state binding energy ε0(rv) of the two interacting particle system in otherwise free space satisfies the equation rv. This expression is easier to verify numerically than (7). That is, for negative scattering lengths shorter than the mean separation between interacting atoms, lower variational energies are obtained using the lengthscaled ground-state solution of the non-interacting problem (which is a product of Slater determinants) multiplied by a Jastrow correlation function β,λJ = FλJJ A↑. =1 unit for measuring the scattering length and potential range in the interacting many-body problem
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