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arXiv:1205.4815 · Physical Review A

Effective-range dependence of resonantly interacting fermions

Michael McNeil Forbes, Stefano Gandolfi, Alexandros Gezerlis

Published Nov 6, 2012• 65 citations
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Abstract

We extract the leading effective-range corrections to the equation of state of the unitary Fermi gas from ab initio fixed-node quantum Monte Carlo (QMC) calculations in a periodic box using a density functional theory (DFT), and show them to be universal by considering several two-body interactions. Furthermore, we find that the DFT is consistent with the best available unbiased QMC calculations, analytic results, and experimental measurements of the equation of state. We also discuss the asymptotic effective-range corrections for trapped systems and present QMC results with the correct asymptotic scaling.

DOI
10.1103/physreva.86.053603

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We extract the leading effective-range corrections to the equation of state of the unitary Fermi gas from ab initio fixed-node quantum Monte Carlo (QMC) calculations in a periodic box using a density functional theory (DFT), and show them to be universal by considering several two-body interactions. Furthermore, we find that the DFT is consistent with the best available unbiased QMC calculations, analytic results, and experimental measurements of the equation of state. We also discuss the asymptotic effective-range corrections for trapped systems and present QMC results with the correct asymptotic scaling.
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We consider interacting Fermi systems close to the unitary regime and compute the corrections to the energy density that are due to a large scattering length and a small effective range. Our approach exploits the universality of the density functional and determines the corrections from the analyical results for the harmonically trapped two-body system. The corrections due to the finite scattering length compare well with the result of Monte Carlo simulations. We also apply our results to symmetric neutron matter.
We calculate the ground-state properties of an unpolarized two-component Fermi gas with the aid of the diffusion quantum Monte Carlo (DMC) methods. Using an extrapolation to the zero effective range of the attractive two-particle interaction, we find E/EfreeE/{E}_{\mathrm{free}}E/Efree​ in the unitary limit to be 0.212(2), 0.407(2), 0.409(3), and 0.398(3) for 4, 14, 38, and 66 atoms, respectively. Our calculations indicate that the dependence of the total energy on the effective range of the interaction Reff{R}_{\mathrm{eff}}Reff​ is sizable and the extrapolation to Reff=0{R}_{\mathrm{eff}}=0Reff​=0 is therefore important for reaching the true unitary limit. To test the quality of nodal surfaces and to estimate the impact of the fixed-node approximation, we perform released-node DMC calculations for 4 and 14 atoms. Analysis of the released-node and the fixed-node results suggests that the main sources of the fixed-node errors are long-range correlations, which are difficult to sample in the released-node approaches due to the fast growth of the bosonic noise. Besides energies, we evaluate the two-body density matrix and the condensate fraction. We find that the condensate fraction for the 66-atom system converges to 0.56(1) after the extrapolation to the zero interaction range.
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−−22N=2--22N=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 ${a}_{\mathit{TF}}{E}_{\mathit{TF}}+\ensuremath{\Delta}\phantom{\rule{0.2em}{0ex}}\mathrm{mod}(N,2)$ where ETF{E}_{\mathit{TF}}ETF​ is the Thomas-Fermi energy of a noninteracting gas in the trap and $\ensuremath{\Delta}$ 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 $\ensuremath{\Delta}=0.7\ifmmode\pm\else\textpm\fi{}0.2\ensuremath{\omega}$ 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.