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arXiv:1608.08411 · Physical Review A

Density-functional theory for resonantly interacting fermions with effective range and neutron matter

Denis Lacroix A

Published Oct 7, 2016• 26 citations
arXivHTMLPDF

Abstract

A density-functional theory is proposed for strongly interacting fermions with arbitrary large negative scattering length. The functional has only two parameters that are directly fixed to reproduce the universal properties of unitary gas: the so-called Bertsch parameter ${\ensuremath{\xi}}_{0}$ and a parameter ${\ensuremath{\eta}}_{e}$ related to the possible influence of the effective range ${r}_{e}$ at infinite scattering length $a$. Using most recent quantum Monte Carlo (QMC) estimates of these two parameters, it is shown that the functional properly reproduces the experimental measurements of interacting Fermi systems not only at unitarity but also away from this limit over a wide range of ${(a{k}_{F})}^{\ensuremath{-}1}$ values. The functional is applied to obtain an expression of Tan's contact parameter including the effect of ${r}_{e}$. Application is finally made to neutron matter. It is shown that most recent QMC results are well reproduced.

DOI
10.1103/physreva.94.043614

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A density-functional theory is proposed for strongly interacting fermions with arbitrary large negative scattering length. The functional has only two parameters that are directly fixed to reproduce the universal properties of unitary gas: the so-called Bertsch parameter \ensuremathξ0{\ensuremath{\xi}}_{0}\ensuremathξ0​ and a parameter \ensuremathηe{\ensuremath{\eta}}_{e}\ensuremathηe​ related to the possible influence of the effective range re{r}_{e}re​ at infinite scattering length aaa. Using most recent quantum Monte Carlo (QMC) estimates of these two parameters, it is shown that the functional properly reproduces the experimental measurements of interacting Fermi systems not only at unitarity but also away from this limit over a wide range of (akF)\ensuremath−1{(a{k}_{F})}^{\ensuremath{-}1}(akF​)\ensuremath−1 values. The functional is applied to obtain an expression of Tan's contact parameter including the effect of re{r}_{e}re​. Application is finally made to neutron matter. It is shown that most recent QMC results are well reproduced.
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 further progress along the line of Ref. [D. Lacroix, Phys. Rev. A 94, 043614 (2016)] where a functional for Fermi systems with anomalously large sss-wave scattering length as{a}_{s}as​ was proposed that has no free parameters. The functional is designed to correctly reproduce the unitary limit in Fermi gases together with the leading-order contributions in the sss- and ppp-wave channels at low density. The functional is shown to be predictive up to densities $\ensuremath{\sim}0.01 {\mathrm{fm}}^{\ensuremath{-}3}$ that is much higher densities compared to the Lee-Yang functional, valid for $\ensuremath{\rho}<{10}^{\ensuremath{-}6} {\mathrm{fm}}^{\ensuremath{-}3}$. The form of the functional retained in this work is further motivated. It is shown that the new functional corresponds to an expansion of the energy in (askF)({a}_{s}{k}_{F})(as​kF​) and (rekF)({r}_{e}{k}_{F})(re​kF​) to all orders, where re{r}_{e}re​ is the effective range and kF{k}_{F}kF​ is the Fermi momentum. One conclusion from the present work is that, except in the extremely low-density regime, nuclear systems can be treated perturbatively in $\ensuremath{-}{({a}_{s}{k}_{F})}^{\ensuremath{-}1}$ with respect to the unitary limit. Starting from the functional, we introduce density-dependent scales and show that scales associated with the bare interaction are strongly renormalized by medium effects. As a consequence, some of the scales at play around saturation are dominated by the unitary gas properties and not directly by low-energy constants. For instance, we show that the scale in the sss-wave channel around saturation is proportional to the so-called Bertsch parameter ${\ensuremath{\xi}}_{0}$ and becomes independent of as{a}_{s}as​. We also point out that these scales are of the same order of magnitude than those empirically obtained in the Skyrme energy density functional. We finally propose a slight modification of the functional such that it becomes accurate up to the saturation density $\ensuremath{\rho}\ensuremath{\simeq}0.16 {\mathrm{fm}}^{\ensuremath{-}3}$.
We review the properties of neutron matter in the low-density regime. In particular, we revise its ground state energy and the superfluid neutron pairing gap and analyze their evolution from the weak to the strong coupling regime. The calculations of the energy and the pairing gap are performed, respectively, within the Brueckner–Hartree–Fock (BHF) approach of nuclear matter and the Bardeen–Cooper–Schrieffer (BCS) theory using the chiral nucleon-nucleon interaction of Entem and Machleidt at N3LO and the Argonne V18 phenomenological potential. Results for the energy are also shown for a simple Gaussian potential with a strength and range adjusted to reproduce the1S0neutron-neutron scattering length and effective range. Our results are compared with those of quantum Monte Carlo (QMC) calculations for neutron matter and cold atoms. The Tan contact parameter in neutron matter is also calculated, finding a reasonable agreement with experimental data from ultra-cold atoms only at very low densities. We find that low-density neutron matter exhibits a behavior close to that of a Fermi gas at the unitary limit, although, this limit is actually never reached. We also review the properties (energy, effective mass, and quasiparticle residue) of a spin-down neutron impurity immersed in a low-density free Fermi gas of spin-up neutrons already studied by the author in a recent work where it was shown that these properties are very close to those of an attractive Fermi polaron in the unitary limit.
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.
We elucidate universal many-body properties of a one-dimensional, two-component ultracold Fermi gas near the ppp-wave Feshbach resonance. The low-energy scattering in this system can be characterized by two parameters, that is, ppp-wave scattering length and effective range. At the unitarity limit where the ppp-wave scattering length diverges and the effective range is reduced to zero without conflicting with the causality bound, the system obeys universal thermodynamics as observed in a unitary Fermi gas with contact sss-wave interaction in three dimensions. It is in contrast to a Fermi gas with the ppp-wave resonance in three dimensions in which the effective range is inevitably finite. We present the universal equation of state in this unitary ppp-wave Fermi gas within the many-body TTT-matrix approach as well as the virial expansion method. Moreover, we examine the single-particle spectral function in the high-density regime where the virial expansion is no longer valid. On the basis of the Hartree-like self-energy shift at the divergent scattering length, we conjecture that the equivalence of the Bertsch parameter across spatial dimensions holds even for a one-dimensional unitary ppp-wave Fermi gas.
In the unitary regime, fermions interact strongly via two-body potentials that exhibit a zero range and a (negative) infinite scattering length. The energy density is proportional to the energy density of the free Fermi gas with a proportionality constant $\ensuremath{\xi}$. The author uses a simple density functional parametrized by an effective mass and the universal constant $\ensuremath{\xi}$, and employs Kohn-Sham density-functional theory to obtain the parameters from fit to one exactly solvable two-body problem. This yields $\ensuremath{\xi}=0.42$ and a rather large effective mass. The form of the density functional is checked by similar Kohn-Sham calculations for the exactly solvable Calogero model.