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arXiv:nucl-th/0509018 · Physical Review C

Cold dilute neutron matter on the lattice. II. Results in the unitary limit

Dean Lee, Thomas Schäfer

Published Jan 18, 2006• 73 citations
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Abstract

This is the second of two articles that investigate cold dilute neutron matter on the lattice using pionless effective field theory. In the unitary limit, where the effective range is zero and scattering length is infinite, simple scaling relations relate thermodynamic functions at different temperatures. When the second virial coefficient is properly tuned, we find that the lattice results obey these scaling relations. We compute the energy per particle, pressure, spin susceptibility, dineutron correlation function, and an upper bound for the superfluid critical temperature.

DOI
10.1103/physrevc.73.015202

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This is the second of two articles that investigate cold dilute neutron matter on the lattice using pionless effective field theory. In the unitary limit, where the effective range is zero and scattering length is infinite, simple scaling relations relate thermodynamic functions at different temperatures. When the second virial coefficient is properly tuned, we find that the lattice results obey these scaling relations. We compute the energy per particle, pressure, spin susceptibility, dineutron correlation function, and an upper bound for the superfluid critical temperature.
We study cold dilute neutron matter on the lattice using an effective field theory. We work in the unitary limit in which the scattering length is much larger than the interparticle spacing. In this article we focus on the equation of state at temperatures above the Fermi temperature and compare lattice simulations to the virial expansion on the lattice and in the continuum. We find that in the unitary limit lattice discretization errors in the second virial coefficient are significantly enhanced. As a consequence the equation of state does not show the universal scaling behavior expected in the unitary limit. We suggest that scaling can be improved by tuning the second virial coefficient rather than the scattering length.
Thermal properties of low-density neutron matter are investigated by determinantal quantum Monte Carlo lattice calculations on 3+1 dimensional cubic lattices. Nuclear effective field theory (EFT) is applied using the pionless single- and two-parameter neutron-neutron interactions, determined from the 1S0{}^{1}{S}_{0}1S0​ scattering length and effective range. The determination of the interactions and the calculations of neutron matter are carried out consistently by applying EFT power counting rules. The thermodynamic limit is taken by the method of finite-size scaling, and the continuum limit is examined in the vanishing lattice filling limit. The 1S0{}^{1}{S}_{0}1S0​ pairing gap at $T\ensuremath{\approx}0$ is computed directly from the off-diagonal long-range order of the spin pair-pair correlation function and is found to be approximately 30% smaller than BCS calculations with the conventional nucleon-nucleon potentials. The critical temperature Tc{T}_{c}Tc​ of the normal-to-superfluid phase transition and the pairing temperature scale T∗{T}^{*}T∗ are determined, and the temperature-density phase diagram is constructed. The physics of low-density neutron matter is clearly identified as being a BCS-Bose-Einstein condensation crossover.
We compute continuum and infinite volume limit extrapolations of the structure factors of neutron matter at finite temperature and density. Using a lattice formulation of leading-order pionless effective field theory, we compute the momentum dependence of the structure factors at finite temperature and at densities beyond the reach of the virial expansion. The Tan contact parameter is computed and the result agrees with the high momentum tail of the vector structure factor. All errors, statistical and systematic, are controlled for. This calculation is a first step towards a model-independent understanding of the linear response of neutron matter at finite temperature.
We report results of fully nonperturbative, path integral Monte Carlo calculations for dilute neutron matter. The neutron-neutron interaction in the sss channel is parameterized by the scattering length and the effective range. We calculate the energy and the chemical potential as a function of temperature at density $\ensuremath{\rho}=0.003 {\text{fm}}^{\ensuremath{-}3}$. The critical temperature Tc{T}_{c}Tc​ for the superfluid-normal phase transition is estimated from the finite size scaling of the condensate fraction. At low temperatures we extract the spectral weight function $A(\mathbf{p},\ensuremath{\omega})$ from the imaginary time propagator using the methods of maximum entropy and singular value decomposition. We determine the quasiparticle spectrum, which can be accurately parameterized by three parameters: an effective mass m∗{m}^{*}m∗, a mean-field potential UUU, and a gap $\ensuremath{\Delta}$. Large values of $\ensuremath{\Delta}/{T}_{c}$ indicate that the system is not a BCS-type superfluid at low temperatures.
The unitary Fermi gas is a many-body system of two-component fermions with zero-range interactions tuned to infinite scattering length. Despite much activity and interest in unitary Fermi gas and its universal properties, there have been great difficulties in performing accurate calculations of the superfluid condensate fraction and pairing wave function. In this paper, we present auxiliary-field lattice Monte Carlo simulations using a lattice interaction which accelerates the approach to the continuum limit, thereby allowing for robust calculations of these difficult observables. As a benchmark test, we compute the ground-state energy of 33 spin-up and 33 spin-down particles. As a fraction of the free Fermi gas energy EFG{E}_{\text{FG}}EFG​, we find E0/EFG=0.369(2),0.372(2),{E}_{0}/{E}_{\text{FG}}=0.369(2),0.372(2),E0​/EFG​=0.369(2),0.372(2), using two different definitions of the finite-system energy ratio, in agreement with the latest theoretical and experimental results. We then determine the condensate fraction by measuring off-diagonal long-range order in the two-body density matrix. We find that the fraction of condensed pairs is $\ensuremath{\alpha}=0.43(2)$. We also extract the pairing wave function and find the pair correlation length to be ${\ensuremath{\zeta}}_{p}{k}_{F}=1.8(3)\ensuremath{\hbar}$, where kF{k}_{F}kF​ is the Fermi momentum. Provided that the simulations can be performed without severe sign oscillations, the methods we present here can be applied to superfluid neutron matter as well as more exotic PPP-wave and DDD-wave superfluids.
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.