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arXiv:0708.2524 · Physical Review C

From low-density neutron matter to the unitary limit

Takumi Abe, R. Seki

Published May 11, 2009• 37 citations
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Abstract

Various quantities of an attractively interacting fermion system at the unitary limit are determined by extrapolating Monte Carlo results of low-density neutron matter. Smooth extrapolation in terms of $1/({k}_{F}{a}_{0})$ (${k}_{F}$ is the Fermi momentum, and ${a}_{0}$ is the ${}^{1}{S}_{0}$ scattering length) is found with the quantities examined: the ground-state energy, the pairing gap at $T\ensuremath{\approx}0$, and the critical temperature of the normal-to-superfluid phase transition. We emphasize proximity of the physics of low-density neutron matter to that at the unitary limit. The extrapolated quantities are in a reasonable agreement with those in the literature.

DOI
10.1103/physrevc.79.054003

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Various quantities of an attractively interacting fermion system at the unitary limit are determined by extrapolating Monte Carlo results of low-density neutron matter. Smooth extrapolation in terms of 1/(kFa0)1/({k}_{F}{a}_{0})1/(kF​a0​) (kF{k}_{F}kF​ is the Fermi momentum, and a0{a}_{0}a0​ is the 1S0{}^{1}{S}_{0}1S0​ scattering length) is found with the quantities examined: the ground-state energy, the pairing gap at T\ensuremath≈0T\ensuremath{\approx}0T\ensuremath≈0, and the critical temperature of the normal-to-superfluid phase transition. We emphasize proximity of the physics of low-density neutron matter to that at the unitary limit. The extrapolated quantities are in a reasonable agreement with those in the literature.
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 study neutron matter at and near the unitary limit using a low-momentum ring diagram approach. By slightly tuning the meson-exchange CD-Bonn potential, neutron-neutron potentials with various 1S0{}^{1}{S}_{0}1S0​ scattering lengths such as ${a}_{s}=\ensuremath{-}12070$ fm and +21+21+21 fm are constructed. Such potentials are renormalized with rigorous procedures to give the corresponding as{a}_{s}as​-equivalent low-momentum potentials Vlow-k{V}_{\text{low-}k}Vlow-k​, with which the low-momentum particle-particle hole-hole ring diagrams are summed up to all orders, giving the ground state energy E0{E}_{0}E0​ of neutron matter for various scattering lengths. At the limit of ${a}_{s}\ensuremath{\rightarrow}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}$, our calculated ratio of E0{E}_{0}E0​ to that of the noninteracting case is found remarkably close to a constant of 0.44 over a wide range of Fermi-momenta. This result reveals an universality that is well consistent with the recent experimental and Monte Carlo computational study on low-density cold Fermi gas at the unitary limit. The overall behavior of this ratio obtained with various scattering lengths is presented and discussed. Ring-diagram results obtained with Vlow-k{V}_{\text{low-}k}Vlow-k​ and those with GGG-matrix interactions are compared.
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.
Experiments with cold Fermi atoms can be tuned to probe strongly-interacting fluids that are very similar to the low-density neutron matter found in the crusts of neutron stars. In contrast to traditional superfluids and superconductors, matter in this regime is very strongly paired, with gaps of the order of the Fermi energy. We compute the T=0T=0T=0 equation of state and pairing gap for cold atoms and low-density neutron matter as a function of the Fermi momentum times the scattering length. Results of quantum Monte Carlo calculations show that the equations of state are very similar. The neutron matter pairing gap at low densities is found to be very large but, except at the smallest densities, significantly suppressed relative to cold atoms because of the finite effective range in the neutron-neutron interaction.
We study the equation of state of neutron matter using a family of unitarity potentials, all of which are constructed to have infinite 1S0{}^{1}{S}_{0}1S0​ scattering lengths as{a}_{s}as​. For such a system, a quantity of much interest is the ratio $\ensuremath{\xi}={E}_{0}/{E}_{0}^{\mathrm{free}}$, where E0{E}_{0}E0​ is the true ground-state energy of the system and E0free{E}_{0}^{\mathrm{free}}E0free​ is that for the noninteracting system. In the limit of ${a}_{s}\ensuremath{\rightarrow}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}$, often referred to as the unitary limit, this ratio is expected to approach a universal constant, namely $\ensuremath{\xi}~0.44(1)$. In the present work we calculate this ratio $\ensuremath{\xi}$ using a family of hard-core square-well potentials whose as{a}_{s}as​ can be exactly obtained, thus enabling us to have many potentials of different ranges and strengths, all with infinite as{a}_{s}as​. We have also calculated $\ensuremath{\xi}$ using a unitarity CDBonn potential obtained by slightly scaling its meson parameters. The ratios $\ensuremath{\xi}$ given by these different unitarity potentials are all close to each other and also remarkably close to 0.44, suggesting that the above ratio $\ensuremath{\xi}$ is indifferent to the details of the underlying interactions as long as they have infinite scattering length. A sum-rule and scaling constraint for the renormalized low-momentum interaction in neutron matter at the unitary limit is studied. The importance of pairing in our all-order ring diagram and model-space HF calculations of neutron matter is discussed.
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.