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arXiv:1308.0939 · Physical Review C

BEC-BCS crossover in neutron matter with renormalization-group-based effective interactions

S. Ramanan, Mi­chael Urban

Published Nov 18, 2013• 24 citations
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Abstract

We study pure neutron matter in the BEC-BCS crossover regime using renormalization-group-based low-momentum interactions within the Nozi\`eres-Schmitt-Rink framework. This is an attempt to go beyond the mean-field description for low-density matter. We work in the basis of so-called Weinberg eigenvectors where the operator ${G}_{0}V$ is diagonal, which proves to be an excellent choice that allows one to use nonlocal interactions in a very convenient way. We study the importance of correlations as a function of density. We notice that there is a significant reduction of the BCS critical temperature at low densities as the neutron matter approaches the unitary limit.

DOI
10.1103/physrevc.88.054315

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We study pure neutron matter in the BEC-BCS crossover regime using renormalization-group-based low-momentum interactions within the Nozi`eres-Schmitt-Rink framework. This is an attempt to go beyond the mean-field description for low-density matter. We work in the basis of so-called Weinberg eigenvectors where the operator G0V{G}_{0}VG0​V is diagonal, which proves to be an excellent choice that allows one to use nonlocal interactions in a very convenient way. We study the importance of correlations as a function of density. We notice that there is a significant reduction of the BCS critical temperature at low densities as the neutron matter approaches the unitary limit.
Due to the large neutron–neutron scattering length, dilute neutron matter resembles the unitary Fermi gas, which lies half-way in the crossover from the BCS phase of weakly coupled Cooper pairs to the Bose–Einstein condensate of dimers. We discuss crossover effects in analogy with the T-matrix theory used in the physics of ultracold atoms, which we generalize to the case of a non-separable finite-range interaction. A problem of the standard Nozières–Schmitt-Rink approach and different ways to solve it are discussed. It is shown that in the strong-coupling regime, the spectral function exhibits a pseudo-gap at temperatures above the critical temperature Tc. The effect of the correlated density on the density dependence of Tc is found to be rather weak, but a possibly important effect due to the reduced quasiparticle weight is identified.
The density, spin, and isospin correlation functions in nuclear matter with a neutron-proton condensate are calculated to study the possible signatures of the BEC-BCS crossover in the low-density region. It is shown that the criterion of the crossover (Phys. Rev. Lett. 95 , 090402 (2005)), consisting in the change of the sign of the density correlation function at low momentum transfer, fails to describe correctly the density-driven BEC-BCS transition at finite isospin asymmetry or finite temperature. The presence (BCS regime) or absence (BEC regime) of the singularity in the momentum distribution of the quasiparticle density of states can be used as an unambiguous signature of the BEC-BCS transition.
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 generalize the Bardeen-Cooper-Schrieffer-Bose-Einstein-condensation (BCS-BEC) crossover of two-component fermions, which is realized by tuning the sss-wave scattering length aaa between the fermions, to the case of an arbitrary effective range rer_{\rm e}re​. By using the Nozières-Schmitt-Rink (NSR) approach, we show another crossover by changing rer_{\rm e}re​ and present several similarities and differences between these two crossovers. Furthermore, the region (re>a/2r_{\rm e}>a/2re​>a/2) where the effective range expansion breaks down and the Hamiltonian becomes non-Hermitian is found, being consistent with the Wigner's causality bound. Our results are universal for low-density interacting fermions with low-energy constants aaa and rer_{\rm e}re​ and are directly relevant to ultracold Fermi atomic gases as well as dilute neutron matter.
Based on the relativistic Hartree-Bogoliubov theory, the influence of the pairing interaction strength on the dineutron correlations and the crossover from superfluidity of neutron Cooper pairs in the 1S0{}^{1\phantom{\rule{-0.16em}{0ex}}}{S}_{0}1S0​ channel to Bose-Einstein condensation of dineutron pairs is systematically investigated in the nuclear matter. The bare nucleon-nucleon interaction Bonn-B is taken in the particle-particle channel with an effective factor to simulate the medium effects and take into account the possible ambiguity of pairing force, and the effective interaction PK1 is used in the particle-hole channel. If the effective factor is larger than 1.101.101.10, a dineutron Bose-Einstein condensation (BEC) state appears in the low-density limit, and if it is smaller than 0.850.850.85, the neutron Cooper pairs are found totally in the weak coupling Bardeen-Cooper-Schrieffer (BCS) region. The reference values of several characteristic quantities which characterize the BCS-BEC crossover are obtained, respectively, from the dimensionless parameter 1/(kFna)1/({k}_{\mathrm{Fn}}a)1/(kFn​a) with aaa the scattering length and kFn{k}_{\mathrm{Fn}}kFn​ the neutron Fermi momentum, the zero-momentum transfer density correlation function D(0)D(0)D(0), and the effective chemical potential ${\ensuremath{\nu}}_{\mathrm{n}}$.
Abstract We compute singlet pairing gaps and critical temperatures in pure neutron matter with different many-body approximations. Medium effects tend to reduce gaps and critical temperatures compared to the standard BCS ansatz. In the mean-field approximation, the ratio of these two quantities remains constant across a wide range of densities. This constant ratio is close to the universal prediction of BCS theory, whether three-neutron interactions are included or not. Using a more sophisticated many-body approach that incorporates the effect of short-range correlations in pairing properties, we find that the gap to critical temperature ratio in the low-density regime is substantially larger than the BCS prediction, independently of the interaction. In this region, our results are relatively close to experiments and theoretical calculations from the unitary Fermi gas. We also find evidence for a different density dependence of zero-temperature gaps and critical temperatures in neutron matter.