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Physical Review C

BCS variation method and universal behavior of trapped neutron systems at unitary limit

Y. Y. Cheng, Y. M. Zhao

Published May 14, 2024• 1 citation
Publisher

Abstract

In this work we study unitary neutron systems confined in a harmonic-oscillator trap, based on the BCS theory and the quasiparticle ring-diagram method. The low-momentum interaction ${V}_{\text{low-}k}$ renormalized from a fine-tuned CD-Bonn potential with the $^{1}S_{0}$ scattering length approaching the infinity is adopted. The energy ratios of trapped unitary neutron systems with various particle numbers, both those with magic numbers and those with nonmagic ones, to corresponding noninteracting systems, are shown to all flow to a constant value. The BCS wave functions for these trapped unitary systems, with different trap parameters and different particle numbers, manifest themselves with universal patterns: the effect of harmonic-oscillator shells is significant, and moreover, the orbits of the lowest valence shell are occupied with a uniform probability. We also discuss the self-energy given by the BCS calculation, and present a weighted sum rule satisfied by the monopole components of the unitary neutron-neutron interaction. In terms of these, the constant energy ratio independent of the trap parameter and the particle number can be interpreted.

DOI
10.1103/physrevc.109.054314

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In this work we study unitary neutron systems confined in a harmonic-oscillator trap, based on the BCS theory and the quasiparticle ring-diagram method. The low-momentum interaction Vlow-k{V}_{\text{low-}k}Vlow-k​ renormalized from a fine-tuned CD-Bonn potential with the 1S0^{1}S_{0}1S0​ scattering length approaching the infinity is adopted. The energy ratios of trapped unitary neutron systems with various particle numbers, both those with magic numbers and those with nonmagic ones, to corresponding noninteracting systems, are shown to all flow to a constant value. The BCS wave functions for these trapped unitary systems, with different trap parameters and different particle numbers, manifest themselves with universal patterns: the effect of harmonic-oscillator shells is significant, and moreover, the orbits of the lowest valence shell are occupied with a uniform probability. We also discuss the self-energy given by the BCS calculation, and present a weighted sum rule satisfied by the monopole components of the unitary neutron-neutron interaction. In terms of these, the constant energy ratio independent of the trap parameter and the particle number can be interpreted.
We study systems of finite-number neutrons in a harmonic trap at the unitary limit. Two very different types of neutron-neutron interactions are applied, namely, the meson-theoretic CD-Bonn potential and hard-core square-well interactions, all tuned to possess an infinite scattering length, and with effective ranges comparable to the trap size. The potentials are renormalized to equivalent, scattering-length preserving low-momentum potentials, Vlow−k, with which the particle-particle hole-hole ring diagrams are summed to all orders to yield the ground-state energy E0 of the finite neutron system. We find the ratio E0/E0free (where E0free denotes the ground-state energy of the corresponding non-interacting system) to be remarkably independent from variations of the harmonic trap parameter, the type and effective range of the unitarity interaction, and the decimation momentum of Vlow−k. Our results support a special linear scaling relation of E0 versus the trap parameter ħω. Certain properties of Landau's quasi-particles for trapped neutrons at the unitary limit are also discussed.
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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 apply the general principles of effective field theories to the construction of effective interactions suitable for few- and many-body calculations in a no-core shell model framework. We calculate the spectrum of systems with three and four two-component fermions in a harmonic trap. In the unitary limit, we find that three-particle results are within 10% of known semianalytical values even in small model spaces. The method is very general, and can be readily extended to other regimes, more particles, different species (e.g., protons and neutrons in nuclear physics), or more-component fermions (as well as bosons). As an illustration, we present calculations of the lowest-energy three-fermion states away from the unitary limit and find a possible inversion of parity in the ground state in the limit of trap size large compared to the scattering length. Furthermore, we investigate the lowest positive-parity states for four fermions, although we are limited by the dimensions we can currently handle in this case.
We calculate energy levels of two and three bosons trapped in a harmonic oscillator potential with oscillator length aosc{a}_{\mathrm{o}\mathrm{s}\mathrm{c}}aosc​. The atoms are assumed to interact through a short-range potential with a scattering length aaa, and the short-distance behavior of the three-body wave function is characterized by a parameter $\ensuremath{\theta}$. For large positive a/aosca/{a}_{\mathrm{o}\mathrm{s}\mathrm{c}}a/aosc​, the energies of states that, in the absence of the trap, correspond to three free atoms approach values independent of aaa and $\ensuremath{\theta}$. For other states, the $\ensuremath{\theta}$ dependence of the energy is strong, but the energy is independent of aaa for $|a/{a}_{\mathrm{o}\mathrm{s}\mathrm{c}}|\ensuremath{\gg}1$.
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