Abstract

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.

Highlights

  • The scenario of “unitary limit” was originally formulated by Bertsch in 1999, asking what will be the ground-state properties of a spin-1/2 fermion system with an interaction of infinite scattering length [1]

  • Our results support a special linear scaling relation of E0, which is shown to be analytically consistent with the unitary-limit virial theorems of Refs. [26, 29]

  • We have studied systems of finite-number neutrons in a harmonic trap at the unitary limit

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Summary

Introduction

The scenario of “unitary limit” was originally formulated by Bertsch in 1999, asking what will be the ground-state properties of a spin-1/2 fermion system with an interaction of infinite scattering length [1]. With impressive advances of cold-atom experimental techniques, such unitary Fermi systems became experimentally accessible at the atomic level, and have attracted intensive attention This limit is of great interest to nuclear systems, because the 1S0 scattering lengths of realistic nucleon-nucleon interactions are all fairly large, such as −18.97 fm in the CD-Bonn potential [2], in comparison with other length scales in the nuclear system. In this work we find the ratio E0/E0free for a trapped unitary system with re comparable to or larger than ah.o., is remarkably invariant with variations of the harmonic trap parameter, the number of fermions, the decimation momentum of the low-momentum interaction Vlow−k [31,32,33,34,35,36,37], and the type and effective range of the inter-fermion potential, suggesting a universal nature of these systems. Is the oscillator single-particle (s.p.) energy of the orbit a, and kF is the top filled s.p. orbit

We also define an intrinsic ratio Ri as
Results and discussions
From the above equation we readily have ω d dω
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