Correlations in the low-density Fermi gas: Fermi-liquid state, dimerization, and Bardeen-Cooper-Schrieffer pairing

Abstract

We present ground state calculations for low-density Fermi gases described by two model interactions, an attractive square-well potential and a Lennard-Jones potential, of varying strength. We use the optimized Fermi-Hypernetted Chain integral equation method which has been proved to provide, in the density regimes of interest here, an accuracy better than one percent. We first examine the low-density expansion of the energy and compare with the exact answer by Huang and Yang (H. Huang and C. N. Yang, Phys. Rev. 105, 767 (1957)). It is shown that a locally correlated wave function of the Jastrow-Feenberg type does not recover the quadratic term in the expansion of the energy in powers of a0kF, where a0 is the vacuum s-wave scattering length and kF the Fermi wave number. The problem is cured by adding second-order perturbation corrections in a correlated basis. Going to higher densities and/or more strongly coupled systems, we encounter an instability of the normal state of the system which is characterized by a divergence of the in-medium scattering length. We interpret this divergence as a phonon-exchange driven dimerization of the system, similar to what one has at zero density when the vacuum scattering length a0 diverges. We then study, in the stable regime, the superfluid gap and its dependence on the density and the interaction strength. We identify two different corrections to low-density expansions: One is medium corrections to the pairing interaction, and the other one finite-range corrections. We show that the most important finite-range corrections are a direct manifestation of the many-body nature of the system.

$$^1S_0$$ 1 S 0 Pairing in Neutron Matter

“…8,16. A recent in-depth study of correlations in the low-density Fermi gas 18 , with emphasis on the presence of Cooper pairing and dimerization, documents the power of the Euler-Lagrange FHNC approach adopted in the present work, especially when coupled with CBF perturbation theory.…”

“…As already pointed out, the maximum gap value obtained by Pavlou et al with AV 18 for their two optimized correlation functions differ by nearly a factor two for the Benhar choice) -reflecting the extreme sensitivity of the gap to inputs for the effective interaction. Recognizing that the induced interaction and energynumerator terms are absent in these two calculations, the information provided by Fig.…”

“…Closer inspection 18 reveals that our approach corresponds to a BCS theory formulated in terms of the scattering matrix 40 . The correlation operator F serves here to tame the short-range dynamical correlations.…”

“…The most interesting result of previous 18 work along these lines is the appearance of a divergence of solutions of the FHNC-EL equations that occurs, as a function of potential strength, well before the divergence of the vacuum scattering length a 0 of the interaction potential. This divergence of solutions of the FHNC-EL equations is analogous to the spinodal instability often discussed in earlier literature, with the principled and practical conclusion that the FHNC-EL equations for the homogeneous system have no solutions if F s 0 < −1, i.e.…”

“…In Ref. 18, divergence of the FHNC-EL equations in the case of a diverging in-medium scattering length gave evidence that the ground state is unstable against dimerization. The appearance of such instabilities whenever the assumptions on the state of the system fail -here, assumption of a non-dimerized phase; in the case of particle-hole instabilities, of a uniform system -is a unique feature of theories such as FHNC-EL that enjoy the topological completeness of parquet diagrams.…”