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arXiv:0705.0944

Spin 1/2 Fermions in the Unitary Limit.I

H. S. Køhler

Published May 7, 2007• 0 citations
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Abstract

This report concerns the energy of a zero-temperature many-body system of spin 1/2 fermions interacting via a two-body potential with a free space infinite scattering length and zero effective range; the Unitary limit. Given the corresponding phase-shift $δ(k)=π/2$ a one-term separable potential is obtained by inverse scattering assuming a momentum cut-off $Λ$ such that $δ(k)=0$ for $k>Λ$. The \it effective \rm interaction in the many-body system is calculated in a pp-ladder approximation with Pauli-blocking but neglecting mean-field (dispersion) corrections; effective mass $m^{*}=1$. Using only the zero relative momentum component of this interaction the total energy is $ξ=4/9$ (in units of the fermigas), a result reported by several previous authors. Integrating the momentum dependent interaction over the Fermi sea this energy is revised to $ξ=0.24.$ This result is independent of density and of the cut-off $Λ$ if $Λ> \sim 3k_{f}$. With $m^{*}\neq 1$ there is however a strong dependence on this cut-off. Including hh-ladders estimates give $ξ=0.4\leftrightarrow 0.6$, but a reliable result would in this case require a Green's function calculation.

DOI
10.48550/arxiv.0705.0944

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This report concerns the energy of a zero-temperature many-body system of spin 1/2 fermions interacting via a two-body potential with a free space infinite scattering length and zero effective range; the Unitary limit. Given the corresponding phase-shift δ(k)=π/2δ(k)=π/2δ(k)=π/2 a one-term separable potential is obtained by inverse scattering assuming a momentum cut-off ΛΛΛ such that δ(k)=0δ(k)=0δ(k)=0 for k>Λk>Λk>Λ. The \it effective \rm interaction in the many-body system is calculated in a pp-ladder approximation with Pauli-blocking but neglecting mean-field (dispersion) corrections; effective mass m∗=1m^{*}=1m∗=1. Using only the zero relative momentum component of this interaction the total energy is ξ=4/9ξ=4/9ξ=4/9 (in units of the fermigas), a result reported by several previous authors. Integrating the momentum dependent interaction over the Fermi sea this energy is revised to ξ=0.24.ξ=0.24.ξ=0.24. This result is independent of density and of the cut-off ΛΛΛ if Λ>∼3kfΛ> \sim 3k_{f}Λ>∼3kf​. With m∗≠1m^{*}\neq 1m∗=1 there is however a strong dependence on this cut-off. Including hh-ladders estimates give ξ=0.4↔0.6ξ=0.4\leftrightarrow 0.6ξ=0.4↔0.6, but a reliable result would in this case require a Green's function calculation.
In scattering theory, the unitary limit is defined by an infinite scattering-length and a zero effective range, corresponding to a phase-shift π/2, independent of energy. This condition is satisfied by a rank-1 separable potential V(k,k')=-v(k)v(k') with v^{2}(k)=(4π)^{2}(Λ^{2}-k^{2})^{-1/2}, Λbeing the cut-off in momentum space.Previous calculations using a Pauli-corrected ladder summation to calculate the energy of a zero temperature many body system of spin 1/2 fermions with this interaction gave ξ=0.24 (in units of kinetic energy) independent of density and with Λ---->infinity. This value of ξis appreciably smaller than the experimental and that obtained from other calculations, most notably from Monte Carlo, which in principle would be the most reliable. Our previous work did however also show a strong dependence on effective range r_0 (with r_0=0 at unitarity). With an increase to r_0=1.0 the energy varied from ξ~0.38 at k_f=0.6 1/fm to ~0.45 at k_f=1.8 1/fm which is somewhat closer to the Monte-Carlo results. These previous calculations are here extended by including the effect of the previously neglected mean-field propagation, the dispersion correction. This is repulsive and found to increase drastically with decreasing effective range. It is large enough to suggest a revised value of ξ~0.4 ~0.5 independent of r_0. Off-shell effects are also investigated by introducing a rank-2 (phase-shift equivalent) separable potential. Effects of 10% or more in energy could be demonstrated for r_0>0. It is pointed out that a computational cut-off in momentum-space brings in another scale in the in otherwise scale-less unitary problem.
The energy shift due to the interaction of two particles in a large box is proportional to the free particle scattering phase-shift. This provides an approximation to the effective interaction referred to as the phase-shift approximation. For a many-body fermion system this effective interaction has to be corrected for the Pauli-blocking. It is used to calculate the energy of a spin 1/2 fermion system as a function of the two-body scattering length and effective range. In the unitary limit with scattering length going to infinity and effective range going to zero the energy is 0.540 (in the units of the non-interactiong fermi gas energy) with a pp-ladder summation. Including hh-ladders the energy is 0.570. A smooth crossover from the BCS to the BEC region is observed. Pairing is not included.
In the unitary regime, fermions interact strongly via two-body potentials that exhibit a zero range and a (negative) infinite scattering length. The energy density is proportional to the energy density of the free Fermi gas with a proportionality constant $\ensuremath{\xi}$. The author uses a simple density functional parametrized by an effective mass and the universal constant $\ensuremath{\xi}$, and employs Kohn-Sham density-functional theory to obtain the parameters from fit to one exactly solvable two-body problem. This yields $\ensuremath{\xi}=0.42$ and a rather large effective mass. The form of the density functional is checked by similar Kohn-Sham calculations for the exactly solvable Calogero model.
Abstract We revisit the properties of the two-component Fermi gas with short-range interactions in three dimensions, in the limit where the s -wave scattering length diverges. Such a unitary Fermi gas possesses universal thermodynamic and dynamical observables that are independent of any interaction length scale. Focusing on trapped systems of N fermions, where N ≤ 10 , we investigate how well we can determine the zero-temperature behavior of the many-body system from published few-body data on the ground-state energy and the contact. For the unpolarized case, we find that the Bertsch parameters extracted from trapped few-body systems all lie within 15% of the established value. Furthermore, the few-body values for the contact are well within the range of values determined in the literature for the many-body system. In the limit of large spin polarization, we obtain a similar accuracy for the polaron energy, and we estimate the polaron’s effective mass from the dependence of its energy on N . We also compute an upper bound for the squared wave-function overlap between the unitary Fermi system and the non-interacting ground state, both for the trapped and uniform cases. This allows us to prove that the trapped unpolarized ground state at unitarity has zero overlap with its non-interacting counterpart in the many-body limit N → ∞ .
The momentum density, n(k) of interacting many-body Fermionic systems is studied (for k>kF) using examples of several well-known two-body interaction models. It is shown that n(k) can be approximated by a zero-range model for momenta k less than about 0.1/re, where re the effective range. If the scattering length is large and one includes the effects of a fixed value of re≠0, n(k) is almost universal for momenta k up to about 2/re. However, n(k) can not be approximated by a zero-range model for momenta k greater than about 1/(are2)1/3, where a is the scattering length, and if one wishes to maintain a sum rule that relates the energy of a two component Fermi-gas to an integral involving the density. We also show that the short separation distance, s, behavior of the pair density varies as s6.
We analyze the universality of the Unitary Fermi Gas in its ground state from a Wilsonian renormalization point of view and compute the effective range dependence of the Bertsch parameter ξξξ exactly. To this end we construct an effective block-diagonal two-body separable interaction with the Fermi momentum as a cut-off which reduces the calculation to the mean field level. The interaction is separable in momentum space and is determined by Tabakin's inverse scattering formula. For a vanishing effective range we get ξ=1769π−173=0.56ξ= \frac{176}{9 π}-\frac{17}{3} = 0.56ξ=9π176​−317​=0.56. By using phase-equivalent similarity transformations we can show that there is a class of exact solutions with any value in the range $ 0.56 \ge ξ\ge -1/3$.