Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1712.05931

An exact solution to the Bertsch problem and the non-universality of the Unitary Fermi Gas

E. Ruiz Arriola, S. Szpigel, V. S. Timóteo

Published Dec 16, 2017• 2 citations
arXivHTMLPDF

Abstract

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 $ξ$ 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 $ξ= \frac{176}{9 π}-\frac{17}{3} = 0.56$. 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$.

DOI
10.48550/arxiv.1712.05931

Related papers

Universality of the unitary Fermi gas: a few-body perspective

Jesper Levinsen, Pietro Massignan, Shimpei Endo +1 more

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 → ∞ .

arXiv:1612.02131•Journal of Physics B Atomic Molecular and Optical Physics•Jan 18, 2017•31 citations

Unitary p -wave Fermi gas in one dimension

Hiroyuki Tajima, Shoichiro Tsutsui, Takahiro M. Doi +1 more

We elucidate universal many-body properties of a one-dimensional, two-component ultracold Fermi gas near the $p$-wave Feshbach resonance. The low-energy scattering in this system can be characterized by two parameters, that is, $p$-wave scattering length and effective range. At the unitarity limit where the $p$-wave scattering length diverges and the effective range is reduced to zero without conflicting with the causality bound, the system obeys universal thermodynamics as observed in a unitary Fermi gas with contact $s$-wave interaction in three dimensions. It is in contrast to a Fermi gas with the $p$-wave resonance in three dimensions in which the effective range is inevitably finite. We present the universal equation of state in this unitary $p$-wave Fermi gas within the many-body $T$-matrix approach as well as the virial expansion method. Moreover, we examine the single-particle spectral function in the high-density regime where the virial expansion is no longer valid. On the basis of the Hartree-like self-energy shift at the divergent scattering length, we conjecture that the equivalence of the Bertsch parameter across spatial dimensions holds even for a one-dimensional unitary $p$-wave Fermi gas.

arXiv:2106.12909•Physical Review A•Aug 23, 2021•13 citations

Spin 1/2 Fermions in the Unitary Limit.I

H. S. Køhler

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.

arXiv:0705.0944•May 7, 2007•0 citations

Spin 1/2 Fermions in the Unitary Limit.III

H. S. Køhler

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.

arXiv:1008.3884•Aug 23, 2010•0 citations

Density-functional theory for fermions in the unitary regime

T. Papenbrock

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.

arXiv:cond-mat/0507183•Physical Review A•Oct 11, 2005•58 citations

Density-functional theory for resonantly interacting fermions with effective range and neutron matter

Denis Lacroix A

A density-functional theory is proposed for strongly interacting fermions with arbitrary large negative scattering length. The functional has only two parameters that are directly fixed to reproduce the universal properties of unitary gas: the so-called Bertsch parameter ${\ensuremath{\xi}}_{0}$ and a parameter ${\ensuremath{\eta}}_{e}$ related to the possible influence of the effective range ${r}_{e}$ at infinite scattering length $a$. Using most recent quantum Monte Carlo (QMC) estimates of these two parameters, it is shown that the functional properly reproduces the experimental measurements of interacting Fermi systems not only at unitarity but also away from this limit over a wide range of ${(a{k}_{F})}^{\ensuremath{-}1}$ values. The functional is applied to obtain an expression of Tan's contact parameter including the effect of ${r}_{e}$. Application is finally made to neutron matter. It is shown that most recent QMC results are well reproduced.

arXiv:1608.08411•Physical Review A•Oct 7, 2016•26 citations
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≥ξ≥−1/3 0.56 \ge ξ\ge -1/30.56≥ξ≥−1/3.
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 → ∞ .
We elucidate universal many-body properties of a one-dimensional, two-component ultracold Fermi gas near the ppp-wave Feshbach resonance. The low-energy scattering in this system can be characterized by two parameters, that is, ppp-wave scattering length and effective range. At the unitarity limit where the ppp-wave scattering length diverges and the effective range is reduced to zero without conflicting with the causality bound, the system obeys universal thermodynamics as observed in a unitary Fermi gas with contact sss-wave interaction in three dimensions. It is in contrast to a Fermi gas with the ppp-wave resonance in three dimensions in which the effective range is inevitably finite. We present the universal equation of state in this unitary ppp-wave Fermi gas within the many-body TTT-matrix approach as well as the virial expansion method. Moreover, we examine the single-particle spectral function in the high-density regime where the virial expansion is no longer valid. On the basis of the Hartree-like self-energy shift at the divergent scattering length, we conjecture that the equivalence of the Bertsch parameter across spatial dimensions holds even for a one-dimensional unitary ppp-wave Fermi gas.
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.
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.
A density-functional theory is proposed for strongly interacting fermions with arbitrary large negative scattering length. The functional has only two parameters that are directly fixed to reproduce the universal properties of unitary gas: the so-called Bertsch parameter ${\ensuremath{\xi}}_{0}$ and a parameter ${\ensuremath{\eta}}_{e}$ related to the possible influence of the effective range re{r}_{e}re​ at infinite scattering length aaa. Using most recent quantum Monte Carlo (QMC) estimates of these two parameters, it is shown that the functional properly reproduces the experimental measurements of interacting Fermi systems not only at unitarity but also away from this limit over a wide range of ${(a{k}_{F})}^{\ensuremath{-}1}$ values. The functional is applied to obtain an expression of Tan's contact parameter including the effect of re{r}_{e}re​. Application is finally made to neutron matter. It is shown that most recent QMC results are well reproduced.