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

Journal of Mathematical Physics

Microscopic theory of superfluid Fermi systems. II. Application to low density hard sphere system

L. E. Reichl

Published Mar 1, 1979• 0 citations
Publisher

Abstract

This is the second of two papers in which the binary expansion is used to study properties of hard core Fermi systems with broken gauge symmetry. In this paper, we make contact with the phenomenology of superfluid Fermi systems commonly used in field theory. This requires, however, that each scattering event is approximated by its forward scattering value, an approximation which, at best is valid at very low densities. Within this approximation we show that a hard sphere Fermi fluid can exhibit superfluidity.

DOI
10.1063/1.524084

Related papers

Microscopic theory of superfluid Fermi systems. I. Binary expansion

L. E. Reichl

This is the first of two papers in which the binary expansion is used to study the properties of hard core fermi systems with broken gauge symmetry. In this paper, which is primarily formal, symmetry-breaking source terms are introduced into the generating function for thermodynamic quantities, and an exact expression is obtained for the generating function in terms of single-particle and two-particle sources and the reaction matrices for the two-body problem. In a subsequent paper, the theory is applied to a dilute hard sphere fluid.

Journal of Mathematical Physics•Mar 1, 1979•1 citation

Ground state energy of a hard-sphere Fermi fluid

George Allen Baker Jr., L. P. Benofy, Mauricio Fortes +1 more

A new method of computing the ground state energy of a Fermi fluid is developed and applied to the hard-sphere Fermi fluids of two and four components. We find the results to be highly accurate for low densities. In this paper we compute and report the perturbation series for the ground state energy for the soft, repulsive, square-well problem at a number of densities for a four-component Fermi system as well as the ladder approximation energies for a hard-sphere potential at the same densities.

Physical Review C•Aug 1, 1986•7 citations

Resonance superfluidity: Renormalization of resonance scattering theory

S. J. J. M. F. Kokkelmans, J. N. Milstein, M. L. Chiofalo +2 more

We derive a theory of superfluidity for a dilute Fermi gas that is valid when scattering resonances are present. The treatment of a resonance in many-body atomic physics requires a novel mean-field approach starting from an unconventional microscopic Hamiltonian. The mean-field equations incorporate the microscopic scattering physics, and the solutions to these equations reproduce the energy-dependent scattering properties. This theory describes the high-${T}_{c}$ behavior of the system, and predicts a value of ${T}_{c}$ that is a significant fraction of the Fermi temperature. It is shown that this mean-field approach does not break down for typical experimental circumstances, even at detunings close to resonance. As an example of the application of our theory, we investigate the feasibility for achieving superfluidity in an ultracold gas of fermionic ${}^{6}\mathrm{Li}.$

arXiv:cond-mat/0112283•Physical Review A•May 14, 2002•184 citations

Theory of a Superfluid Fermi Liquid. I. General Formalism and Static Properties

A. J. Leggett

The microscopic theory of a superfluid Fermi liquid at finite temperature is developed for the case of a pure system with $S$-wave pairing, and applied to the calculation of the static properties. As a function of $\ensuremath{\theta}\ensuremath{\equiv}\frac{T}{{T}_{c}}$ these properties are determined entirely by the Landau parameters ${F}_{0}$, ${F}_{1}$, ${Z}_{0}$, etc., characterizing quasiparticle interactions in the normal phase. In particular the spin susceptibility $\ensuremath{\chi}$ and the density of the normal component ${\ensuremath{\rho}}_{n}$ are given by $\frac{\ensuremath{\chi}(\ensuremath{\theta})}{\ensuremath{\chi}(1)}=(1+\frac{1}{4}{Z}_{0})\frac{f(\ensuremath{\theta})}{[1+\frac{1}{4}{Z}_{0}f(\ensuremath{\theta})]},$ $\frac{{\ensuremath{\rho}}_{n}}{\ensuremath{\rho}}=(1+\frac{1}{3}{F}_{1})\frac{f(\ensuremath{\theta})}{[1+\frac{1}{3}{F}_{1}f(\ensuremath{\theta})]},$ where the universal function $f(\ensuremath{\theta})\ensuremath{\equiv}\ensuremath{-}{[\ensuremath{\nu}(0)]}^{\ensuremath{-}1}{\ensuremath{\Sigma}}_{\mathrm{p}}(\frac{\mathrm{dn}}{d{E}_{\mathrm{p}}})$ is the "effective density of states near the Fermi surface" relative to its value $\ensuremath{\nu}(0)$ in the normal phase. Thus the often-quoted expression ${\ensuremath{\rho}}_{n}=\frac{1}{3}{\ensuremath{\Sigma}}_{\mathrm{p}}{\mathrm{p}}^{2}(\frac{\mathrm{dn}}{d{E}_{\mathrm{p}}})$ is valid for an interacting system only in the limit $T\ensuremath{\rightarrow}0$. In the latter part of the paper a simple phenomenological theory of "Fermi-liquid" effects on $\ensuremath{\chi}$ and ${\ensuremath{\rho}}_{n}$ is developed for arbitrary conditions (including the presence of impurities and pairing with $l\ensuremath{\ne}0$); it is found that under most circumstances explicit expressions for $\ensuremath{\chi}$ and ${\ensuremath{\rho}}_{n}$ may be obtained which involve only the Landau parameters and a suitably generalized effective density of states. The theory should apply to the possible superfluid phase of ${\mathrm{He}}^{3}$ and to most superconductors. It is suggested that the Knight shift in nontransition-metal superconductors should display some "Fermi-liquid" effects. The weak-field dc penetration depth $\ensuremath{\lambda}(T)$ is shown to be insensitive to such effects both in the Pippard limit and near ${T}_{c}$; however, in a London superconductor at lower temperatures the correction to $\ensuremath{\lambda}(T)$ should be observable and yield a direct estimate of ${F}_{1}$.

Physical Review•Dec 13, 1965•284 citations

Quantum Hard-Sphere Gas. II

Marshall Luban

Using a method due to the author which correctly reproduces the scattering of two hard spheres for all relative momenta and all partial waves, we study the properties of a low-density hard-sphere boson gas at $T=0$\ifmmode^\circ\else\textdegree\fi{}K. The system is described by a Hamiltonian $\mathcal{H}$ defined for all relative distances of the $N$ particles. The terms of $\mathcal{H}$ corresponding to the forward motion of a particle through the system lead to the notion of an effective mass ${m}^{*}$ for each particle in excess of its actual mass $m$, with $\frac{{m}^{*}}{m}$ given by a monotonically increasing function of $\ensuremath{\rho}{a}^{3}$, where $\ensuremath{\rho}$ and $a$ are the number of particles per unit volume and the hard-sphere diameter, respectively. The excitation spectrum is linear at low momenta, displays a roton-like behavior for momenta $p\ensuremath{\gtrsim}\frac{\ensuremath{\hbar}}{a}$, possesses further extrema at higher momenta, and asymptotically approaches $\frac{{p}^{2}}{2{m}^{*}}$. The pair distribution function vanishes in the low-density limit for particle separations $r\ensuremath{\le}a$.

Physical Review•May 17, 1965•6 citations

Quantum-Mechanical Many-Body Problem with Hard-Sphere Interaction

Kerson Huang, Chongming Yang

The system under consideration is an $N$-particle quantum-mechanical system enclosed in a volume $V$, in which the particles interact via two-body hard-sphere potentials, with hard-sphere diameter $a$. The two-body hard-sphere problem is first discussed by a generalization of Fermi's pseudopotential by means of which the problem is formulated entirely in terms of the scattering phase shifts. It is then shown that a pseudopotential for the $N$-body problem can be introduced, and leads to an expansion of the energy levels of the system in powers of $a$. The first order energy levels of a Bose and a Fermi system are calculated. For the Bose system, the first order energy formula exhibits an "energy gap" above the ground state, leading to properties of the system not dissimilar to that of a superfluid. The ground state energy for a Bose system is calculated to order ${a}^{3}$ and that for the Fermi system, to order ${a}^{2}$. The physical interpretation and validity of these results are discussed.

Physical Review•Feb 1, 1957•841 citations
This is the second of two papers in which the binary expansion is used to study properties of hard core Fermi systems with broken gauge symmetry. In this paper, we make contact with the phenomenology of superfluid Fermi systems commonly used in field theory. This requires, however, that each scattering event is approximated by its forward scattering value, an approximation which, at best is valid at very low densities. Within this approximation we show that a hard sphere Fermi fluid can exhibit superfluidity.
This is the first of two papers in which the binary expansion is used to study the properties of hard core fermi systems with broken gauge symmetry. In this paper, which is primarily formal, symmetry-breaking source terms are introduced into the generating function for thermodynamic quantities, and an exact expression is obtained for the generating function in terms of single-particle and two-particle sources and the reaction matrices for the two-body problem. In a subsequent paper, the theory is applied to a dilute hard sphere fluid.
We derive a theory of superfluidity for a dilute Fermi gas that is valid when scattering resonances are present. The treatment of a resonance in many-body atomic physics requires a novel mean-field approach starting from an unconventional microscopic Hamiltonian. The mean-field equations incorporate the microscopic scattering physics, and the solutions to these equations reproduce the energy-dependent scattering properties. This theory describes the high-Tc{T}_{c}Tc​ behavior of the system, and predicts a value of Tc{T}_{c}Tc​ that is a significant fraction of the Fermi temperature. It is shown that this mean-field approach does not break down for typical experimental circumstances, even at detunings close to resonance. As an example of the application of our theory, we investigate the feasibility for achieving superfluidity in an ultracold gas of fermionic 6Li.{}^{6}\mathrm{Li}.6Li.
The microscopic theory of a superfluid Fermi liquid at finite temperature is developed for the case of a pure system with SSS-wave pairing, and applied to the calculation of the static properties. As a function of $\ensuremath{\theta}\ensuremath{\equiv}\frac{T}{{T}_{c}}$ these properties are determined entirely by the Landau parameters F0{F}_{0}F0​, F1{F}_{1}F1​, Z0{Z}_{0}Z0​, etc., characterizing quasiparticle interactions in the normal phase. In particular the spin susceptibility $\ensuremath{\chi}$ and the density of the normal component ${\ensuremath{\rho}}_{n}$ are given by $\frac{\ensuremath{\chi}(\ensuremath{\theta})}{\ensuremath{\chi}(1)}=(1+\frac{1}{4}{Z}_{0})\frac{f(\ensuremath{\theta})}{[1+\frac{1}{4}{Z}_{0}f(\ensuremath{\theta})]},$ $\frac{{\ensuremath{\rho}}_{n}}{\ensuremath{\rho}}=(1+\frac{1}{3}{F}_{1})\frac{f(\ensuremath{\theta})}{[1+\frac{1}{3}{F}_{1}f(\ensuremath{\theta})]},$ where the universal function $f(\ensuremath{\theta})\ensuremath{\equiv}\ensuremath{-}{[\ensuremath{\nu}(0)]}^{\ensuremath{-}1}{\ensuremath{\Sigma}}_{\mathrm{p}}(\frac{\mathrm{dn}}{d{E}_{\mathrm{p}}})$ is the "effective density of states near the Fermi surface" relative to its value $\ensuremath{\nu}(0)$ in the normal phase. Thus the often-quoted expression ${\ensuremath{\rho}}_{n}=\frac{1}{3}{\ensuremath{\Sigma}}_{\mathrm{p}}{\mathrm{p}}^{2}(\frac{\mathrm{dn}}{d{E}_{\mathrm{p}}})$ is valid for an interacting system only in the limit $T\ensuremath{\rightarrow}0$. In the latter part of the paper a simple phenomenological theory of "Fermi-liquid" effects on $\ensuremath{\chi}$ and ${\ensuremath{\rho}}_{n}$ is developed for arbitrary conditions (including the presence of impurities and pairing with $l\ensuremath{\ne}0$); it is found that under most circumstances explicit expressions for $\ensuremath{\chi}$ and ${\ensuremath{\rho}}_{n}$ may be obtained which involve only the Landau parameters and a suitably generalized effective density of states. The theory should apply to the possible superfluid phase of He3{\mathrm{He}}^{3}He3 and to most superconductors. It is suggested that the Knight shift in nontransition-metal superconductors should display some "Fermi-liquid" effects. The weak-field dc penetration depth $\ensuremath{\lambda}(T)$ is shown to be insensitive to such effects both in the Pippard limit and near Tc{T}_{c}Tc​; however, in a London superconductor at lower temperatures the correction to $\ensuremath{\lambda}(T)$ should be observable and yield a direct estimate of F1{F}_{1}F1​.
Using a method due to the author which correctly reproduces the scattering of two hard spheres for all relative momenta and all partial waves, we study the properties of a low-density hard-sphere boson gas at T=0T=0T=0\ifmmode^\circ\else\textdegree\fi{}K. The system is described by a Hamiltonian H\mathcal{H}H defined for all relative distances of the NNN particles. The terms of H\mathcal{H}H corresponding to the forward motion of a particle through the system lead to the notion of an effective mass m∗{m}^{*}m∗ for each particle in excess of its actual mass mmm, with m∗m\frac{{m}^{*}}{m}mm∗​ given by a monotonically increasing function of $\ensuremath{\rho}{a}^{3}$, where $\ensuremath{\rho}$ and aaa are the number of particles per unit volume and the hard-sphere diameter, respectively. The excitation spectrum is linear at low momenta, displays a roton-like behavior for momenta $p\ensuremath{\gtrsim}\frac{\ensuremath{\hbar}}{a}$, possesses further extrema at higher momenta, and asymptotically approaches p22m∗\frac{{p}^{2}}{2{m}^{*}}2m∗p2​. The pair distribution function vanishes in the low-density limit for particle separations $r\ensuremath{\le}a$.
The system under consideration is an NNN-particle quantum-mechanical system enclosed in a volume VVV, in which the particles interact via two-body hard-sphere potentials, with hard-sphere diameter aaa. The two-body hard-sphere problem is first discussed by a generalization of Fermi's pseudopotential by means of which the problem is formulated entirely in terms of the scattering phase shifts. It is then shown that a pseudopotential for the NNN-body problem can be introduced, and leads to an expansion of the energy levels of the system in powers of aaa. The first order energy levels of a Bose and a Fermi system are calculated. For the Bose system, the first order energy formula exhibits an "energy gap" above the ground state, leading to properties of the system not dissimilar to that of a superfluid. The ground state energy for a Bose system is calculated to order a3{a}^{3}a3 and that for the Fermi system, to order a2{a}^{2}a2. The physical interpretation and validity of these results are discussed.