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:cond-mat/0311437 · Physical Review Letters

Exact Pairing Correlations for One-Dimensionally Trapped Fermions with Stochastic Mean-Field Wave Functions

Olivier Juillet, Francesca Gulminelli, Ph. Chomaz

Published Apr 19, 2004• 21 citations
arXivHTMLPDF

Abstract

The canonical thermodynamic properties of a one-dimensional system of interacting spin-1/2 fermions with an attractive zero-range pseudopotential are investigated within an exact approach. The density operator is evaluated as the statistical average of dyadics formed from a stochastic mean-field propagation of independent Slater determinants. For a harmonically trapped Fermi gas and for fermions confined in a 1D-like torus, we observe the transition to a quasi-BCS state with Cooper-like momentum correlations and an algebraic long-range order. For a few trapped fermions in a rotating torus, a dominant superfluid component with quantized circulation can be isolated.

DOI
10.1103/physrevlett.92.160401

Related papers

Pairing correlations in a trapped one-dimensional Fermi gas

Stephen Kudla, Dominique Gautreau, Daniel E. Sheehy

We use a BCS-type variational wave function to study attractively interacting quasi-one-dimensional fermionic atomic gases, motivated by cold-atom experiments that access the one-dimensional regime using an anisotropic harmonic trapping potential (with trapping frequencies ${\ensuremath{\omega}}_{x}={\ensuremath{\omega}}_{y}\ensuremath{\gg}{\ensuremath{\omega}}_{z}$) that confines the gas to a cigar-shaped geometry. To handle the presence of the trap along the $z$ direction, we construct our variational wave function from the harmonic oscillator Hermite functions, which are the eigenstates of the single-particle problem. Using an analytic determination of the effective interaction among harmonic oscillator states along with a numerical solution of the resulting variational equations, we make specific experimental predictions for how pairing correlations would be revealed in experimental probes, such as the local density and the momentum correlation function.

arXiv:1404.4081•Physical Review A•Apr 9, 2015•5 citations

Superfluid instability in ultra-cold gas of fermionic atoms with attractive potential in a one-dimensional trap

P. Schlottmann, Andrei A. Zvyagin

In the context of ultracold fermionic atoms with effective spin S confined to an elongated trap we study the one-dimensional gas interacting via an attractive δ-function potential using the Bethe ansatz solution. There are N = 2 S + 1 fundamental states: The particles can either be unpaired or clustered in bound states of 2, 3, ..., 2 S and 2S + 1 atoms. In a magnetic field, the rich ground state phase diagram consists of these N states and various mixed phases in which combinations of the fundamental states coexist. The phase diagram simplifies considerably in zero-field, where only bound states of N atoms can exist. Due to the harmonic confinement and within the local density approximation, the density profile of bound states decreases along the tube from the center of the trap to its boundaries. In an array of tubes with weak Josephson tunneling superfluid order may arise. In zero-field the response functions determining the superfluid and density wave order are calculated using conformal field theory and the exact Bethe ansatz solution. The response function for superfluidity consists of a power law with distance, while the correlation function for density waves is a power law of distance times a sinusoidal factor oscillating with distance with a period given by two times the Fermi momentum. For S = 1/2 superfluidity is a possibility for all densities and density waves can be excluded. For S ≥ 3/2 superfluidity may occur at low densities but at high densities it gives way to density waves. We discuss the scenario of phase separation where for S ≥ 3/2 the system has superfluid long-range order toward the trap boundaries and density waves at its center.

Journal of Physics Conference Series•Aug 14, 2014•5 citations

Unconventional pairing in few-fermion systems at finite temperature

Daniel Pęcak, Tomasz Sowiński

Attractively interacting two-component mixtures of fermionic particles confined in a one-dimensional harmonic trap are investigated. Properties of balanced and imbalanced systems are systematically explored with the exact diagonalization approach, focusing on the finite-temperature effects. Using single- and two-particle density distributions, specific non-classical pairing correlations are analyzed in terms of the noise correlations-quantity directly accessible in state-of-the-art experiments with ultra-cold atoms. It is shown that along with increasing temperature, any imbalanced system hosting Fulde-Ferrel-Larkin-Ovchinnikov pairs crossovers to a standard Bardeen-Cooper-Schrieffer one characterized by zero net momentum of resulting pairs. By performing calculations for systems with different imbalances, the approximate boundary between the two phases on a phase diagram is determined.

arXiv:2202.07639•Scientific Reports•Oct 19, 2022•6 citations

Langevin equations for interacting fermions and Cooper-like pairing in trapped one-dimensional fermions

Lev I. Plimak, Matthew J. Collett, M. K. Olsen

Momentum correlations in a one-dimensional equilibrium ensemble of trapped fermions, with a point interaction between particles of opposite spin have been studied. In the degenerate regime correlations were observed between fermions with opposite spins and momenta, similar to Cooper pairing. These correlations appear as soon as the temperature is below the Fermi energy, which is a much less stringent condition than that of the BCS transition proper. Calculations are carried out in both perturbative and non-perturbative regimes. To achieve the latter, it is shown that interacting fermionic dynamics may be solved as a stochastic linear transformation of Grassmann algebra generators, much in the way random c-number paths are introduced in the conventional quantum stochastics of bosons. Importantly, the method thus emerging is inherently free of the sign problem.

Physical Review A•Nov 16, 2001•28 citations

Fermi gas with attractive potential and spin S=32 in a one-dimensional trap: Response functions for superfluidity and FFLO signatures

P. Schlottmann, Andrei A. Zvyagin

In the context of a gas of ultracold atoms with effective spin $S=3/2$ confined to an elongated trap, we study the one-dimensional Fermi gas interacting via an attractive $\ensuremath{\delta}$-function potential within the grand-canonical ensemble. The particles can be either unbound or clustered in bound states of two, three, and four fermions. The rich $\ensuremath{\mu}$ versus $H$ ground-state phase diagram ($\ensuremath{\mu}$ is the chemical potential and $H$ the external magnetic field) consists of the four basic states and the various possible mixed phases in which some these states coexist. Extending the analysis of K. Yang [Phys. Rev. B 63, 140511(R) (2001)] for $S=1/2$, we study the correlation functions of the generalized Cooper clusters of bound states of two, three, and four particles using conformal field theory and the exact Bethe Ansatz solution. The correlation functions consist of a power law with distance times a sinusoidal term oscillating with distance. In an array of tubes with weak Josephson tunneling, the type of superfluid order is determined by these correlation functions. The wavelength of the oscillations is related to the periodicity of a generalized Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state for higher spin particles. All the relevant states are analyzed for $S=3/2$.

Physical Review B•May 18, 2012•18 citations

Energetic and structural properties of two-dimensional trapped mesoscopic Fermi gases

Emma Laird, Brendan C. Mulkerin, Jia Wang +1 more

We theoretically investigate equal-mass spin-balanced two-component Fermi gases in which pairs of atoms with opposite spins interact via a short-range isotropic model potential. We probe the distinction between two-dimensional and quasi-two-dimensional harmonic confinement by tuning the effective range parameter within two-dimensional scattering theory. Our approach, which yields numerically exact energetic and structural properties, combines a correlated Gaussian basis-set expansion with the stochastic variational method. For systems containing up to six particles, we: 1) Present the ground- and excited-state energy spectra; 2) Study non-local correlations by analysing the one- and two-body density matrices, extracting from these the occupation numbers of natural orbitals, the momentum distributions of atoms and pairs, and the molecular `condensate fraction’; 3) Study local correlations by computing the radial and pair distribution functions. This paper extends current theoretical knowledge on the properties of trapped few-fermion systems as realised in state-of-the-art cold-atom experiments.

arXiv:2506.20891•SciPost Physics Core•Apr 9, 2026•0 citations
The canonical thermodynamic properties of a one-dimensional system of interacting spin-1/2 fermions with an attractive zero-range pseudopotential are investigated within an exact approach. The density operator is evaluated as the statistical average of dyadics formed from a stochastic mean-field propagation of independent Slater determinants. For a harmonically trapped Fermi gas and for fermions confined in a 1D-like torus, we observe the transition to a quasi-BCS state with Cooper-like momentum correlations and an algebraic long-range order. For a few trapped fermions in a rotating torus, a dominant superfluid component with quantized circulation can be isolated.
We use a BCS-type variational wave function to study attractively interacting quasi-one-dimensional fermionic atomic gases, motivated by cold-atom experiments that access the one-dimensional regime using an anisotropic harmonic trapping potential (with trapping frequencies ${\ensuremath{\omega}}_{x}={\ensuremath{\omega}}_{y}\ensuremath{\gg}{\ensuremath{\omega}}_{z}$) that confines the gas to a cigar-shaped geometry. To handle the presence of the trap along the zzz direction, we construct our variational wave function from the harmonic oscillator Hermite functions, which are the eigenstates of the single-particle problem. Using an analytic determination of the effective interaction among harmonic oscillator states along with a numerical solution of the resulting variational equations, we make specific experimental predictions for how pairing correlations would be revealed in experimental probes, such as the local density and the momentum correlation function.
In the context of ultracold fermionic atoms with effective spin S confined to an elongated trap we study the one-dimensional gas interacting via an attractive δ-function potential using the Bethe ansatz solution. There are N = 2 S + 1 fundamental states: The particles can either be unpaired or clustered in bound states of 2, 3, ..., 2 S and 2S + 1 atoms. In a magnetic field, the rich ground state phase diagram consists of these N states and various mixed phases in which combinations of the fundamental states coexist. The phase diagram simplifies considerably in zero-field, where only bound states of N atoms can exist. Due to the harmonic confinement and within the local density approximation, the density profile of bound states decreases along the tube from the center of the trap to its boundaries. In an array of tubes with weak Josephson tunneling superfluid order may arise. In zero-field the response functions determining the superfluid and density wave order are calculated using conformal field theory and the exact Bethe ansatz solution. The response function for superfluidity consists of a power law with distance, while the correlation function for density waves is a power law of distance times a sinusoidal factor oscillating with distance with a period given by two times the Fermi momentum. For S = 1/2 superfluidity is a possibility for all densities and density waves can be excluded. For S ≥ 3/2 superfluidity may occur at low densities but at high densities it gives way to density waves. We discuss the scenario of phase separation where for S ≥ 3/2 the system has superfluid long-range order toward the trap boundaries and density waves at its center.
Attractively interacting two-component mixtures of fermionic particles confined in a one-dimensional harmonic trap are investigated. Properties of balanced and imbalanced systems are systematically explored with the exact diagonalization approach, focusing on the finite-temperature effects. Using single- and two-particle density distributions, specific non-classical pairing correlations are analyzed in terms of the noise correlations-quantity directly accessible in state-of-the-art experiments with ultra-cold atoms. It is shown that along with increasing temperature, any imbalanced system hosting Fulde-Ferrel-Larkin-Ovchinnikov pairs crossovers to a standard Bardeen-Cooper-Schrieffer one characterized by zero net momentum of resulting pairs. By performing calculations for systems with different imbalances, the approximate boundary between the two phases on a phase diagram is determined.
Momentum correlations in a one-dimensional equilibrium ensemble of trapped fermions, with a point interaction between particles of opposite spin have been studied. In the degenerate regime correlations were observed between fermions with opposite spins and momenta, similar to Cooper pairing. These correlations appear as soon as the temperature is below the Fermi energy, which is a much less stringent condition than that of the BCS transition proper. Calculations are carried out in both perturbative and non-perturbative regimes. To achieve the latter, it is shown that interacting fermionic dynamics may be solved as a stochastic linear transformation of Grassmann algebra generators, much in the way random c-number paths are introduced in the conventional quantum stochastics of bosons. Importantly, the method thus emerging is inherently free of the sign problem.
In the context of a gas of ultracold atoms with effective spin S=3/2S=3/2S=3/2 confined to an elongated trap, we study the one-dimensional Fermi gas interacting via an attractive $\ensuremath{\delta}$-function potential within the grand-canonical ensemble. The particles can be either unbound or clustered in bound states of two, three, and four fermions. The rich $\ensuremath{\mu}$ versus HHH ground-state phase diagram ($\ensuremath{\mu}$ is the chemical potential and HHH the external magnetic field) consists of the four basic states and the various possible mixed phases in which some these states coexist. Extending the analysis of K. Yang [Phys. Rev. B 63, 140511(R) (2001)] for S=1/2S=1/2S=1/2, we study the correlation functions of the generalized Cooper clusters of bound states of two, three, and four particles using conformal field theory and the exact Bethe Ansatz solution. The correlation functions consist of a power law with distance times a sinusoidal term oscillating with distance. In an array of tubes with weak Josephson tunneling, the type of superfluid order is determined by these correlation functions. The wavelength of the oscillations is related to the periodicity of a generalized Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state for higher spin particles. All the relevant states are analyzed for S=3/2S=3/2S=3/2.
We theoretically investigate equal-mass spin-balanced two-component Fermi gases in which pairs of atoms with opposite spins interact via a short-range isotropic model potential. We probe the distinction between two-dimensional and quasi-two-dimensional harmonic confinement by tuning the effective range parameter within two-dimensional scattering theory. Our approach, which yields numerically exact energetic and structural properties, combines a correlated Gaussian basis-set expansion with the stochastic variational method. For systems containing up to six particles, we: 1) Present the ground- and excited-state energy spectra; 2) Study non-local correlations by analysing the one- and two-body density matrices, extracting from these the occupation numbers of natural orbitals, the momentum distributions of atoms and pairs, and the molecular `condensate fraction’; 3) Study local correlations by computing the radial and pair distribution functions. This paper extends current theoretical knowledge on the properties of trapped few-fermion systems as realised in state-of-the-art cold-atom experiments.