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:0907.3198 · Physical Review Letters

Analytic Thermodynamics and Thermometry of Gaudin-Yang Fermi Gases

Erhai Zhao, Xi‐Wen Guan, W. Vincent Liu, Murray T. Batchelor, Masaki Oshikawa

Published Sep 30, 2009• 46 citations
arXivHTMLPDF

Abstract

We study the thermodynamics of a one-dimensional attractive Fermi gas (the Gaudin-Yang model) with spin imbalance. The exact solution has been known from the thermodynamic Bethe ansatz for decades, but it involves an infinite number of coupled nonlinear integral equations whose physics is difficult to extract. Here the solution is analytically reduced to a simple, powerful set of four algebraic equations. The simplified equations become universal and exact in the experimental regime of strong interaction and relatively low temperature. Using the new formulation, we discuss the qualitative features of finite-temperature crossover and make quantitative predictions on the density profiles in traps. We propose a practical two-stage scheme to achieve accurate thermometry for a trapped spin-imbalanced Fermi gas.

DOI
10.1103/physrevlett.103.140404

Related papers

Thermodynamics, contact, and density profiles of the repulsive Gaudin-Yang model

Ovidiu I. Pâţu, Andreas Klümper

We address the problem of computing the thermodynamic properties of the repulsive one-dimensional two-component Fermi gas with contact interaction, also known as the Gaudin-Yang model. Using a specific lattice embedding and the quantum transfer matrix we derive an exact system of only two nonlinear integral equations for the thermodynamics of the homogeneous model which is valid for all temperatures and values of the chemical potential, magnetic field, and coupling strength. This system allows for an easy and extremely accurate calculation of thermodynamic properties circumventing the difficulties associated with the truncation of the thermodynamic Bethe ansatz system of equations. We present extensive results for the densities, polarization, magnetic susceptibility, specific heat, interaction energy, Tan contact, and local correlation function of opposite spins. Our results show that at low and intermediate temperatures the experimentally accessible contact is a nonmonotonic function of the coupling strength. As a function of the temperature the contact presents a pronounced local minimum in the Tonks-Girardeau regime which signals an abrupt change of the momentum distribution in a small interval of temperature. The density profiles of the system in the presence of a harmonic trapping potential are computed using the exact solution of the homogeneous model coupled with the local density approximation. We find that at finite temperature the density profile presents a double shell structure (partially polarized center and fully polarized wings) only when the polarization in the center of the trap is above a critical value which is monotonically increasing with temperature.

arXiv:1512.07485•Physical Review A•Mar 9, 2016•24 citations

Exactly solvable models and ultracold Fermi gases

M T Batchelor, Angela Foerster, X-W Guan +1 more

Exactly solvable models of ultracold Fermi gases are reviewed via their thermodynamic Bethe ansatz solution. Analytical and numerical results are obtained for the thermodynamics and ground state properties of two- and three-component one-dimensional attractive fermions with population imbalance. New results for the universal finite temperature corrections are given for the two-component model. For the three-component model, numerical solution of the dressed energy equations confirms that the analytical expressions for the critical fields and the resulting phase diagrams at zero temperature are highly accurate in the strong coupling regime. The results provide a precise description of the quantum phases and universal thermodynamics which are applicable to experiments with cold fermionic atoms confined to one-dimensional tubes.

arXiv:1011.3259•Journal of Statistical Mechanics Theory and Experiment•Dec 6, 2010•17 citations

10 Exactly solvable models and ultracold Fermi gases

Murray T. Batchelor, Angela Foerster, X. -w. Guan

Exactly solvable models of ultracold Fermi gases are reviewed via their thermodynamic Bethe Ansatz solution. Analytical and numerical results are obtained for the thermodynamics and ground state properties of two- and three-component one-dimensional attractive fermions with population imbalance. New results for the universal finite temperature corrections are given for the two-component model. For the three-component model, numerical solution of the dressed energy equations confirm that the analytical expressions for the critical fields and the resulting phase diagrams at zero temperature are highly accurate in the strong coupling regime. The results provide a precise description of the quantum phases and universal thermodynamics which are applicable to experiments with cold fermionic atoms confined to one-dimensional tubes.

•Aug 16, 2016•10 citations

Paired states in spin-imbalanced atomic Fermi gases in one dimension

Paata Kakashvili, C. J. Bolech

A growing expertise to engineer, manipulate, and probe different cold-atom analogs of electronic condensed-matter systems allows to probe properties of exotic pairing. We study paired states of spin-imbalanced ultracold atomic system of fermions with attractive short-range interactions in one-dimensional traps. Calculations are done using the Bethe ansatz technique, and the trap is incorporated into the solution via a local-density approximation. The thermodynamic Bethe ansatz equations are solved numerically and different local-density profiles are calculated for zero and finite temperatures. A procedure to identify the homogeneous-system phase diagram using local-density profiles in the trap is also proposed. Such scheme would be immediately useful for the experiments.

arXiv:0812.2963•Physical Review A•Apr 15, 2009•68 citations

Fermi gases in one dimension: From Bethe ansatz to experiments

Xi‐Wen Guan, Murray T. Batchelor, Chaohong Lee

This article reviews theoretical and experimental developments for one-dimensional Fermi gases. Specifically, the experimentally realized two-component delta-function interacting Fermi gas---the Gaudin-Yang model---and its generalizations to multicomponent Fermi systems with larger spin symmetries is discussed. The exact results obtained for Bethe ansatz integrable models of this kind enable the study of the nature and microscopic origin of a wide range of quantum many-body phenomena driven by spin population imbalance, dynamical interactions, and magnetic fields. This physics includes Bardeen-Cooper-Schrieffer-like pairing, Tomonaga-Luttinger liquids, spin-charge separation, Fulde-Ferrel-Larkin-Ovchinnikov-like pair correlations, quantum criticality and scaling, polarons, and the few-body physics of the trimer state (trions). The fascinating interplay between exactly solved models and experimental developments in one dimension promises to yield further insight into the exciting and fundamental physics of interacting Fermi systems.

arXiv:1301.6446•Reviews of Modern Physics•Nov 27, 2013•544 citations

Dark-soliton-like excitations in the Yang–Gaudin gas of attractively interacting fermions

Sophie S Shamailov, Joachim Brand

Yrast states are the lowest energy states at given non-zero momentum and provide a natural extension of the concept of dark solitons to strongly interacting one-dimensional quantum gases. Here we study the yrast states of the balanced spin- Fermi gas with attractive delta-function interactions in one-dimension with the exactly solvable Yang–Gaudin model. The corresponding Bethe-ansatz equations are solved for finite particle number and in the thermodynamic limit. Properties corresponding to the soliton-like nature of the yrast excitations are calculated including the missing particle number, phase step, and inertial and physical masses. The inertial to physical mass ratio, which is related to the frequency of oscillations in a trapped gas, is found to be unity in the limits of strong and weak attraction and falls to in the crossover regime. This result is contrasted by one-dimensional mean field theory, which predicts a divergent mass ratio in the weakly attractive limit. By means of an exact mapping our results also predict the existence and properties of dark-soliton-like excitations in the super Tonks–Girardeau gas. The prospects for experimental observations are briefly discussed.

arXiv:1603.04864•New Journal of Physics•Jul 1, 2016•27 citations
We study the thermodynamics of a one-dimensional attractive Fermi gas (the Gaudin-Yang model) with spin imbalance. The exact solution has been known from the thermodynamic Bethe ansatz for decades, but it involves an infinite number of coupled nonlinear integral equations whose physics is difficult to extract. Here the solution is analytically reduced to a simple, powerful set of four algebraic equations. The simplified equations become universal and exact in the experimental regime of strong interaction and relatively low temperature. Using the new formulation, we discuss the qualitative features of finite-temperature crossover and make quantitative predictions on the density profiles in traps. We propose a practical two-stage scheme to achieve accurate thermometry for a trapped spin-imbalanced Fermi gas.
We address the problem of computing the thermodynamic properties of the repulsive one-dimensional two-component Fermi gas with contact interaction, also known as the Gaudin-Yang model. Using a specific lattice embedding and the quantum transfer matrix we derive an exact system of only two nonlinear integral equations for the thermodynamics of the homogeneous model which is valid for all temperatures and values of the chemical potential, magnetic field, and coupling strength. This system allows for an easy and extremely accurate calculation of thermodynamic properties circumventing the difficulties associated with the truncation of the thermodynamic Bethe ansatz system of equations. We present extensive results for the densities, polarization, magnetic susceptibility, specific heat, interaction energy, Tan contact, and local correlation function of opposite spins. Our results show that at low and intermediate temperatures the experimentally accessible contact is a nonmonotonic function of the coupling strength. As a function of the temperature the contact presents a pronounced local minimum in the Tonks-Girardeau regime which signals an abrupt change of the momentum distribution in a small interval of temperature. The density profiles of the system in the presence of a harmonic trapping potential are computed using the exact solution of the homogeneous model coupled with the local density approximation. We find that at finite temperature the density profile presents a double shell structure (partially polarized center and fully polarized wings) only when the polarization in the center of the trap is above a critical value which is monotonically increasing with temperature.
Exactly solvable models of ultracold Fermi gases are reviewed via their thermodynamic Bethe ansatz solution. Analytical and numerical results are obtained for the thermodynamics and ground state properties of two- and three-component one-dimensional attractive fermions with population imbalance. New results for the universal finite temperature corrections are given for the two-component model. For the three-component model, numerical solution of the dressed energy equations confirms that the analytical expressions for the critical fields and the resulting phase diagrams at zero temperature are highly accurate in the strong coupling regime. The results provide a precise description of the quantum phases and universal thermodynamics which are applicable to experiments with cold fermionic atoms confined to one-dimensional tubes.
Exactly solvable models of ultracold Fermi gases are reviewed via their thermodynamic Bethe Ansatz solution. Analytical and numerical results are obtained for the thermodynamics and ground state properties of two- and three-component one-dimensional attractive fermions with population imbalance. New results for the universal finite temperature corrections are given for the two-component model. For the three-component model, numerical solution of the dressed energy equations confirm that the analytical expressions for the critical fields and the resulting phase diagrams at zero temperature are highly accurate in the strong coupling regime. The results provide a precise description of the quantum phases and universal thermodynamics which are applicable to experiments with cold fermionic atoms confined to one-dimensional tubes.
A growing expertise to engineer, manipulate, and probe different cold-atom analogs of electronic condensed-matter systems allows to probe properties of exotic pairing. We study paired states of spin-imbalanced ultracold atomic system of fermions with attractive short-range interactions in one-dimensional traps. Calculations are done using the Bethe ansatz technique, and the trap is incorporated into the solution via a local-density approximation. The thermodynamic Bethe ansatz equations are solved numerically and different local-density profiles are calculated for zero and finite temperatures. A procedure to identify the homogeneous-system phase diagram using local-density profiles in the trap is also proposed. Such scheme would be immediately useful for the experiments.
This article reviews theoretical and experimental developments for one-dimensional Fermi gases. Specifically, the experimentally realized two-component delta-function interacting Fermi gas---the Gaudin-Yang model---and its generalizations to multicomponent Fermi systems with larger spin symmetries is discussed. The exact results obtained for Bethe ansatz integrable models of this kind enable the study of the nature and microscopic origin of a wide range of quantum many-body phenomena driven by spin population imbalance, dynamical interactions, and magnetic fields. This physics includes Bardeen-Cooper-Schrieffer-like pairing, Tomonaga-Luttinger liquids, spin-charge separation, Fulde-Ferrel-Larkin-Ovchinnikov-like pair correlations, quantum criticality and scaling, polarons, and the few-body physics of the trimer state (trions). The fascinating interplay between exactly solved models and experimental developments in one dimension promises to yield further insight into the exciting and fundamental physics of interacting Fermi systems.
Yrast states are the lowest energy states at given non-zero momentum and provide a natural extension of the concept of dark solitons to strongly interacting one-dimensional quantum gases. Here we study the yrast states of the balanced spin- Fermi gas with attractive delta-function interactions in one-dimension with the exactly solvable Yang–Gaudin model. The corresponding Bethe-ansatz equations are solved for finite particle number and in the thermodynamic limit. Properties corresponding to the soliton-like nature of the yrast excitations are calculated including the missing particle number, phase step, and inertial and physical masses. The inertial to physical mass ratio, which is related to the frequency of oscillations in a trapped gas, is found to be unity in the limits of strong and weak attraction and falls to in the crossover regime. This result is contrasted by one-dimensional mean field theory, which predicts a divergent mass ratio in the weakly attractive limit. By means of an exact mapping our results also predict the existence and properties of dark-soliton-like excitations in the super Tonks–Girardeau gas. The prospects for experimental observations are briefly discussed.