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

TL;DR

This paper develops an exact and efficient method to compute thermodynamic properties of the one-dimensional repulsive Gaudin-Yang Fermi gas, revealing temperature-dependent behaviors of contact and density profiles relevant for experiments.

Contribution

It introduces a simplified system of two nonlinear integral equations for the model's thermodynamics, enabling accurate calculations across all parameters and temperatures.

Findings

Contact is a non-monotonic function of coupling at low/intermediate temperatures.

The contact exhibits a local minimum in the Tonks-Girardeau regime with temperature.

Density profiles show a double shell structure at finite temperature under certain polarization conditions.

Abstract

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 non-monotonic 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.