Isothermal sweep theorems for ultra-cold quantum gases in a canonical ensemble

TL;DR

This paper derives isothermal sweep theorems for ultra-cold quantum gases in a canonical ensemble, linking thermodynamic quantities and the contact parameter, with applications to Fermi gases in BCS and BEC regimes.

Contribution

It introduces the isothermal Hellmann-Feynman theorem for mixed states and applies it to derive sweep theorems and the Virial theorem for quantum gases in a canonical ensemble.

Findings

Temperature dependence of the contact parameter can be determined from entropy or specific heat variations.

Sweep theorems relate changes in thermodynamic quantities to scattering length variations.

Virial theorem is extended to canonical ensembles for quantum gases.

Abstract

After deriving the isothermal Hellmann-Feynman theorem (IHFT) that is suitable for mixed states in a canonical ensemble, we use this theorem to obtain the isothermal magnetic-field sweep theorems for the free, average and trapping energies, and for the entropy, specific heat, pressure and atomic compressibility of strongly-correlated ultra-cold quantum gases. In particular, we apply the sweep theorems to two-component Fermi gases in the weakly-interacting BCS and BEC limits, showing that the temperature dependence of the contact parameter can be determined by the variation of either the entropy or specific heat with respect to the scattering length. We also use the IHFT to obtain the Virial theorem in a canonical ensemble, and discuss its implications for quantum gases.