Large-$N$ scaling of Tan's contact for the harmonically trapped Tonks--Girardeau gas at finite temperature

TL;DR

This paper derives the large-N scaling of Tan's contact for harmonically trapped Tonks-Girardeau bosons at finite temperature, revealing universal behaviors and providing explicit phase-space integral representations.

Contribution

It introduces the subleading coefficient of the scaling law, with explicit universal phase-space integral representations and asymptotic limits, advancing understanding of finite-temperature quantum gases.

Findings

The leading scaling coefficient matches the local-density-approximation result.

The subleading coefficient is explicitly represented in terms of universal phase-space integrals.

The ratio of subleading to leading coefficients is universal in the high-temperature limit.

Abstract

We derive the canonical-ensemble scaling of Tan's contact for $N$ harmonically trapped Tonks--Girardeau bosons at finite temperature in the large-$N$ limit. The leading scaling coefficient reproduces the local-density-approximation result and is obtained from a contour-integral representation of the canonical partition function followed by a saddle-point reduction to a phase-space integral with a self-consistent scaled chemical potential. The subleading coefficient is the central new object of this work: it admits an explicit representation in terms of universal phase-space integrals of the Fermi factor, has closed-form Sommerfeld and virial limits, and is identified with the canonical-versus-grand-canonical ensemble difference at fixed mean particle number. In the high-temperature Boltzmann regime the ratio of subleading to leading coefficients collapses to a universal value, traceable to the Poissonian particle-number statistics of the dilute grand-canonical gas. We construct Pad\'e approximants for both scaling functions that interpolate uniformly between the low-temperature Sommerfeld and high-temperature virial regimes; for the subleading coefficient we report a form that is uniformly accurate on our working range of temperatures and asymptotically correct beyond. The scaling law is verified against canonical contour-integration data across the full temperature range.