Canonical-ensemble scaling of Tan's contact in the trapped Tonks--Girardeau gas
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, having 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 Padé 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, and the experimental visibility of the ensemble correction is quantified: for the mesoscopic particle numbers and temperatures of current one-dimensional experiments it reaches the several-percent level.
Versions
Related
- Interaction-Assisted Spin Coherence in a Multimode Trapped Atomic Ensemble
- Tan's Contact for Trapped Spin-Orbit-Coupled Fermi Gases at Finite Temperature
- Large Deviations for the Bose--Einstein Condensate of the Ideal Gas in the Canonical Ensemble
- Particle-Number-Preserving Quantum Signal Processing for Thermal Properties: Canonical-Ensemble Estimation and Grand-Canonical Reconstruction
- Canonical-Ensemble Stability of the Quantum Oppenheimer-Snyder Black-Hole Exterior
BibTeX
@misc{santana2026canonicalensemble,
title = {{Canonical-ensemble scaling of Tan's contact in the trapped Tonks--Girardeau gas}},
author = {Felipe Taha Sant'Ana},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2605.15994},
eprint = {2605.15994},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.15994},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Canonical-ensemble scaling of Tan's contact in the trapped Tonks--Girardeau gas
AU - Felipe Taha Sant'Ana
PY - 2026
DA - 2026/08/25
PB - arXiv
DO - 10.48550/arxiv.2605.15994
UR - https://arxiv.org/abs/2605.15994
AB - 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, having 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 Padé 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, and the experimental visibility of the ensemble correction is quantified: for the mesoscopic particle numbers and temperatures of current one-dimensional experiments it reaches the several-percent level.
ER -