Large-$N$ scaling of Tan's contact for the harmonically trapped Tonks--Girardeau gas at finite temperature
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arXiv
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| Formato: | Preprint |
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2026
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| _version_ | 1866916046506557440 |
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| author | Sant'Ana, Felipe Taha |
| author_facet | Sant'Ana, Felipe Taha |
| contents | 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 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_15994 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Large-$N$ scaling of Tan's contact for the harmonically trapped Tonks--Girardeau gas at finite temperature Sant'Ana, Felipe Taha Quantum Gases Statistical Mechanics 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 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. |
| title | Large-$N$ scaling of Tan's contact for the harmonically trapped Tonks--Girardeau gas at finite temperature |
| topic | Quantum Gases Statistical Mechanics |
| url | https://arxiv.org/abs/2605.15994 |