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

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 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.

• The paper identifies a two-term scaling law for Tan’s contact, with leading N^(5/2) and subleading N^(3/2) contributions.
• It employs a saddle-point reduction and contour-integral method to derive closed-form expressions for the scaling functions A(τ) and B(τ) across degenerate and dilute regimes.
• The work resolves canonical versus grand-canonical discrepancies and offers precise predictions for experimental observations in 1D cold-atom systems.

Large-$N$ Scaling of Tan's Contact in the Harmonically Trapped Tonks–Girardeau Gas at Finite Temperature

Overview and Motivation

This work establishes a rigorous large-$N$ scaling law for Tan's contact in a harmonically trapped Tonks–Girardeau (TG) gas at finite temperature, formulated in the canonical ensemble. The TG gas arises as the infinite repulsion limit of the 1D Lieb–Liniger model, mapping strongly interacting bosons onto noninteracting fermionic wavefunctions via the Girardeau transformation. The central observable, Tan's contact $\mathcal{C}$, quantifies short-range pair correlations and governs the universal $k^{-4}$ tail in the momentum distribution, connecting microscopic physics to thermodynamic identities and interaction energies. Previous treatments either relied on grand-canonical ensemble (GCE) approximations (notably via the local density approximation, LDA) or studied small particle numbers, thereby leaving the subleading finite-$N$ corrections and canonical-vs-grand-canonical distinctions insufficiently resolved.

The principal technical achievement of this paper is a two-term scaling law for the canonical contact:

$$\mathcal{C}_N(\tau) = A(\tau) N^{5/2} + B(\tau) N^{3/2},$$

where $\tau=T/T_F$ is the reduced temperature ($T_F$ being the Fermi temperature), $A(\tau)$ recovers the LDA/GCE result, and $B(\tau)$ is a novel universal function representing the ensemble correction—quantifying the difference between canonical and grand-canonical treatments at fixed mean particle number. Closed forms are derived for $N$0 and $N$1 in the deep degenerate and dilute regimes, and uniformly accurate Padé approximants are constructed for intermediate temperatures.

Canonical Contact: Formalism and Scaling Derivation

The TG gas in a harmonic potential is described by the Hamiltonian

$N$2

in harmonic oscillator units. In the $N$3 limit, bosonic eigenstates are mapped onto antisymmetrized fermionic Slater determinants, ensuring identical local observables (e.g. density, energy) but distinctive off-diagonal correlations.

Tan's contact is defined as the coefficient in the large-momentum tail of the momentum distribution:

$N$4

where $N$5 is the thermal one-body density matrix.

The canonical ensemble evaluation is built from a contour-integral representation of the partition function,

$N$6

with $N$7 the grand partition function, and the contact integrand utilizes a kernel functional averaged along this contour.

Figure 1: Canonical-ensemble contact as a function of reduced temperature $N$8, evaluated from the contour-integral representation for various particle numbers.

The large-$N$9 expansion employs a saddle-point reduction leading to phase-space integrals involving the self-consistent scaled chemical potential $\mathcal{C}$0. In this limit, the Fermi occupation function becomes independent of $\mathcal{C}$1 at leading order, parameterized by $\mathcal{C}$2 and $\mathcal{C}$3:

$\mathcal{C}$4

The universal scaling law emerges as:

$\mathcal{C}$5

where

$\mathcal{C}$6

with $\mathcal{C}$7, $\mathcal{C}$8 denoting phase-space moments of $\mathcal{C}$9.

Analytic Limits: Degenerate and Boltzmann Regimes

In the degenerate ($k^{-4}$0) regime, the leading coefficient $k^{-4}$1 admits a Sommerfeld expansion

$k^{-4}$2

while the subleading $k^{-4}$3 is linear in $k^{-4}$4 with boundary layer (Airy) corrections negligible for $k^{-4}$5:

$k^{-4}$6

In the high-temperature, dilute ($k^{-4}$7) regime, the coefficients reduce to

$k^{-4}$8

so the ratio $k^{-4}$9, which reflects the Poissonian particle-number statistics of the GCE, with the canonical contact below its grand-canonical counterpart by a universal finite-$N$0 correction.

Figure 2: Low-$N$1 verification of the scaling law, illustrating convergence of the ratio $N$2 to unity with increasing $N$3 and demonstrating the accuracy of the analytic coefficients.

Figure 3: High-$N$4 verification, showing scaling collapse and confirming canonical corrections in the Boltzmann regime.

Universal Scaling Functions: Numerical Evaluation and Interpolation

For intermediate $N$5, no analytic simplification is available, and the universal functions $N$6, $N$7 are evaluated numerically via direct quadrature of their integral representations. Additionally, Padé approximants are constructed to interpolate between analytic regimes and provide uniformly accurate closed forms across the physical temperature window.

Figure 4: Universal scaling functions $N$8 and $N$9 computed numerically (solid blue), Padé approximants (black dashed), and analytic asymptotes (colored dotted lines), spanning $$\mathcal{C}_N(\tau) = A(\tau) N^{5/2} + B(\tau) N^{3/2},$$0.

Physical Origin of the Subleading Correction: Ensemble Correspondence

A detailed comparison of the canonical and grand-canonical treatments reveals that $$\mathcal{C}_N(\tau) = A(\tau) N^{5/2} + B(\tau) N^{3/2},$$1 quantifies the ensemble correction to the contact—i.e., the part arising due to the fixed-$$\mathcal{C}_N(\tau) = A(\tau) N^{5/2} + B(\tau) N^{3/2},$$2 constraint in the canonical ensemble and absent in GCE calculations. The explicit form of $$\mathcal{C}_N(\tau) = A(\tau) N^{5/2} + B(\tau) N^{3/2},$$3 is obtained from universal phase-space integrals of the Fermi function and the global particle-number cumulants of the GCE:

$$\mathcal{C}_N(\tau) = A(\tau) N^{5/2} + B(\tau) N^{3/2},$$4

with $$\mathcal{C}_N(\tau) = A(\tau) N^{5/2} + B(\tau) N^{3/2},$$5 built from local and global fluctuations. This correction becomes non-negligible at order $$\mathcal{C}_N(\tau) = A(\tau) N^{5/2} + B(\tau) N^{3/2},$$6 and is strictly negative.

Numerical Verification and Padé Approximants

The scaling law is verified numerically for $$\mathcal{C}_N(\tau) = A(\tau) N^{5/2} + B(\tau) N^{3/2},$$7 up to $$\mathcal{C}_N(\tau) = A(\tau) N^{5/2} + B(\tau) N^{3/2},$$8 across $$\mathcal{C}_N(\tau) = A(\tau) N^{5/2} + B(\tau) N^{3/2},$$9 using canonical contour-integration, confirming both the scaling exponents and the accuracy of the universal coefficients. Padé approximants for $\tau=T/T_F$0 and $\tau=T/T_F$1 enable efficient evaluation and interpolation for practical comparison with experiment.

Figure 5: Full-$\tau=T/T_F$2 scaling verification: ratio $\tau=T/T_F$3 using numerical and Padé scaling functions, confirming the scaling law across the full temperature range.

Implications and Future Directions

The explicit canonical scaling law provides a quantitative prediction for Tan's contact in trapped TG gases at finite temperature. The identification of $\tau=T/T_F$4 as the ensemble correction resolves the canonical-vs-grand-canonical ambiguity and delineates the physical origin of subleading corrections. For practical settings, such as experiments measuring Tan's contact in 1D Bose gases [Huang et al 2025], these results provide stringent benchmarks across the degenerate to classical crossover.

The theoretical framework and contour-integral methodology are extensible to zero-temperature edge corrections (where non-commuting limits produce $\tau=T/T_F$5 scaling in strict $\tau=T/T_F$6), multi-component fermionic mixtures, and finite-coupling Lieb–Liniger gases, where canonical-vs-grand-canonical distinctions are expected to dictate subleading scaling terms. Further work will elucidate thermal suppression of edge corrections and extend the theory to more general settings.

Conclusion

This paper rigorously formulates and analytically derives the canonical large-$\tau=T/T_F$7 scaling law for Tan's contact in the harmonically trapped TG gas at finite temperature, establishing the universality of the leading coefficient and precisely quantifying the subleading ensemble correction. Closed-form expressions and efficient Padé interpolants bridge all physical regimes, and numerical verification affirms the scaling law's accuracy and practical relevance for quantitative comparison with modern cold-atom experiments.