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REVIEW 1 major objections 4 minor 68 references

Scaling properties of the Tan's contact: embedding pairs and correlation effect in the Tonks limit

T0 review · 1 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read In the strongly interacting regime, every N-particle contact ratio collapses onto the two-boson curve.

desk verdict Useful, mostly solid paper with a real high-temperature inconsistency in the advertised explicit formula; the canonical scaling collapse for N≤5 is credible. read the letter →

arxiv 1908.08714 v2 pith:5U75O4K2 submitted 2019-08-23 cond-mat.quant-gas

F. T. Sant'Ana , F. Hébert , V. Rousseau , M. Albert , P. Vignolo This is my paper · ORCID
classification cond-mat.quant-gas MSC 81V7082B1082B80 PACS 67.85.-d05.30.Jp
keywords Tan'scontactLieb-LinigergasTonks-Girardeaulimitcanonicalensemblefinitetemperatureone-dimensionalbosonsscalingfunctionquantumMonteCarlo
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Strongly interacting one-dimensional trapped Bose gases are usually studied one system at a time, with a fixed particle number, interaction, and temperature. This paper claims that in the strongly interacting regime the canonical Tan's contact for any number $N$ of bosons is fixed by two simpler quantities: the exactly computable two-boson contact and the $N$-body contact at infinite repulsion, the Tonks-Girardeau limit. Concretely, the ratio $C_N(g,T)/C_N(\infty,T)$ is claimed to be a universal function of the rescaled interaction strength $z>1$ and rescaled temperature $\tau$, with no remaining $N$ dependence. The paper also conjectures an explicit formula for the $N$ dependence of the Tonks-Girardeau contact at any temperature, verified numerically for $N=2$ to $5$. If the claim holds, the full contact of a few-atom experiment is obtained from the two-boson curve and the known Tonks limit, with correlations and pair effects separated cleanly.

What carries the argument

The mechanism is a chain of exact reductions. The canonical contact is obtained from Tan's sweep relation, $C_N^c = -(m^2/\pi\hbar^4)\partial F/\partial g^{-1}$, equivalently $C_N^c = (g m^2/\pi\hbar^4)\langle H_{\mathrm{int}}\rangle$. For two bosons the relative spectrum comes from the implicit equation $\Gamma(-\nu/2)/\Gamma(-\nu/2+1/2) = -\sqrt{2}\,|a_{1\mathrm D}|/a_{\mathrm{ho}}$, giving the exact curve $C_2^c(z,\tau)$ and its Tonks-Girardeau limit. For $N$ bosons at infinite repulsion the contact is written through the fermionic two-body density matrix, $C_N^c(\infty,T) = (2/\pi)\int dx\,F(x)$, and for finite interactions the authors use quantum Monte Carlo on a discretized Hubbard model. The identity that carries the claim is $f_N(z>1,\tau)\simeq f_2(z>1,\tau)$, and the conjectured $N$-dependent factor $s(N)=N^{5/2}-N^{3/4}(1+e^{-2/\tau})$ is what makes the Tonks-Girardeau data for different $N$ collapse onto a single curve.

What would settle it

Evaluate the exact fermionic density-matrix expression for the canonical Tonks-Girardeau contact, Eq. (7), for $N=6$ and $N=7$ over a range of $\tau$; if $C_N^c(\infty,\tau)/h_2(\infty,\tau)$ departs from $s(N)$ or gains a visible $\tau$ dependence, Eq. (23) is wrong. A second check is to compute $f_N(z=2.5,\tau)$ for $N=6$ by quantum Monte Carlo and ask whether it still lies on the $N=2$ curve.

Watch

Extended reading notes

Core claim

The central claim is the scaling law $f_N(z>1,\tau) \simeq f_2(z>1,\tau)$, where $f_N = C_N^c(z,\tau)/C_N^c(\infty,\tau)$ is the canonical contact rescaled by its value at infinite repulsion, $z = a_{\mathrm{ho}}/(|a_{1\mathrm D}|\sqrt{N})$ is the rescaled interaction, and $\tau=T/T_F$ the rescaled temperature. Equivalently, all the non-trivial particle-number dependence of the contact is embedded in the Tonks-Girardeau contact $C_N^c(\infty,\tau)$. For that quantity the paper conjectures $C_N^c(\infty,\tau) = h_2(\infty,\tau)\,[N^{5/2}-N^{3/4}(1+e^{-2/\tau})]$, with $h_2$ fixed by the exact two-boson formula; Monte Carlo data for $N=2$ to $5$ collapse on this curve within about $5\%$ at $z=1$ and $1\%$ at $z=2.5$. The same collapse is shown to fail in the grand-canonical ensemble at intermediate temperatures, so the universal form is specific to fixed particle number.

Load-bearing premise

The load-bearing premise is the conjectured formula for the Tonks-Girardeau contact, $C_N^c(\infty,\tau)=h_2(\infty,\tau)(N^{5/2}-N^{3/4}(1+e^{-2/\tau}))$, whose exponents come from a zero-temperature fit and large-temperature pair counting, whose exponential crossover is put in by hand, and which is verified only for $N=2$ to $5$.

Editorial extensions

If this is right

  • For every $N\ge 2$ and every temperature in the regime $z>1$, the canonical contact is $C_N^c(z,\tau)=f_2(z,\tau)\,C_N^c(\infty,\tau)$, so no separate $N$-body calculation is required once the two-boson curve and the Tonks-Girardeau contact are known.
  • The conjectured formula $C_N^c(\infty,\tau)=h_2(\infty,\tau)(N^{5/2}-N^{3/4}(1+e^{-2/\tau}))$ supplies an explicit bridge from few-atom samples to the thermodynamic limit, where $N^{5/2}$ scaling is recovered.
  • At high temperature the formula reduces to the pair-counting result $N(N-1)/2$ times the two-boson contact, while at low temperature it carries the correlation-induced $N^{5/2}-N^{3/4}$ dependence.
  • The universal collapse does not survive in the grand-canonical ensemble at intermediate temperatures, so fixed-$N$ (canonical) experiments are required to observe it.
  • The authors note that an analogous universal ratio follows from recent results for the homogeneous gas, suggesting the scaling extends beyond harmonic traps.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • I would expect the $3/4$ exponent to be a fingerprint of the harmonic confinement, so a box-trapped version of this construction should show a different finite-$N$ factor; checking the homogeneous analogue cited in the paper would settle that.
  • The factor $1+e^{-2/\tau}$ is an interpolation between the low- and high-temperature limits; deriving it from the fermionic density-matrix sums might extend the collapse to weaker interactions ($z\le 1$) or to multi-component mixtures.
  • A practical experimental test is to measure the momentum-tail coefficient in a few-atom 1D tube for two or three particle numbers at fixed $z$ and $\tau$, and compare the ratio $C_N/C_N(\infty)$ with the two-boson ratio.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. The paper studies the canonical Tan's contact of N repulsive Lieb-Liniger bosons in a one-dimensional harmonic trap at finite temperature, with a focus on small N. The authors compute the contact exactly for two bosons at arbitrary interaction and for N bosons in the Tonks-Girardeau (TG) limit, and use quantum Monte Carlo simulations for intermediate interactions. Their central claim is that in the strongly interacting regime (z > 1) the ratio f_N(z, τ) = C_N(g(z), T(τ))/C_N(∞, T(τ)) is approximately independent of N, so that all nontrivial particle-number dependence is carried by the TG contact. To make this explicit they conjecture an analytic N-dependence for the TG contact, Eq. (23), with an interpolating factor s(N) that bridges the zero-temperature N^{3/4} correction and the high-temperature pair-counting behavior. The paper also compares canonical and grand-canonical contacts and shows that the proposed scaling fails in the grand-canonical ensemble at intermediate interactions.

Significance. If the central universality claim were established, it would be a practically useful result: the canonical contact for a trapped Lieb-Liniger gas in the strong-coupling regime could be obtained from the exactly computable two-boson curve and the N-body TG contact. The paper has genuine strengths: the two-boson calculation and the TG contact evaluation are clean, the QMC data appear to have small errors, and the comparison between canonical and grand-canonical ensembles is informative. The proposed scaling relation is interesting and plausible. However, the main quantitative formula for the TG contact, Eq. (23), is internally inconsistent with the paper's own high-temperature limit, and the central universality claim is tested only in a narrow parameter window. These issues currently prevent the paper from supporting the advertised "any N, any T" conclusion.

major comments (1)
  1. [Sec. III.C, Eq. (23)] The central scaling hypothesis (25) is tested only for N ≤ 5, z = 1 and 2.5, and rescaled temperatures roughly in the range 0.1 ≲ τ ≲ 1 in Figs. 4(d) and 5(d). The abstract and introduction state "we show" and "for any number of particles and temperature," which overstates the evidence. Since Eq. (25) is presented as a conjecture, the language should be qualified throughout, and the conclusion should clearly delimit the tested regime. A direct test at larger N or wider τ, even for the TG contact where exact results are available, would materially strengthen the claim.
minor comments (4)
  1. [Sec. III.C, Eq. (24) and Fig. 3] The dashed line in Fig. 3 is labeled as the high-temperature limit h_2(∞, τ ≫ 1), but the plotted object appears to be h_N(∞, τ ≫ 1) of Eq. (20), which is not the τ → ∞ limit of h_2(∞, τ) as defined in Eq. (24). This labeling should be corrected to avoid confusion, especially given the issue raised in the first major comment.
  2. [Sec. II.B and Sec. III.C] There are several typos: "g is ininite" should be "g is infinite," and "can de derived" should be "can be derived." These should be corrected.
  3. [Figs. 4 and 5] The horizontal axes of Figs. 4 and 5 are labeled only "0.1 1"; the tick labels and the intended τ range should be made explicit so the reader can see exactly which temperatures are covered.
  4. [References] Reference [3] (Yang and Yang) appears to have an incorrect journal and volume: the standard citation is J. Math. Phys. 10, 1115 (1969). Please verify all bibliographic entries.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the universal scaling law (Eq. 25) is tested against independent exact two-body and exact Tonks-Girardeau inputs, and Eq. (23) is explicitly labeled a conjecture rather than a prediction derived from its own inputs.

full rationale

The central scaling claim is Eq. (25), f_N(z>1,τ)=C_N(g,T)/C_N(∞,T)≃f_2(z>1,τ). The denominator C_N(∞,T) is computed exactly from the fermionic two-body density matrix (Eqs. 7-9), the numerator is independent QMC data obtained from Eq. (3), and f_2 is the exact two-boson contact from Eq. (5). Figures 4(d) and 5(d) therefore compare three independent inputs, so the observed collapse is not forced by any definition. Equation (23) is explicitly presented as a conjecture for the N-dependence of the Tonks-Girardeau contact, and although h_2 is defined as C_2/s(2), making the N=2 case an identity, the N=3 to 5 verification against exact Eq. (7) is a genuine consistency check; the later QMC test of Eq. (25) uses the exact C_N(∞) in the denominator, not the conjectured s(N). The self-citations ([37], [45], [46]) provide motivation and analytic ingredients, but the main result is benchmarked internally against exact two-body and TG data, so no load-bearing circular step is present. The apparent high-temperature limit issue in Eq. (23), which does not reduce to the pair-counting result Eq. (19), and the restriction of tests to N≤5, τ≤5 are correctness and extrapolation risks rather than circularity.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

No new physical entities are postulated. The central claim depends on two fitted or hand-chosen parameters (the zero-T exponent from prior work and the crossover coefficient), two standard physical mappings, and the explicit conjectures in Eqs. (23) and (25).

free parameters (2)
  • Zero-temperature correlation exponent eta = 3/4
    Taken from the fit in the authors' prior work [37]; enters s(N) as N^{3/4} and controls the low-tau scaling used in the collapse.
  • Crossover coefficient in s(N) = 2 (in exp(-2/tau))
    Chosen by hand so s(N) interpolates between N^{5/2} - N^{3/4} at tau -> 0 and N^{5/2} - N^{3/2} at tau -> infinity; it is not derived and is tuned to make the QMC and exact TG data collapse.
assumptions (6)
  • domain assumption Tan's sweep relation (Eq. 2) gives the contact from the free energy derivative in the canonical ensemble.
    Standard Tan theorem, assumed valid for finite N, trapped, finite-temperature Lieb-Liniger bosons.
  • domain assumption The two-body energy spectrum satisfies the implicit relation f(nu) = -sqrt(2)|a1D|/aho (Eq. 4).
    Taken from Busch et al. [12]; used for the exact two-boson contact.
  • domain assumption In the Tonks-Girardeau limit the contact can be computed from the fermionic two-body density matrix (Eqs. 7-9).
    Requires the fermionization mapping at infinite repulsion; standard but not derived in this paper.
  • domain assumption Stochastic Green function QMC on the discretized Hubbard model (Delta = 0.1) gives exact canonical averages within quoted errors.
    The method is cited [47, 48]; systematic discretization errors are checked on some simulations only.
  • ad hoc to paper The scaling ansatz f_N(z > 1, tau) approx f_2(z > 1, tau) holds for all N and temperatures (Eq. 25).
    This is the central conjecture; it is not derived and is supported only by QMC data for N <= 5 and a limited tau range.
  • ad hoc to paper The interpolating form s(N) = N^{5/2} - N^{3/4}(1 + exp(-2/tau)) reproduces the exact TG contact at all temperatures (Eq. 23).
    Proposed to match zero-T and large-T limits; the exponential crossover is not derived from the model.

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Cite this review

Pith. "Pith review of Scaling properties of the Tan's contact: embedding pairs and correlation effect in the Tonks limit." pith.science (2026). https://pith.science/paper/5U75O4K2

@misc{pith2026190808714,
  author       = {Pith},
  title        = {Pith review of: Scaling properties of the Tan's contact: embedding pairs and correlation effect in the Tonks limit},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5U75O4K2}},
  note         = {Machine review of arXiv:1908.08714}
}
read the original abstract

We study the Tan's contact of a one dimensional quantum gas of N repulsive identical bosons confined in a harmonic trap at finite temperature. This canonical ensemble framework corresponds to the experimental conditions, the number of particles being fixed for each experimental sequence. We show that, in the strongly interacting regime, the contact rescaled by the contact at the Tonks-Girardeau limit is an universal function of two parameters, the rescaled interaction strength and temperature. This means that all pair and correlation effects in the Tan's contact are embedded in the Tan's contact in the Tonks-Girardeau limit.

Figures

Figures reproduced from arXiv: 1908.08714 by the authors.

Figure 2
Figure 2. FIG. 2: Canonical (empty symbols) [Eq. (7)] and grand [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Canonical contact in the Tonks-Girardeau limit [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4: Panels (a), (b) and (c) [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figures from the paper (3 more)
Figure 5
Figure 5. Figure 5: FIG. 5: Panels (a), (b) and (c) [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Canonical (empty symbols) and grand-canonical con [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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