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REVIEW 2 major objections 5 minor 57 references

Exact Results for the Boundary Energy of One-Dimensional Bosons

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Pith's one-line read Exact boundary energy of a 1D Bose gas is fixed by a corrected integral equation

desk verdict A genuinely useful correction of Gaudin's boundary-energy formula, with a new exact integral equation; the one real caveat is a terse continuum-limit step, but the result is well supported and deserves peer review. read the letter →

arxiv 1908.08172 v2 pith:A2L3PO6W submitted 2019-08-22 cond-mat.quant-gas cond-mat.stat-mechmath-phmath.MPquant-ph

Benjamin Reichert , Grigori E. Astrakharchik , Aleksandra Petkovic , Zoran Ristivojevic This is my paper · ORCID
classification cond-mat.quant-gascond-mat.stat-mechmath-phmath.MPquant-ph
keywords Lieb-LinigermodelboundaryenergyBetheansatzhard-wallboxintegralequationdarksolitonone-dimensionalbosonsquantumMonteCarlo
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

This paper claims to settle the boundary energy of the one-dimensional Lieb-Liniger Bose gas confined by hard walls: the energy cost of the walls' presence is fixed exactly, for any strength of inter-particle repulsion, by a single linear integral equation. Gaudin's 1971 Bethe-ansatz treatment contained an equation of the same form but with the wrong right-hand side; it actually computes the energy of a moving dark soliton, not the boundary energy. With the corrected right-hand side the authors obtain the weak-coupling expansion $E_B = \frac{8}{3}\epsilon\sqrt{\gamma}\,[1 - \frac{3}{16}\sqrt{\gamma} + O(\gamma)]$ and the strong-coupling expansion $E_B = \frac{\pi^2}{2}\epsilon\,[1 - \frac{4}{3\gamma} - \frac{4}{\gamma^2} + O(\gamma^{-3})]$, and verify the result against diffusion Monte Carlo simulations. Because the boundary energy is the leading finite-size correction in box-trapped atomic gases, an exact formula is a direct check on experiments and on approximate treatments of inhomogeneous one-dimensional quantum gases.

What carries the argument

The load-bearing object is the linear integral equation for the odd function $g(k) = L\rho(k)(\bar{k}_i - k_i)$, obtained by subtracting the Bethe ansatz equations for zero and periodic boundary conditions and replacing the discrete quasimomentum sums by integrals over the Lieb density $\rho(k)$. The correct source term $r(k) = \operatorname{sgn}(k)/4 + \frac{1}{2\pi}\arctan\frac{2k}{c}$ carries the boundary condition's effect; it is the only place where Gaudin's equation is altered. The paper then reformulates the problem through the symmetric Green's function of the Lieb equation, defining $\sigma(k)$ that satisfies the same integral equation with source $k$, so that $E_B = \int \sigma(k) r(k)\, dk$. This Green's-function form is what makes the weak- and strong-coupling asymptotic expansions tractable.

What would settle it

Solve the Bethe-ansatz equations (6) and (8) numerically for $N = 100, 1000, 10000$ at several interaction strengths (e.g., $\gamma = 0.1, 1, 10$), compute $\Delta k_i$, and form $g_N(k) = L\rho(k)\Delta k_i$; if $g_N(k)$ does not approach the solution of Eq. (14) with corrections that vanish as $1/N$ uniformly in $\gamma$, then the integral equation is not exact. A finite-$N$ extrapolation of $E_B$ from the Bethe equations that disagrees with Eq. (15) at any $\gamma$ would also settle it.

Watch

Extended reading notes

Core claim

The central result is the integral equation $g(k) - \frac{c}{\pi}\int_{-Q}^{Q} dk'\, \frac{g(k')}{c^2 + (k'-k)^2} = r(k)$ with the source term $r(k) = \frac{\operatorname{sgn}(k)}{4} + \frac{1}{2\pi}\arctan\frac{2k}{c}$, together with the energy formula $E_B = \frac{\hbar^2}{m}\int_{-Q}^{Q} k\, g(k)\, dk$. The paper argues that this pair gives the exact thermodynamic-limit boundary energy for every repulsion strength $c > 0$, because the difference between the zero- and periodic-boundary Bethe ansatz quasimomenta becomes, in the thermodynamic limit, a smooth odd function $g(k)$ solving this equation. The correction to Gaudin is the extra $\frac{1}{2\pi}\arctan\frac{2k}{c}$ term: Gaudin's source $r_G(k) = \operatorname{sgn}(k)/2$ is only the $c\to 0$ limit. Consequently, what Gaudin computed is exactly the energy of Lieb's type-II excitation at momentum $\pi\hbar n$ (the dark soliton in the weak-coupling limit), which is always larger than the true boundary energy. The paper also shows that the first two terms of the strong-coupling expansion are universal, matching the exact hard-rod gas result when written in terms of the gas parameter.

Load-bearing premise

The derivation replaces the discrete Bethe-ansatz equations by a continuum integral equation, assuming that the quasimomentum shifts $\Delta k_i$ are $O(1/L)$ and converge to a smooth odd function $g(k)$ with a remainder that vanishes uniformly as $N\to\infty$ for every interaction strength $c>0$; if that remainder is not uniform in $c$, the claimed exactness at arbitrary interaction would be lost.

Editorial extensions

If this is right

  • The exact boundary energy can be evaluated numerically for any $\gamma$, providing a benchmark for finite-size corrections in box-trapped one-dimensional Bose gases.
  • The subleading term in the weak-coupling boundary energy gives a precise beyond-mean-field prediction that can be compared with experiments and with Gross-Pitaevskii-based treatments.
  • Gaudin's dark-soliton identification is corrected: the dark soliton and the boundary energy are distinct except at leading weak-coupling order, settling a long-standing confusion.
  • The universality of the first two strong-coupling coefficients extends to hard-rod gases and other short-range-interacting one-dimensional systems, so the boundary energy depends only on the gas parameter at low density.
  • The asymptotic expansions together with the integral equation supply a controlled interpolation across the crossover from Tonks-Girardeau to mean-field regimes.

Reading between the lines

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

  • If the equation is exact, finite-$N$ corrections to the boundary energy could be computed systematically by expanding around the thermodynamic-limit $g(k)$, giving a route to quantitative finite-size scaling in one-dimensional traps.
  • The distinction between the type-II excitation and the boundary energy suggests that other 'boundary' quantities defined through Bethe ansatz comparisons may similarly hide a soliton energy; the same subtract-and-expand method could be applied to spin chains or Fermi gases in hard-wall boxes.
  • A testable extension: measure the density profile near a hard wall in an ultracold one-dimensional Bose gas; the boundary energy's dependence on $\gamma$ implies a specific healing-length correction that time-of-flight imaging could resolve.
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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

2 major / 5 minor

Summary. The paper studies the ground-state boundary energy EB of one-dimensional bosons with contact interaction (Lieb-Liniger model) in a hard-wall box. Starting from Gaudin's Bethe-Ansatz equations for zero and periodic boundary conditions, the authors subtract the two sets of equations and, after a discrete-to-continuum step, obtain a linear integral equation (14) for an odd function g(k) that determines EB through Eq. (15). This is claimed to be the exact thermodynamic-limit boundary energy for arbitrary interaction c>0. The paper then solves the integral equation in the weakly and strongly interacting limits, obtaining EB=(8/3)ε√γ[1−3√γ/16+O(γ)] and EB=(π²/2)ε[1−4/(3γ)−4/γ²+O(γ⁻³)], and compares the results with quantum Monte Carlo data and with a dual-fermion perturbation calculation. It also shows that Gaudin's earlier boundary-energy expression coincides with the energy of the zero-velocity type-II excitation and overestimates the true boundary energy, and it argues for a universal leading form of EB in terms of the gas parameter using the hard-rod gas as a comparison.

Significance. If the central derivation is accepted, the paper settles a long-standing question: it provides a compact integral-equation characterization of the boundary energy of the Lieb-Liniger model, corrects Gaudin's 1971 result beyond leading order, and exposes the distinction between the boundary energy and the dark-soliton/type-II-excitation energy. The result is elegant and should be useful for finite-size analyses of one-dimensional Bose gases in box potentials and for impurity problems with an infinite repulsive wall. The manuscript is strengthened by independent numerical checks (diffusion Monte Carlo) and by an analytic dual-fermion check of the strong-coupling expansion. The derivations are algebraic and the integral equations used are standard for the Lieb-Liniger model, and no free parameters are fitted to the boundary energy.

major comments (2)
  1. [Equations (11)-(14)] The central exactness claim is not fully established because the transition from the finite-N Bethe-Ansatz equations to the integral equation (14) replaces sums over the N positive quasimomenta by integrals against the continuum density 2Lρ(k) with no estimate of the remainder. Equation (11) carries an O(1/N) symbol, but the error in approximating Σ_j [θ′(k_i−k_j)−θ′(k_i+k_j)]Δk_j by the corresponding integral is not bounded. This is load-bearing rather than cosmetic because EB is an O(1) quantity obtained from an O(1/N) difference of two large energies, so an uncontrolled O(1/N) remainder could in principle contribute. At weak coupling the Lieb density vanishes as (Q−k)^{1/2} near the Fermi edge, so the local quasimomentum spacing there is much larger than 1/N and standard Euler-Maclaurin-style estimates fail; the limit N→∞ may also be nonuniform in γ. To support the statement that Eq. (14) is exact for all c>0, the authors should either supply a justified remainder estimate (or cite a known theorem for this continuum limit), or explicitly restate the status of Eq. (14) as an assumption/conjecture rather than a proven exact result. The same issue affects the weak-coupling expansion (19) if the interchange of N→∞ and γ→0 is not controlled.
  2. [Supplemental Material, Eqs. (S6)-(S9)] The weak-coupling solution ρ(x) and σ(x) used to derive Eq. (19) is stated to be valid only for 1−x²≫λ, yet the derivation integrates these expressions over the entire interval including the Fermi-edge region. The text asserts that this limitation is not important, but gives no estimate of the boundary-layer contribution. Because Eq. (19) contains a genuine subleading correction (the −3√γ/16 term), the authors should show explicitly that the region 1−x²≲λ contributes only at higher order in γ to λ, σ, and ultimately EB; otherwise the quoted first correction is not justified by the presented calculation.
minor comments (5)
  1. [Equation (11)] The two terms in the square bracket, θ′(k_i−k_j)(Δk_i−Δk_j) and θ′(k_i+k_j)(Δk_i+Δk_j), appear without a visible plus sign between them; please correct the typography.
  2. [Acknowledgment] The name in the acknowledgment is typeset as "V . Yurovky"; it should read "V. A. Yurovsky".
  3. [Figure 1 and QMC paragraph] The statement that the N=21 and N=41 quantum Monte Carlo results agree with the Bethe-Ansatz curve would be more persuasive with a quantitative statement of the finite-N difference and, if possible, an extrapolation to the thermodynamic limit; the current text relies on visual inspection of the figure.
  4. [Hard-rod universality section] The notation γ=−2/na in Eq. (22) reuses the symbol γ, which was earlier defined as c/n>0. Please clarify that this is a different, negative gas parameter for the hard-rod mapping, so that the comparison with Eq. (20) is unambiguous.
  5. [Last paragraph] The sentence about the regime L≲ξ would benefit from a more explicit statement of the order of limits: the derivation of Eq. (15) takes L→∞ with density fixed, and the very-weak-coupling finite-size regime γ≲1/N² is a separate limit that the paper deliberately does not address.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: boundary-energy integral equation follows from Bethe ansatz with no fitted inputs; remaining continuum-limit gap is a rigor issue, not circularity.

full rationale

This paper's derivation chain is self-contained rather than circular. The boundary energy is defined by Eq. (10) as the thermodynamic limit of the difference between zero- and periodic-boundary Bethe ansatz ground-state energies. By subtracting the Bethe ansatz equations (6) and (8), and using only the Lieb density integral equation (12) to replace the discrete sums, the paper obtains the integral equation (14) with an explicit inhomogeneity r(k) = sgn(k)/4 + arctan(2k/c)/(2pi). Nothing in Eq. (14) or Eq. (15) is fitted to the boundary energy; Q, lambda, and rho(k) are fixed by density normalization, and the expansions (19)-(20) are obtained by solving the integral equation. The independent checks (QMC, dual-fermion perturbation theory, hard-rod excluded volume) do not feed values back into the Bethe ansatz derivation. The one author self-citation that plays a technical role, Ref. [35], supplies a published perturbative solution of the Lieb integral equation for the excitation spectrum; it does not contain the boundary energy as an input and is therefore independent support rather than circularity. The step between Eqs. (11) and (14) replaces discrete sums by integrals against rho(k) with an unproved O(1/N) remainder; this is a mathematical-rigor or correctness risk, not a circular reduction, because the equations are not equivalent to their inputs by construction. Hence no circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard Bethe-ansatz integrability, Gaudin's zero-boundary equations, and the thermodynamic-limit continuum approximation for the quasimomentum density. No free parameters are fitted to the boundary-energy data, and no new physical entities are introduced. The asymptotic expansions rely on cited previous solutions of the Lieb integral equation, which are independent of the boundary-energy result itself.

assumptions (4)
  • domain assumption The Lieb-Liniger Hamiltonian (4) is exactly solvable by Bethe ansatz, and Eq. (6) gives all ground-state quasimomenta for periodic boundary conditions.
    Invoked at Eqs. (5)-(6); this is the established exact solution of Lieb and Liniger, taken as input rather than rederived in this paper.
  • domain assumption Gaudin's zero-boundary Bethe ansatz equations, Eqs. (7)-(8), are exact for the ground state of the Lieb-Liniger model in a hard-wall box.
    Taken from Ref. [22]; the paper relies on these equations without rederiving them.
  • standard math In the thermodynamic limit, discrete Bethe-ansatz sums can be replaced by integrals weighted by the periodic Lieb density ρ(k) defined in Eqs. (12)-(13), with O(1/N) corrections vanishing.
    This continuum-limit replacement is used between Eqs. (11) and (14); the paper does not provide a rigorous error bound in the main text.
  • domain assumption The weak-coupling density solution (S6) from Popov (Ref. [33]) and the strong-coupling expansion of Ref. [35] are correct to the stated orders.
    These external results enter the asymptotic expansions (19) and (20) and are cited rather than rederived in this paper.

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Pith. "Pith review of Exact Results for the Boundary Energy of One-Dimensional Bosons." pith.science (2026). https://pith.science/paper/A2L3PO6W

@misc{pith2026190808172,
  author       = {Pith},
  title        = {Pith review of: Exact Results for the Boundary Energy of One-Dimensional Bosons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A2L3PO6W}},
  note         = {Machine review of arXiv:1908.08172}
}
read the original abstract

We study bosons in a one-dimensional hard-wall box potential. In the case of contact interaction, the system is exactly solvable by the Bethe ansatz, as first shown by Gaudin in 1971. Although contained in the exact solution, the boundary energy in the thermodynamic limit for this problem is only approximately calculated by Gaudin, who found the leading order result at weak repulsion. Here we derive an exact integral equation that enables one to calculate the boundary energy in the thermodynamic limit at an arbitrary interaction. We then solve such an equation and find the asymptotic results for the boundary energy at weak and strong interactions. The analytical results obtained from the Bethe ansatz are in agreement with the ones found by other complementary methods, including quantum Monte Carlo simulations. We study the universality of the boundary energy in the regime of a small gas parameter by making a comparison with the exact solution for the hard rod gas.

Figures

Figures reproduced from arXiv: 1908.08172 by the authors.

Figure 1
Figure 1. The boundary energy EB in units of  as a function of the interaction strength γ. The lower (black) dots represent the exact numerically obtained results, while the two asymptotic behaviors at small and large γ are given by formulas (19) and (20). The upper (brown) dots represent the result of Gaudin [22] and coincides with the energy of Lieb’s type-II excitation with zero velocity (momentum π¯hn) in the model with … view at source ↗

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    G. E. Astrakharchik and S. Giorgini, “Correlation functions and momentum distribution of one-dimensional Bose systems,” Physical Review A 68, 031602 (2003)

    2003

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    Beyond the Tonks-Girardeau Gas: Strongly Correlated Regime in Quasi-One-Dimensional Bose Gases,

    G. E. Astrakharchik, J. Boronat, J. Casulleras, and S. Giorgini, “Beyond the Tonks-Girardeau Gas: Strongly Correlated Regime in Quasi-One-Dimensional Bose Gases,” Physical Re- view Letters 95, 190407 (2005)

    2005

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    Ultradilute Low- Dimensional Liquids,

    D. S. Petrov and G. E. Astrakharchik, “Ultradilute Low- Dimensional Liquids,” Physical Review Letters 117, 100401 (2016)

    2016

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    Spin Dy- namics and Andreev-Bashkin Effect in Mixtures of One- Dimensional Bose Gases,

    L. Parisi, G. E. Astrakharchik, and S. Giorgini, “Spin Dy- namics and Andreev-Bashkin Effect in Mixtures of One- Dimensional Bose Gases,” Physical Review Letters 121, 025302 (2018)

    2018

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    Liquid State of One-Dimensional Bose Mixtures: A Quantum Monte Carlo Study,

    L. Parisi, G. E. Astrakharchik, and S. Giorgini, “Liquid State of One-Dimensional Bose Mixtures: A Quantum Monte Carlo Study,” Physical Review Letters122, 105302 (2019)

    2019

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    Phonons and the Properties of a Bose System,

    L. Reatto and G. V . Chester, “Phonons and the Properties of a Bose System,” Physical Review155, 88 (1967)

    1967

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    Low-dimensional weakly interacting Bose gases: Nonuniversal equations of state,

    G. E. Astrakharchik, J. Boronat, I. L. Kurbakov, Yu. E. Lo- zovik, and F. Mazzanti, “Low-dimensional weakly interacting Bose gases: Nonuniversal equations of state,” Physical Review A 81, 013612 (2010)

    2010

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    Beyond-Luttinger-liquid thermody- namics of a one-dimensional Bose gas with repulsive contact interactions,

    Giulia De Rosi, Pietro Massignan, Maciej Lewenstein, and Grigori E. Astrakharchik, “Beyond-Luttinger-liquid thermody- namics of a one-dimensional Bose gas with repulsive contact interactions,” Physical Review Research1, 033083 (2019)

    2019

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    Ground-state properties of a one-dimensional system of dipoles,

    A. S. Arkhipov, G. E. Astrakharchik, A. V . Belikov, and Yu. E. Lozovik, “Ground-state properties of a one-dimensional system of dipoles,” Journal of Experimental and Theoretical Physics Letters 82, 39 (2005)

    2005

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    Evi- dence of Luttinger-liquid behavior in one-dimensional dipolar quantum gases,

    R. Citro, E. Orignac, S. De Palo, and M. L. Chiofalo, “Evi- dence of Luttinger-liquid behavior in one-dimensional dipolar quantum gases,” Physical Review A75, 051602 (2007)

    2007

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    Super-Tonks- Girardeau State in an Attractive One-Dimensional Dipolar Gas,

    M. D. Girardeau and G. E. Astrakharchik, “Super-Tonks- Girardeau State in an Attractive One-Dimensional Dipolar Gas,” Physical Review Letters109, 235305 (2012)

    2012

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    Phase diagram of Rydberg atoms with repulsive van der Waals interaction,

    O. N. Osychenko, G. E. Astrakharchik, Y . Lutsyshyn, Yu. E. Lozovik, and J. Boronat, “Phase diagram of Rydberg atoms with repulsive van der Waals interaction,” Physical Review A 84, 063621 (2011)

    2011

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    One-Dimensional Liquid 4He: Dynamical Properties be- yond Luttinger-Liquid Theory,

    G. Bertaina, M. Motta, M. Rossi, E. Vitali, and D. E. Galli, “One-Dimensional Liquid 4He: Dynamical Properties be- yond Luttinger-Liquid Theory,” Physical Review Letters 116, 135302 (2016)

    2016

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    Luttinger-liquid behav- ior of one-dimensional 3He,

    G. E. Astrakharchik and J. Boronat, “Luttinger-liquid behav- ior of one-dimensional 3He,” Physical Review B 90, 235439 (2014)

    2014

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    Realization of an excited, strongly correlated quantum gas phase,

    Elmar Haller, Mattias Gustavsson, Manfred J. Mark, Johann G. Danzl, Russell Hart, Guido Pupillo, and Hanns-Christoph N¨agerl, “Realization of an excited, strongly correlated quantum gas phase,” Science 325, 1224 (2009)

    2009

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    S7 Exact Results for the Boundary Energy of One-Dimensional Bosons –Supplemental material– Benjamin Reichert1, Grigori E

    Michel Gaudin, The Bethe Wavefunction (Cambridge Univer- sity Press, 2014). S7 Exact Results for the Boundary Energy of One-Dimensional Bosons –Supplemental material– Benjamin Reichert1, Grigori E. Astrakharchik2, Aleksandra Petkovi´c1, and Zoran Ristivojevic1 1Laboratoire de ...

    2014

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.