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arxiv: 2604.24784 · v1 · submitted 2026-04-24 · ❄️ cond-mat.quant-gas · cond-mat.stat-mech· math-ph· math.MP

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The Lieb-Liniger model

Zoran Ristivojevic
Authors on Pith no claims yet

Pith reviewed 2026-05-08 08:52 UTC · model grok-4.3

classification ❄️ cond-mat.quant-gas cond-mat.stat-mechmath-phmath.MP
keywords Lieb-Liniger modelBethe ansatzone-dimensional bosonscontact interactionsground-state energyexcitation spectrumboundary energyintegrable quantum gases
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The pith

The Lieb-Liniger model for one-dimensional bosons with contact interactions admits an exact Bethe ansatz solution that yields explicit expressions for energies and spectra in terms of the interaction parameter.

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

The paper reviews how the Lieb-Liniger Hamiltonian for bosons in one dimension is solved exactly by the Bethe ansatz, which determines the allowed momenta and wavefunctions for any interaction strength. It compiles known exact and perturbative results while deriving new explicit formulas for the ground-state energy series, high-energy excitations, and boundary energy, all expressed directly through the interaction parameter. A sympathetic reader cares because this system models real ultracold-atom experiments in tight waveguides, where exact results can be compared to data without uncontrolled approximations. The review also emphasizes identities that link different physical quantities and explains the technical steps needed to extract numbers from the formal solution.

Core claim

The Lieb-Liniger model describes one-dimensional bosons with contact interactions. This many-body system admits an exact solution in terms of the Bethe ansatz. Some of the exact and perturbative results for this model are reviewed. Particular attention is devoted to the explicit evaluation, in terms of the interaction parameter, of physical quantities that can be formally exactly extracted from the Bethe ansatz solution. Another goal of this review is to stress exact relations between various quantities. The technical developments are explained in detail. The most relevant experimental realisations of the studied problems are eventually discussed. This review also contains several new ics:

What carries the argument

The Bethe ansatz, which constructs the exact eigenstates from superpositions of plane waves whose rapidities satisfy a set of transcendental equations parameterized by the dimensionless interaction strength gamma.

If this is right

  • Explicit series expansions for the ground-state energy become available in both weak- and strong-coupling limits and can be used directly for thermodynamics.
  • Exact identities connect the energy to other observables such as pressure and sound velocity without further computation.
  • The boundary energy for open or periodic systems follows from the same Bethe equations and can be evaluated numerically to arbitrary precision.
  • High-energy states with large rapidities produce a continuous spectrum whose density can be compared to time-of-flight images in experiments.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The closed-form expressions at arbitrary coupling may enable analytic studies of correlation functions or quench dynamics that remain inaccessible in non-integrable models.
  • The demonstrated convergence of the strong-coupling series supplies a benchmark that can guide variational or numerical methods applied to nearby integrable or nearly integrable systems.
  • Direct comparison of the predicted boundary energy with finite-size cold-atom data could test the completeness of the Bethe-ansatz description under realistic trap conditions.

Load-bearing premise

The Bethe ansatz supplies a complete set of eigenstates for the Lieb-Liniger Hamiltonian, so that all subsequent explicit evaluations and new calculations can be carried out without missing sectors or extra approximations.

What would settle it

A high-precision measurement or exact numerical diagonalization of the ground-state energy or the high-energy excitation spectrum at a calibrated value of the interaction parameter that deviates from the explicit Bethe-ansatz formulas beyond numerical error.

Figures

Figures reproduced from arXiv: 2604.24784 by Zoran Ristivojevic.

Figure 1
Figure 1. Figure 1: Plot of the ground-state function e2(γ). The dots represent numerically exact values, while the curves are obtained from the analytical results. In the regime of small γ, the dashed line is drawn using the first three terms and the solid line is drawn using all nine terms of the series (3.13). In the regime of large γ, the dashed line is plotted using the first three terms of the series (4.5) and the solid… view at source ↗
Figure 2
Figure 2. Figure 2: Plot of the Luttinger liquid parameter K as a function of γ that closely parallels the one of figure 1. The dots represent numerically exact values, while the curves are obtained from the analytical results. In the regime of small γ, the dashed line is drawn using the first three terms and the solid line is drawn using nine terms of the corresponding series. In the regime of large γ, the dashed line is plo… view at source ↗
Figure 3
Figure 3. Figure 3: The spectrum of type-I elementary excitations in the Lieb–Liniger model has its characteristic form in several regions of the interaction-momentum plane. The number in parentheses denotes the equation where the spectrum is given. The crossover momenta are p0 ∼ γ 1/2 and p ∗ ∼ γ 3/4 at small γ, and p ∗ ∼ γ at large γ. The asymptotes cross at γc = 1, corresponding to the momentum ℏn. At weak interactions, th… view at source ↗
Figure 4
Figure 4. Figure 4: Plot of the coefficient C4. The dots represent numerically exact values and the curves are obtained from the analytical results (8.50) and (8.51). An alternative form is given by C4 = 29K2K1 5760π 2 + 7γKK′K1 5760π 2 + γK2K′ 1 1152π 2 − K4 180π 2 − 49γK3K′ 5760π 2 − γ 2K2K′2 1536π 2 − 17γ 2K3K′′ 11520π 2 . (8.49) Equation (8.49) is exact and expressed in terms of two dimensionless parameters K and K1. At t… view at source ↗
Figure 5
Figure 5. Figure 5: The dimensionless free energy per particle f(γ, τ) obtained from the analytical expression (8.47) is represented by the curves and the one obtained by numerically solving the Yang–Yang equation is shown by the dots. Four distinct values of τ are considered: 0.1, 1, 2, and 3. For a fixed value of γ, the function f(γ, τ) decreases when τ increases. This is expected as the entropy of the system increases with… view at source ↗
Figure 6
Figure 6. Figure 6: The boundary energy EB in units of ϵ as a function of the interaction strength γ. The dots represent the exact result obtained numerically. The two curves describe the series expansions at small and large γ that are given by equations (9.29) and (9.33), while the dashed line is at π 2/2 corresponding to the value at γ → ∞. For 2 ≲ γ ≲ 9, on the plot there is a visible deviation between the exact result and… view at source ↗
Figure 7
Figure 7. Figure 7: Capacitance C(κ) as a function of the distance between the plates κ. The dots represent the exact result obtained by solving numerically equation (10.3). The solid red line is the κ → 0 expansion to O(κ 6 ) order, while the solid blue line is κ → ∞ expansion to O(1/κ8 ) order. 10.4 The capacitance at small separations Now that all the coefficients cn,m,k can be systematically calculated, we can obtain the … view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of the exact result for the polaron dispersion (the dots) and the obtained parametric form (solid curves) given by equations (11.31) and (11.34) for γ = 0.1. The dashed curve is the dark soliton dispersion. On the plot is only shown the region of momenta 0 ≤ p ≤ πℏn. At other values of p, the dispersion follows from the symmetry (11.16) and the periodicity (11.20). where ϑ = 2η/c is kept fixed. … view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of the exact result for the polaron dispersion (the dots) and the obtained expression (11.39) (solid curve) for γ = 20. The dashed curve represents the dispersion (11.39) at the leading order that only accounts for the term proportional to 1/γ. As a consistency check, we have verified that the substitution of m/m∗ given by equation (11.28) and ν calculated in equation (11.29) into equation (11.4… view at source ↗
read the original abstract

The Lieb-Liniger model describes one-dimensional bosons with contact interactions. This many-body system admits an exact solution in terms of the Bethe ansatz. Some of the exact and perturbative results for this model are reviewed. Particular attention is devoted to the explicit evaluation, in terms of the interaction parameter, of physical quantities that can be formally exactly extracted from the Bethe ansatz solution. Another goal of this review is to stress exact relations between various quantities. The technical developments are explained in detail. The most relevant experimental realisations of the studied problems are eventually discussed. This review also contains several new results such as the study of convergence of the ground-state energy series at strong interactions, the excitation spectrum at high energies, and the evaluation of the boundary energy.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript reviews the Lieb-Liniger model of one-dimensional bosons with contact interactions. It states that the model admits an exact solution via the Bethe ansatz, reviews exact and perturbative results for quantities such as the ground-state energy and excitation spectrum, emphasizes explicit evaluations in terms of the interaction parameter and exact relations between quantities, provides detailed technical explanations, discusses experimental realizations, and presents new results on the convergence of the strong-coupling ground-state energy series, the high-energy excitation spectrum, and the boundary energy.

Significance. If the derivations and new calculations are correct, this review would serve as a useful reference in the field of integrable quantum gases. The Lieb-Liniger model is paradigmatic for Bethe-ansatz techniques with direct relevance to ultracold-atom experiments in one dimension. Compiling explicit expressions, analyzing series convergence at strong coupling, and treating boundary effects extends the practical utility of the solution while the focus on exact relations and technical details aids clarity for theorists and experimentalists.

minor comments (2)
  1. The abstract and introduction should more explicitly delineate which parts of the manuscript contain the new results (convergence study, high-energy spectrum, boundary energy) versus reviews of established material.
  2. Notation for the interaction parameter and related quantities (e.g., c, gamma) should be checked for consistency across sections and equations.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript reviewing the Lieb-Liniger model, its Bethe-ansatz solution, and the new results on series convergence, high-energy spectrum, and boundary energy. We appreciate the recognition of its potential utility as a reference for theorists and experimentalists working with integrable quantum gases. The recommendation is for minor revision, but no specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

This is a review paper whose central premise is the established fact that the Lieb-Liniger Hamiltonian with repulsive contact interactions is exactly solvable by the Bethe ansatz (a result originating in the 1963 Lieb-Liniger paper and subsequent literature). All subsequent explicit evaluations of ground-state energy, excitation spectrum, boundary energy, and series convergence are presented as direct extractions or extensions from that external solution rather than quantities defined in terms of the paper's own fitted outputs or self-referential predictions. No self-definitional steps, fitted-input predictions, or load-bearing self-citation chains appear; the new results remain independent computations within the standard framework.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The review rests on the established Bethe-ansatz solution of the Lieb-Liniger Hamiltonian; no new free parameters, ad-hoc axioms, or invented entities are introduced beyond the standard model definition.

axioms (1)
  • domain assumption The Bethe ansatz yields the complete set of eigenstates and eigenvalues for the Lieb-Liniger Hamiltonian with delta-function interactions.
    Invoked throughout the review as the basis for all exact and perturbative results.

pith-pipeline@v0.9.0 · 5424 in / 1379 out tokens · 33021 ms · 2026-05-08T08:52:25.932996+00:00 · methodology

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