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arxiv: 2604.04627 · v1 · submitted 2026-04-06 · ❄️ cond-mat.stat-mech · hep-th· nlin.SI

Recognition: 2 theorem links

· Lean Theorem

The Roaming Bethe Roots: An Effective Bethe Ansatz Beyond Integrability

Wenlong Zhao , Yunfeng Jiang , Rui-Dong Zhu
Authors on Pith no claims yet

Pith reviewed 2026-05-10 20:14 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech hep-thnlin.SI
keywords effective Bethe ansatzintegrability breakingquantum spin chainsapproximate eigenstatesoff-shell Bethe statesmany-body systemsentanglement entropy
0
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The pith

Renormalizing Bethe roots via cost minimization approximates eigenstates of weakly non-integrable quantum systems

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

The paper develops an effective Bethe ansatz that keeps the standard wave-function structure but lets the roots adjust to account for small integrability-breaking terms. These adjusted roots are found by minimizing cost functions tied to energy, overlap, or entropy. The resulting states match exact eigenstates well when the breaking is weak and over a broad range of parameters, but match poorly when the breaking is strong. This approach thus serves both as a practical tool for describing nearly integrable systems and as a diagnostic for how far a model sits from integrability.

Core claim

The effective Bethe ansatz retains the functional form of the Bethe wave function while renormalizing the Bethe roots to account for integrability-breaking interactions, with the roots determined by minimizing physically motivated cost functions; the resulting off-shell Bethe states then serve as approximate eigenstates whose quality is high for weak breaking but degrades for strong breaking.

What carries the argument

Roaming Bethe roots obtained by minimizing cost functions while preserving the Bethe wave-function form

Load-bearing premise

The functional form of the Bethe wave function remains approximately valid even after integrability is broken, so that only the roots require adjustment.

What would settle it

For the deformed XXZ spin chain, compare the energy variance or state fidelity of the minimized off-shell Bethe state against the exact diagonalization result as the deformation parameter is increased from zero.

Figures

Figures reproduced from arXiv: 2604.04627 by Rui-Dong Zhu, Wenlong Zhao, Yunfeng Jiang.

Figure 1
Figure 1. Figure 1: FIG. 1: Fidelity and Entanglement Entropy computed in two models. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Evolution of the effective Bethe roots of the ground state of two models with [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Fidelity and Entanglement Entropy computed in XXZ model. [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Fidelity computed in two models based on an effective Bethe ansatz generated from the 8-vertex R-matrix. [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
read the original abstract

We propose an effective Bethe ansatz for solving quantum many-body systems near an integrable point. Our approach retains the functional form of the Bethe wave function while renormalizing the Bethe roots to account for integrability-breaking interactions. These effective roots are determined by minimizing physically motivated cost functions. The resulting off-shell Bethe states serve as approximate eigenstates of the non-integrable models. We assess the quality of the approximation using various physical observables, including the energy eigenvalue, state fidelity, and bipartite entanglement entropy. Our tests show that for models with weak integrability-breaking, the effective Bethe ansatz provides a high-quality approximation to the exact eigenstates over a wide range of deformation parameters. In contrast, for models with strong integrability-breaking interactions, the efficacy of the effective Bethe ansatz degrades relatively quickly as the deformation parameter increases. The efficacy of the method thus offers a useful probe for characterizing the strength of integrability breaking. Within its regime of accuracy, it also provides a new representation of the eigenstates of nearly integrable models, enabling one to exploit the algebraic structure inherited from integrability.

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

2 major / 2 minor

Summary. The manuscript proposes an effective Bethe ansatz for quantum many-body systems near integrability by retaining the exact functional form of the Bethe wave function while renormalizing the Bethe roots via minimization of physically motivated cost functions. The resulting off-shell Bethe states are asserted to serve as approximate eigenstates of the deformed non-integrable Hamiltonian. Quality is assessed numerically via energy eigenvalues, state fidelity, and bipartite entanglement entropy, with the claim that the approximation holds over a wide range of deformation parameters for weak integrability breaking but degrades quickly for strong breaking.

Significance. If the numerical support is robust and the method generalizes, this provides a practical representation of eigenstates in nearly integrable models that inherits the algebraic structure of the Bethe ansatz, potentially enabling new calculations of observables and dynamics. It also offers a diagnostic for the strength of integrability breaking. The approach is noteworthy for attempting to extend Bethe-ansatz techniques beyond exact integrability without abandoning the wave-function ansatz entirely.

major comments (2)
  1. [§2 and §4] The central claim that the minimized off-shell Bethe states remain high-fidelity approximate eigenstates for weak breaking rests entirely on numerical tests; no perturbative expansion in the deformation parameter or norm bound on the overlap with the true eigenstate is supplied to show why the Bethe manifold captures the eigenstates to leading order. This is load-bearing for the assertion of a 'wide range' of validity (see the discussion of the method and the numerical results sections).
  2. [§3] The cost functions used to determine the effective roots are described as physically motivated but their explicit relation to the eigenstate condition (e.g., minimizing ||(H−E)|ψ⟩|| or maximizing fidelity) is not derived. If the cost functions are not equivalent to these quantities, the optimized states may reproduce selected observables while still possessing large residuals, weakening the claim that they serve as approximate eigenstates (see the definition of the cost functions and the comparison to exact diagonalization).
minor comments (2)
  1. [Abstract] The abstract would be strengthened by including at least one concrete quantitative metric (e.g., a fidelity value or energy error for a specific model and deformation strength) rather than qualitative statements such as 'high-quality approximation'.
  2. [§2] Notation for the effective roots and the precise form of the cost functions should be introduced with an equation number in the main text to improve readability when comparing to the standard Bethe equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below, indicating revisions where appropriate.

read point-by-point responses

  1. Referee: [§2 and §4] The central claim that the minimized off-shell Bethe states remain high-fidelity approximate eigenstates for weak breaking rests entirely on numerical tests; no perturbative expansion in the deformation parameter or norm bound on the overlap with the true eigenstate is supplied to show why the Bethe manifold captures the eigenstates to leading order. This is load-bearing for the assertion of a 'wide range' of validity.

    Authors: We agree that an analytical argument would strengthen the foundation. In the revised manuscript we add a perturbative analysis (new subsection in §2) showing that, to leading order in the deformation, adjustment of the Bethe roots cancels the first-order correction to the overlap with the exact eigenstate. A general norm bound is not supplied because deriving one is equivalent to solving the full non-integrable eigenvalue problem; we therefore retain the claim of a wide validity range on the basis of the perturbative result together with the existing numerical evidence. revision: partial

  2. Referee: [§3] The cost functions used to determine the effective roots are described as physically motivated but their explicit relation to the eigenstate condition (e.g., minimizing ||(H−E)|ψ⟩|| or maximizing fidelity) is not derived. If the cost functions are not equivalent to these quantities, the optimized states may reproduce selected observables while still possessing large residuals.

    Authors: We thank the referee for highlighting this point. The cost functions were chosen to target energy accuracy and entanglement structure. In the revision we derive that, for weak deformations, minimizing the chosen cost functions is equivalent (up to higher-order terms) to minimizing the residual norm ||(H−E)ψ||. We also add direct numerical comparisons of this residual for the optimized off-shell states, confirming that residuals remain small in the regime where fidelity is high. revision: yes

Circularity Check

0 steps flagged

No significant circularity; variational optimization within Bethe manifold is self-contained

full rationale

The paper proposes an effective Bethe ansatz that retains the exact functional form of the integrable Bethe wave function and determines renormalized roots by minimizing physically motivated cost functions derived from the deformed Hamiltonian. This is a standard restricted variational procedure whose outputs are then validated by direct numerical comparison of observables (energy, fidelity, entanglement) against exact diagonalization on specific models. No derivation chain reduces a claimed first-principles result to its own fitted inputs by construction, no uniqueness theorem is imported, and no self-citation is load-bearing for the central claim. The method is therefore non-circular; its accuracy is an empirical question assessed outside the construction itself.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach introduces effective roots as adjusted parameters but relies on the domain assumption of near-integrability preserving the wave function structure.

free parameters (1)
  • Cost function parameters
    The cost functions used to determine the effective roots may involve tunable parameters chosen to optimize the approximation.
axioms (1)
  • domain assumption The Bethe wave function form remains approximately valid for systems near integrability
    This is the foundational assumption allowing retention of the functional form while adjusting roots.

pith-pipeline@v0.9.0 · 5504 in / 1302 out tokens · 71674 ms · 2026-05-10T20:14:32.440291+00:00 · methodology

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Reference graph

Works this paper leans on

68 extracted references · 44 canonical work pages · 1 internal anchor

  1. [1]

    Such eigenstates can be constructed by repeatedly acting with theB-operator on the pseudovacuum state|Ω⟩=| ↑ L⟩ |{uj}⟩=B(u 1)

    as HXXX ∝ d du logT(u) u=i/2 (14) 7 Consequently, eigenvectors of the transfer matrix are also eigenvectors of the Hamiltonian. Such eigenstates can be constructed by repeatedly acting with theB-operator on the pseudovacuum state|Ω⟩=| ↑ L⟩ |{uj}⟩=B(u 1). . .B(u N)|Ω⟩(15) For this state to be an eigenstate of the transfer ma- trix, the complex parameters{u...

  2. [2]

    On conservation of conditionally periodic motions for a small change in Hamilton’s function,

    A. A. N. Kolmogorov, “On conservation of conditionally periodic motions for a small change in Hamilton’s function,”Proceedings of the USSR Academy of Sciences98(1954) 527–530.https: //api.semanticscholar.org/CorpusID:222404868

    1954

  3. [3]

    Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian,

    V. I. Arnol’d, “Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian,”Russian Math. Surveys18no. 5, (1963) 9–36. http://mi.mathnet.ru/eng/rm6404

    1963

  4. [4]

    On invariant curves of area-preserving mapping of an annulus,

    J. Moser, “On invariant curves of area-preserving mapping of an annulus,”Matematika6no. 5, (1962) 51–68.http://mi.mathnet.ru/mat236

    1962

  5. [5]

    Boosting nearest-neighbour to long-range integrable spin chains

    T. Bargheer, N. Beisert, and F. Loebbert, “Boosting Nearest-Neighbour to Long-Range Integrable Spin Chains,”J. Stat. Mech.0811(2008) L11001, arXiv:0807.5081 [hep-th]

    work page arXiv 2008

  6. [6]

    Long-range deformations for integrable spin chains

    T. Bargheer, N. Beisert, and F. Loebbert, “Long-Range Deformations for Integrable Spin Chains,”J. Phys. A 42(2009) 285205,arXiv:0902.0956 [hep-th]

    work page arXiv 2009

  7. [7]

    Approximate conservation laws in perturbed integrable lattice models,

    M. Mierzejewski, T. Prosen, and P. Prelovˇ sek, “Approximate conservation laws in perturbed integrable lattice models,”Phys. Rev. B92no. 19, (Nov., 2015) 195121,arXiv:1508.06385 [cond-mat.str-el]

    work page arXiv 2015

  8. [8]

    Quasi-conserved quantities in the perturbed XXX spin chain,

    S. Malikis, D. Kurlov, and V. Gritsev, “Quasi-conserved quantities in the perturbed XXX spin chain,”arXiv e-prints(July, 2020) arXiv:2007.01715, arXiv:2007.01715 [cond-mat.stat-mech]

    work page arXiv 2020

  9. [9]

    Finite-size generators for weak integrability breaking perturbations in the Heisenberg chain,

    S. Vanovac, F. M. Surace, and O. I. Motrunich, “Finite-size generators for weak integrability breaking perturbations in the Heisenberg chain,”Phys. Rev. B 110no. 14, (Oct., 2024) 144309,arXiv:2406.08730 [cond-mat.stat-mech]

    work page arXiv 2024

  10. [10]

    Glimmers of a Quantum KAM Theorem: Insights from Quantum Quenches in One-Dimensional Bose Gases,

    G. P. Brandino, J.-S. Caux, and R. M. Konik, “Glimmers of a Quantum KAM Theorem: Insights from Quantum Quenches in One-Dimensional Bose Gases,” Phys. Rev. X5(Dec, 2015) 041043.https: //link.aps.org/doi/10.1103/PhysRevX.5.041043

    work page doi:10.1103/physrevx.5.041043 2015

  11. [11]

    Prethermalization,

    J. Berges, S. Borsanyi, and C. Wetterich, “Prethermalization,”Phys. Rev. Lett.93(2004) 142002, arXiv:hep-ph/0403234

    work page arXiv 2004

  12. [12]

    Generalized Gibbs ensemble prediction of prethermalization plateaus and their relation to nonthermal steady states in integrable systems,

    M. Kollar, F. A. Wolf, and M. Eckstein, “Generalized Gibbs ensemble prediction of prethermalization plateaus and their relation to nonthermal steady states in integrable systems,”Phys. Rev. B84no. 5, (Aug.,

  13. [13]

    054304,arXiv:1102.2117 [cond-mat.str-el]

    work page arXiv

  14. [14]

    Prethermalization and universal dynamics in near-integrable quantum systems,

    T. Langen, T. Gasenzer, and J. Schmiedmayer, “Prethermalization and universal dynamics in near-integrable quantum systems,”J. Stat. Mech.1606 no. 6, (2016) 064009,arXiv:1603.09385 [cond-mat.quant-gas]

    work page arXiv 2016

  15. [15]

    Prethermalization and Thermalization in Isolated Quantum Systems,

    K. Mallayya, M. Rigol, and W. D. Roeck, “Prethermalization and Thermalization in Isolated Quantum Systems,”Phys. Rev. X9no. 2, (2019) 021027,arXiv:1810.12320 [cond-mat.stat-mech]

    work page arXiv 2019

  16. [16]

    Interaction Quench in the Hubbard Model,

    M. Moeckel and S. Kehrein, “Interaction Quench in the Hubbard Model,”Phys. Rev. Lett.100no. 17, (2008) 175702

    2008

  17. [17]

    Spin Transport in a One-Dimensional Anisotropic Heisenberg Model,

    M. ˇZnidariˇ c, “Spin Transport in a One-Dimensional Anisotropic Heisenberg Model,”Phys. Rev. Lett.106 no. 22, (2011) 220601,arXiv:1103.4094 [cond-mat.str-el]

    work page arXiv 2011

  18. [18]

    Multiple relaxation times in perturbed XXZ chain,

    M. Mierzejewski, J. Paw lowski, P. Prelovsek, and J. Herbrych, “Multiple relaxation times in perturbed XXZ chain,”SciPost Physics13no. 2, (Aug., 2022) 013,arXiv:2112.08158 [cond-mat.str-el]

    work page arXiv 2022

  19. [19]

    Spin conductivity in almost integrable spin chains,

    P. Jung and A. Rosch, “Spin conductivity in almost integrable spin chains,”Phys. Rev. B76no. 24, (Dec.,

  20. [20]

    245108,arXiv:0708.1313 [cond-mat.str-el]

    work page arXiv

  21. [21]

    Scaling of electrical and thermal conductivities in an almost integrable chain,

    Y. Huang, C. Karrasch, and J. E. Moore, “Scaling of electrical and thermal conductivities in an almost integrable chain,”Phys. Rev. B88no. 11, (Sept., 2013) 115126,arXiv:1212.0012 [cond-mat.str-el]

    work page arXiv 2013

  22. [22]

    Spin and thermal conductivity of quantum spin chains and ladders,

    C. Karrasch, D. M. Kennes, and F. Heidrich-Meisner, “Spin and thermal conductivity of quantum spin chains and ladders,”Phys. Rev. B91no. 11, (Mar., 2015) 115130,arXiv:1412.6047 [cond-mat.str-el]

    work page arXiv 2015

  23. [23]

    Heat Conductivity of the Heisenberg Spin-1 /2 Ladder: From Weak to Strong Breaking of Integrability,

    R. Steinigeweg, J. Herbrych, X. Zotos, and W. Brenig, “Heat Conductivity of the Heisenberg Spin-1 /2 Ladder: From Weak to Strong Breaking of Integrability,”Phys. Rev. Lett.116no. 1, (Jan., 2016) 017202, arXiv:1503.03871 [cond-mat.str-el]

    work page arXiv 2016

  24. [24]

    Nonequilibrium Dynamics and Weakly Broken Integrability,

    J. Durnin, M. J. Bhaseen, and B. Doyon, “Nonequilibrium Dynamics and Weakly Broken Integrability,”Phys. Rev. Lett.127no. 13, (2021) 130601,arXiv:2004.11030 [cond-mat.stat-mech]

    work page arXiv 2021

  25. [25]

    Hydrodynamics of weak integrability breaking,

    A. Bastianello, A. De Luca, and R. Vasseur, “Hydrodynamics of weak integrability breaking,” Journal of Statistical Mechanics: Theory and Experiment2021no. 11, (Nov., 2021) 114003. http://dx.doi.org/10.1088/1742-5468/ac26b2

    work page doi:10.1088/1742-5468/ac26b2 2021

  26. [26]

    Integrals of Motion and S Matrix of the (Scaled) T=T(c) Ising Model with Magnetic Field,

    A. B. Zamolodchikov, “Integrals of Motion and S Matrix of the (Scaled) T=T(c) Ising Model with Magnetic Field,”Int. J. Mod. Phys. A4(1989) 4235

    1989

  27. [27]

    Delfino, G

    G. Delfino, G. Mussardo, and P. Simonetti, “Nonintegrable quantum field theories as perturbations of certain integrable models,”Nucl. Phys. B473(1996) 469–508,arXiv:hep-th/9603011

    work page arXiv 1996

  28. [28]

    TRUNCATED-FERMIONIC-SPACE APPROACH TO THE CRITICAL 2D ISING MODEL WITH MAGNETIC FIELD,

    V. YUROV and A. ZAMOLODCHIKOV, “TRUNCATED-FERMIONIC-SPACE APPROACH TO THE CRITICAL 2D ISING MODEL WITH MAGNETIC FIELD,”International Journal of Modern Physics A06no. 25, (1991) 4557–4578, https://doi.org/10.1142/S0217751X91002161. https://doi.org/10.1142/S0217751X91002161

    work page doi:10.1142/s0217751x91002161 1991

  29. [29]

    TRUNCATED CONFORMAL SPACE APPROACH TO SCALING LEE-YANG MODEL,

    V. P. Yurov and A. B. Zamolodchikov, “TRUNCATED CONFORMAL SPACE APPROACH TO SCALING LEE-YANG MODEL,”Int. J. Mod. Phys. A5(1990) 3221–3246

    1990

  30. [30]

    Fonseca and A

    P. Fonseca and A. Zamolodchikov, “Ising field theory in a magnetic field: Analytic properties of the free energy,”arXiv:hep-th/0112167

    work page internal anchor Pith review arXiv

  31. [31]

    On the theory of metals. 1. Eigenvalues and eigenfunctions for the linear atomic chain,

    H. Bethe, “On the theory of metals. 1. Eigenvalues and eigenfunctions for the linear atomic chain,”Z. Phys.71 (1931) 205–226

    1931

  32. [32]

    Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State,

    E. H. Lieb and W. Liniger, “Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State,”Phys. Rev.130(May, 1963) 1605–1616. https: //link.aps.org/doi/10.1103/PhysRev.130.1605

    work page doi:10.1103/physrev.130.1605 1963

  33. [33]

    Some exact results for the many body problems in one dimension with repulsive delta function interaction,

    C.-N. Yang, “Some exact results for the many body problems in one dimension with repulsive delta function interaction,”Phys. Rev. Lett.19(1967) 1312–1314

    1967

  34. [34]

    R. J. Baxter,Exactly solved models in statistical 9 mechanics. 1982

    1982

  35. [35]

    In fact, for a sufficiently large deformation parameter, all perturbations lead to strong integrability breaking

    So far, a rigorous definition of weak and strong integrability-breaking perturbations is still lacking. In fact, for a sufficiently large deformation parameter, all perturbations lead to strong integrability breaking. In this context, the distinction between weak and strong should be understood qualitatively. For the same deformation parameter, some pertu...

  36. [36]

    Weak integrability breaking perturbations of integrable models,

    F. M. Surace and O. Motrunich, “Weak integrability breaking perturbations of integrable models,”Phys. Rev. Res.5no. 4, (2023) 043019,arXiv:2302.12804 [cond-mat.stat-mech]

    work page arXiv 2023

  37. [37]

    T¯T-deformation and long range spin chains,

    B. Pozsgay, Y. Jiang, and G. Tak´ acs, “T¯T-deformation and long range spin chains,”JHEP03(2020) 092, arXiv:1911.11118 [hep-th]

    work page arXiv 2020

  38. [38]

    T ¯T Deformations and Integrable Spin Chains,

    E. Marchetto, A. Sfondrini, and Z. Yang, “T ¯T Deformations and Integrable Spin Chains,”Phys. Rev. Lett.124no. 10, (2020) 100601,arXiv:1911.12315 [hep-th]

    work page arXiv 2020

  39. [39]

    The Space of Integrable Systems from Generalised T ¯T-Deformations,

    B. Doyon, J. Durnin, and T. Yoshimura, “The Space of Integrable Systems from Generalised T ¯T-Deformations,”SciPost Phys.13no. 3, (2022) 072, arXiv:2105.03326 [hep-th]

    work page arXiv 2022

  40. [40]

    Transport in Almost Integrable Models: Perturbed Heisenberg Chains,

    P. Jung, R. W. Helmes, and A. Rosch, “Transport in Almost Integrable Models: Perturbed Heisenberg Chains,”Phys. Rev. Lett.96no. 6, (Feb., 2006) 067202, arXiv:cond-mat/0509615 [cond-mat.str-el]

    work page arXiv 2006

  41. [41]

    Prethermalization, thermalization, and Fermi’s golden rule in quantum many-body systems,

    K. Mallayya and M. Rigol, “Prethermalization, thermalization, and Fermi’s golden rule in quantum many-body systems,”Phys. Rev. B104(2021) 184302, arXiv:2109.01705 [cond-mat.stat-mech]

    work page arXiv 2021

  42. [42]

    Mori, T.N

    T. Mori, T. N. Ikeda, E. Kaminishi, and M. Ueda, “Thermalization and prethermalization in isolated quantum systems: a theoretical overview,”J. Phys. B 51no. 11, (2018) 112001,arXiv:1712.08790 [cond-mat.stat-mech]

    work page arXiv 2018

  43. [43]

    Crossover from Poisson to Wigner-Dyson level statistics in spin chains with integrability breaking,

    D. A. Rabson, B. N. Narozhny, and A. J. Millis, “Crossover from Poisson to Wigner-Dyson level statistics in spin chains with integrability breaking,” Phys. Rev. B69(Feb, 2004) 054403.https: //link.aps.org/doi/10.1103/PhysRevB.69.054403

    work page doi:10.1103/physrevb.69.054403 2004

  44. [44]

    Onset of quantum chaos in one-dimensional bosonic and fermionic systems and its relation to thermalization,

    L. F. Santos and M. Rigol, “Onset of quantum chaos in one-dimensional bosonic and fermionic systems and its relation to thermalization,”Phys. Rev. E81no. 3, (2010) 036206

    2010

  45. [45]

    Energy Level Distribution of Perturbed Conformal Field Theories,

    G. P. Brandino, R. M. Konik, and G. Mussardo, “Energy Level Distribution of Perturbed Conformal Field Theories,”J. Stat. Mech.1007(2010) P07013, arXiv:1004.4844 [cond-mat.stat-mech]

    work page arXiv 2010

  46. [46]

    Quasiparticle breakdown in the quasi-one-dimensional Ising ferromagnet CoNb 2O6,

    N. J. Robinson, F. H. L. Essler, I. Cabrera, and R. Coldea, “Quasiparticle breakdown in the quasi-one-dimensional Ising ferromagnet CoNb 2O6,” Phys. Rev. B90no. 17, (Nov., 2014) 174406, arXiv:1407.2794 [cond-mat.str-el]

    work page arXiv 2014

  47. [47]

    Two-dimensional Ising Field Theory in a Magnetic Field: Breakup of the Cut in the Two Point Function,

    B. M. McCoy and T. T. Wu, “Two-dimensional Ising Field Theory in a Magnetic Field: Breakup of the Cut in the Two Point Function,”Phys. Rev. D18(1978) 1259

    1978

  48. [48]

    Ising spectroscopy. I. Mesons at T<T(c),

    P. Fonseca and A. Zamolodchikov, “Ising spectroscopy. I. Mesons at T<T(c),”arXiv:hep-th/0612304

    work page arXiv

  49. [49]

    Kormos, M

    M. Kormos, M. Collura, G. Tak´ acs, and P. Calabrese, “Real-time confinement following a quantum quench to a non-integrable model,”Nature Physics13no. 3, (Mar., 2017) 246–249,arXiv:1604.03571 [cond-mat.stat-mech]

    work page arXiv 2017

  50. [50]

    Confinement of spinons in the XXZ spin-1/2 chain in presence of a transverse magnetic field,

    S. B. Rutkevich, “Confinement of spinons in the XXZ spin-1/2 chain in presence of a transverse magnetic field,”Phys. Rev. B109no. 1, (Jan., 2024) 014411, arXiv:2307.08328 [cond-mat.str-el]

    work page arXiv 2024

  51. [51]

    Quenches and confinement in a Heisenberg–Ising spin ladder,

    G. Lagnese, F. M. Surace, M. Kormos, and P. Calabrese, “Quenches and confinement in a Heisenberg–Ising spin ladder,”J. Phys. A55no. 12, (2022) 124003,arXiv:2110.09472 [cond-mat.stat-mech]

    work page arXiv 2022

  52. [52]

    Mesons in a quantum Ising ladder,

    Y. Gao, Y. Jiang, and J. Wu, “Mesons in a quantum Ising ladder,”JHEP07(2025) 072,arXiv:2502.15463 [hep-th]

    work page arXiv 2025

  53. [53]

    Meson mass spectrum in Ising field theory,

    A. Litvinov, P. Meshcheriakov, and E. Shestopalov, “Meson mass spectrum in Ising field theory,”Phys. Rev. D112no. 8, (2025) 085021,arXiv:2507.15766 [hep-th]

    work page arXiv 2025

  54. [54]

    The Fate of the False Vacuum. 1. Semiclassical Theory,

    S. R. Coleman, “The Fate of the False Vacuum. 1. Semiclassical Theory,”Phys. Rev. D15(1977) 2929–2936. [Erratum: Phys.Rev.D 16, 1248 (1977)]

    1977

  55. [55]

    The Fate of the False Vacuum. 2. First Quantum Corrections,

    C. G. Callan, Jr. and S. R. Coleman, “The Fate of the False Vacuum. 2. First Quantum Corrections,”Phys. Rev. D16(1977) 1762–1768

    1977

  56. [56]

    False vacuum decay in quantum spin chains,

    G. Lagnese, F. M. Surace, M. Kormos, and P. Calabrese, “False vacuum decay in quantum spin chains,”Phys. Rev. B104no. 20, (2021) L201106, arXiv:2107.10176 [cond-mat.stat-mech]

    work page arXiv 2021

  57. [57]

    False vacuum decay in the (1+1)-dimensionalφ4 theory,

    D. Sz´ asz-Schagrin and G. Tak´ acs, “False vacuum decay in the (1+1)-dimensionalφ4 theory,”Phys. Rev. D106 no. 2, (2022) 025008,arXiv:2205.15345 [hep-th]

    work page arXiv 2022

  58. [58]

    On Next-Nearest-Neighbor Interaction in Linear Chain. I,

    C. K. Majumdar and D. K. Ghosh, “On Next-Nearest-Neighbor Interaction in Linear Chain. I,” J. Math. Phys.10no. 8, (1969) 1388

    1969

  59. [59]

    Spontaneous dimerization in the S= 1 2 Heisenberg antiferromagnetic chain with competing interactions,

    F. D. M. Haldane, “Spontaneous dimerization in the S= 1 2 Heisenberg antiferromagnetic chain with competing interactions,”Phys. Rev. B25(Apr, 1982) 4925–4928.https: //link.aps.org/doi/10.1103/PhysRevB.25.4925

    work page doi:10.1103/physrevb.25.4925 1982

  60. [60]

    Ground-State Properties of the One-Dimensional Isotropic Spin-1/2 Heisenberg Antiferromagnet with Competing Interactions,

    T. Tonegawa and I. Harada, “Ground-State Properties of the One-Dimensional Isotropic Spin-1/2 Heisenberg Antiferromagnet with Competing Interactions,”Journal of the Physical Society of Japan56no. 6, (1987) 2153–2167.https://doi.org/10.1143/JPSJ.56.2153

    work page doi:10.1143/jpsj.56.2153 1987

  61. [61]

    Quantum entanglement and quantum phase transitions in frustrated Majumdar–Ghosh model,

    G.-H. Liu, C.-H. Wang, and X.-Y. Deng, “Quantum entanglement and quantum phase transitions in frustrated Majumdar–Ghosh model,”Physica B: Condensed Matter406no. 1, (2011) 100–103. https://www.sciencedirect.com/science/article/ pii/S0921452610009634. [59]±0.5 are Bethe roots in the XXX integrable model for L= 4, M= 1

    2011

  62. [62]

    Exact solution of the one-dimensional isotropic Heisenberg chain with arbitrary spin S,

    H. M. Babujian, “Exact solution of the one-dimensional isotropic Heisenberg chain with arbitrary spin S,”Phys. Lett. A90(1982) 479–482

    1982

  63. [63]

    The picture of low-lying excitations in the isotropic Heisenberg chain of arbitrary spins,

    L. A. Takhtajan, “The picture of low-lying excitations in the isotropic Heisenberg chain of arbitrary spins,” Phys. Lett. A87(1982) 479–482

    1982

  64. [64]

    Rigorous Results on Valence Bond Ground States in Antiferromagnets,

    I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, “Rigorous Results on Valence Bond Ground States in Antiferromagnets,”Phys. Rev. Lett.59(1987) 799

    1987

  65. [65]

    Valence Bond Ground States in Isotropic Quantum Antiferromagnets,

    I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, “Valence Bond Ground States in Isotropic Quantum Antiferromagnets,”Commun. Math. Phys.115(1988) 10 477

    1988

  66. [66]

    Effective Bethe Ansatz for Spin-1 Non-integrable Models,

    Z. Wang and R.-D. Zhu, “Effective Bethe Ansatz for Spin-1 Non-integrable Models,”to appear(2026)

    2026

  67. [67]

    Algebraic Aspects of the Bethe Ansatz

    L. D. Faddeev, “Algebraic aspects of Bethe Ansatz,” Int. J. Mod. Phys. A10(1995) 1845–1878, arXiv:hep-th/9404013

    work page arXiv 1995

  68. [68]

    N. A. Slavnov,Algebraic Bethe ansatz. 4, 2018. arXiv:1804.07350 [math-ph]

    work page arXiv 2018