Integrable classical lattice models such as the Toda chain, the Ishimori spin chain, and the harmonic oscillator chain possess an extensive set of conserved quantities. A recent framework (Vanovac, McCarthy, Surace, and Motrunich, 2026) constructs weak integrability-breaking (WIB) perturbations of such models — perturbations of strength λ\lambda that admit extensive quasi-conserved quantities surviving to order O(λℓ)O(\lambda^\ell) and that delay thermalization from the Fermi-golden-rule time λ−2\lambda^{-2} to a parametrically longer λ−2ℓ\lambda^{-2\ell} — from generators XX of extensive local, boosted, bilocal, and strictly local type, and exhibits a new trilocal generator for the alpha-Fermi-Pasta-Ulam-Tsingou (FPUT) interaction. Two structural questions are open and define the target of this problem. First, whether the three known long-range-deformation generator families (extensive local, boosted, bilocal) exhaust all generators producing extensive, local, translationally invariant WIB perturbations — a list that the paper's own new trilocal generator for the alpha-FPUT interaction, which lies outside these classes, shows to be not closed; the quartic beta-FPUT interaction, whose adiabatic gauge potential variance scales as L4L^4 but for which no generator has been found, is the critical unresolved test case. Second, whether the conjectured hierarchy of relaxation times τ∼λ−2ℓ\tau\sim\lambda^{-2\ell} — quasi-integrals persisting up to O(λ−ℓ)O(\lambda^{-\ell}) and decay to full thermalization over λ−2ℓ\lambda^{-2\ell}, i.e. O(λ−2)O(\lambda^{-2}) and λ−4\lambda^{-4} for the ℓ=2\ell=2 WIBs — can be given a rigorous framework for classical systems. As of September 2026, both strands remain open.

Background

An integrable classical many-body system possesses an extensive set of independent conserved quantities (integrals of motion, IoMs) — in the lattice models considered here, roughly one independent charge per site — which prevent thermalization and force ballistic or anomalously slow transport. Canonical examples are the Toda chain (particles with exponential nearest-neighbor interactions), the Ishimori model (a classical spin chain with commuting conserved charges), and the harmonic oscillator chain (HOC, a discretized linear string). Adding a generic perturbation of strength λ\lambda destroys the IoMs and, by Fermi's golden rule (FGR) type arguments, thermalizes the system on a timescale τ∼λ−2\tau\sim\lambda^{-2}.

Weak integrability breaking (WIB) denotes the special class of perturbations for which extensive quasi-conserved quantities survive: each IoM can be deformed, Qα=Qα(0)+λQα(1)+…Q_\alpha = Q_\alpha^{(0)} + \lambda Q_\alpha^{(1)} + \dots, so that its Poisson bracket with the perturbed Hamiltonian H=H0+λVH = H_0 + \lambda V vanishes to order O(λℓ)O(\lambda^\ell) with ℓ≥2\ell \ge 2. The dynamics is then expected to remain near-integrable up to a prethermal plateau in which the quasi-integrals persist up to O(λ−ℓ)O(\lambda^{-\ell}); the decay to full thermalization is delayed to τ∼λ−2ℓ\tau\sim\lambda^{-2\ell} — for ℓ=2\ell = 2, a quasi-integral persistence window O(λ−2)O(\lambda^{-2}) and thermalization over λ−4\lambda^{-4}, parametrically longer than the FGR time. The concept and a systematic construction were developed for quantum spin chains (Surace and Motrunich 2023): WIB perturbations are produced by generators XX through V=i[X,H0]V = i[X, H_0], with three known families — extensive local generators (sums of local terms), boosted generators (spatial integral of an IoM density, X=∑jj qjX = \sum_j j\, q_j), and bilocal generators built from pairs of IoM densities. The recent classical formulation (Vanovac, McCarthy, Surace, and Motrunich 2026) translates the construction to classical lattice dynamics via the Poisson bracket V={X,H0}V = \{X, H_0\} with quasi-IoM corrections Qα(1)={X,Qα(0)}Q_\alpha^{(1)} = \{X, Q_\alpha^{(0)}\}, adds strictly local (impurity-type) generators, and works out explicit WIB families for the Ishimori, Toda, and HOC models. In free-fermionic quantum chains, a related construction (Pozsgay, Sharipov, Tiutiakina, and Vona 2024) revealed weakly breaking perturbations that lie outside all known generator classes and for which only a subset of the conserved charges can be corrected — evidence that the known classification is incomplete even on the quantum side. A central diagnostic there is the classical adiabatic gauge potential (AGP) — the generator of the canonical transformation connecting nearby invariant tori under an infinitesimal parameter change, evaluated along trajectories — whose time-averaged variance χ\chi saturates for WIBs with polynomial scaling in system size LL (χsat∼L\chi_{\rm sat}\sim L for momentum-conserving extensive local generators, ∼L2\sim L^2 for boosted and bilocal ones, and higher powers when the relevant conservation laws or symmetries are not respected — e.g., L3L^3 for SU(2)-breaking boosted and bilocal perturbations of the Ishimori chain and L4L^4 for momentum-nonconserving extensive local generators in the HOC — or when the generator lies outside the standard classes — L5L^5 for the alpha-FPUT trilocal generator), but grows in time or much faster with LL for generic perturbations.

The central application is the Fermi-Pasta-Ulam-Tsingou (FPUT) problem, the 1953 observation that a nonlinear chain started near its harmonic (HOC) state fails to reach energy equipartition for extremely long times. The cubic (α\alpha-FPUT) and quartic (β\beta-FPUT) nearest-neighbor nonlinearities are the two paradigmatic perturbations of the HOC. The classical WIB paper proves that the α\alpha-FPUT interaction is a genuine WIB of the HOC: using the Toda IoMs it constructs corrections to the entire HOC charge hierarchy, giving an infinite tower of quasi-integrals, and identifies the corresponding AGP as a new trilocal (three-site-support) generator in real space — the first generator outside the previously known classes. It further shows that any cubic, translationally invariant, momentum-conserving perturbation of the HOC admits such a trilocal generator. In contrast, the quartic β\beta-FPUT interaction is not generated by any known class, and the source authors conjecture that it may represent a more generic form of integrability breaking of the HOC than the α\alpha-FPUT perturbation; its AGP variance saturates with scaling χsat∼L4\chi_{\rm sat}\sim L^4 — polynomial, as for a WIB, but the power counting alone does not decide the question, since momentum-nonconserving extensive local generators (a known class) already produce L4L^4 scaling in the HOC, while the proven α\alpha-FPUT WIB has a trilocal generator scaling as L5L^5 — and whether it admits quasi-integral corrections that are extensive and local in real space (as they may be even when the generator itself is nonlocal), for the entire HOC charge hierarchy or only for a subset of the charges (as happens in the free-fermion analogues), is unresolved. Whether the known generator families are exhaustive, how the richer "higher-order" locality structure suggested by the trilocal generator fits into the classification, and whether the delayed thermalization timescales τ∼λ−2ℓ\tau\sim\lambda^{-2\ell} can be derived rigorously for classical systems, are the open structural questions addressed to this framework.

Problem Statement

The problem is to complete the structural theory of weak integrability breaking for classical integrable lattice models, in two linked parts within the framework of Vanovac, McCarthy, Surace, and Motrunich (arXiv:2603.11712). Both parts are required for a resolution: part (1) is resolved by an exhaustiveness claim or no-go result for the complete generator classification, stated relative to an explicitly specified function class of generators and corrections, together with a determination of the β\beta-FPUT weak-integrability-breaking status; part (2) is resolved by the rigorous framework for the relaxation-time hierarchy. A result that settles only one part, or only a sub-case within a part, leaves the remainder of the task open.

(1) Generator-classification completeness. Determine whether the generators XX that produce extensive, local, translationally invariant WIB perturbations V={X,H0}V=\{X,H_0\} (with quasi-integrals Qα(0)+λ{X,Qα(0)}Q_\alpha^{(0)}+\lambda\{X,Q_\alpha^{(0)}\} whose Poisson bracket with the perturbed Hamiltonian vanishes to O(λ2)O(\lambda^2) or higher) in classical integrable lattice models such as the Ishimori chain, the Toda chain, and the harmonic oscillator chain are exhausted by the known generator families — extensive local, boosted, and bilocal together with the "higher-order" multi-local families of the trilocal type — or whether further generator families exist. Strictly local (impurity-type) generators, whose perturbations have support only in a finite region around a single site, lie outside this extensive, translationally invariant target class; the newly identified trilocal generator for the α\alpha-FPUT interaction does produce perturbations in the class but is not one of the three known classes, so the question is whether further generators of this or other "higher-order" multi-local type exist and how they enter an exhaustive classification. Generators are understood modulo the known gauge freedom X→X+f({Qα(0)})X \to X + f(\{Q_\alpha^{(0)}\}) — adding an arbitrary function of the integrals of motion leaves VV unchanged — and any exhaustiveness claim or no-go result must be stated relative to an explicitly specified function class of generators and corrections. The question is sharpened by the critical test case: is the quartic β\beta-FPUT perturbation of the harmonic oscillator chain a WIB — i.e., does it admit quasi-integral corrections that are extensive and local in real space, for the entire HOC charge hierarchy (which would make it a WIB in the strict sense of the Background) or only for a subset of the charges (as happens in the free-fermion analogues, which would count as partial weak integrability breaking rather than a WIB in the strict sense), possibly from a generator of a novel multi-local type — or is it a generic integrability-breaking perturbation; and, given that momentum-nonconserving extensive local generators (a known class) already produce χsat∼L4\chi_{\rm sat}\sim L^4 AGP-variance scaling in the HOC, what does its L4L^4 scaling imply for its long-time dynamics?

(2) Relaxation-time hierarchy. Determine whether the conjectured hierarchy of thermalization timescales τ∼λ−2ℓ\tau\sim\lambda^{-2\ell} (ℓ>1\ell>1) associated with quasi-integrals conserved to order O(λℓ)O(\lambda^\ell) admits a rigorous framework for classical systems in the setting of this WIB construction — in particular whether the O(λ−ℓ)O(\lambda^{-\ell}) prethermal persistence of the quasi-integrals (O(λ−2)O(\lambda^{-2}) for the constructed order-λ2\lambda^2 WIBs), the conjectured λ−2ℓ\lambda^{-2\ell} decay to full thermalization (λ−4\lambda^{-4} for the constructed order-λ2\lambda^2 WIBs), and the general ℓ>2\ell>2 hierarchy can be established, or must be restricted or refuted — with any claimed scaling stating the dependence of the implied constants on system size, energy density, and observable. One proposed route to the λ−4\lambda^{-4} decay is a nearby integrable Hamiltonian approximating the perturbed system to O(λ2)O(\lambda^2); the framework is not required to use it. Directly testing these predictions numerically, e.g., measuring the crossover t⋆∼λ−4t_\star\sim\lambda^{-4}, is a companion open direction that the source couples to this challenge and is a related program rather than part of the required resolution of part (2).

The common objective is a complete account of which weak perturbation structures exist in classical integrable lattice models and how each such structure fixes the timescale on which integrability is lost; the two parts are linked because the timescale hierarchy is governed by exactly the generator/quasi-integral structure that the classification enumerates.

References

  1. Sara Vanovac, Catherine McCarthy, Federica Maria Surace, and Olexei I. Motrunich, 'Weak integrability breaking perturbations in classical integrable models on the lattice', arXiv:2603.11712 (2026), DOI 10.48550/arxiv.2603.11712, https://arxiv.org/abs/2603.11712
  2. Federica Maria Surace and Olexei I. Motrunich, 'Weak integrability breaking perturbations of integrable models', Phys. Rev. Res. 5, 043019 (2023), arXiv:2302.12804, DOI 10.1103/PhysRevResearch.5.043019, https://arxiv.org/abs/2302.12804
  3. Sara Vanovac, Federica Maria Surace, and Olexei I. Motrunich, 'Finite-size generators for weak integrability breaking perturbations in the Heisenberg chain', Phys. Rev. B 110, 144309 (2024), arXiv:2406.08730, DOI 10.1103/PhysRevB.110.144309, https://arxiv.org/abs/2406.08730
  4. Balázs Pozsgay, Rustem Sharipov, Anastasiia Tiutiakina, and István Vona, 'Adiabatic gauge potential and integrability breaking with free fermions', SciPost Phys. 17, 075 (2024), arXiv:2402.12979, DOI 10.21468/SciPostPhys.17.3.075, https://arxiv.org/abs/2402.12979
  5. Alvise Bastianello, Andrea De Luca, and Romain Vasseur, 'Hydrodynamics of weak integrability breaking', J. Stat. Mech. (2021) 114003, arXiv:2103.11997, DOI 10.1088/1742-5468/ac26b2, https://doi.org/10.1088/1742-5468/ac26b2
  6. Cheng Chen, Luca Capizzi, Alice Marché, Guillaume Bornet, Gabriel Emperauger, Thierry Lahaye, Antoine Browaeys, Maurizio Fagotti, and Leonardo Mazza, 'Observing weakly broken conservation laws in a dipolar Rydberg quantum spin chain', arXiv:2602.02251 (2026), https://arxiv.org/abs/2602.02251
  7. Sahil Kumar Singh, Abhishek Dhar, and Sanjay Moudgalya, 'Absence of hidden analytic conserved quantities in harmonically confined rods', arXiv:2607.18872 (2026), https://arxiv.org/abs/2607.18872
  8. Nachiket Karve, Nathan Rose, David Campbell, and Anatoli Polkovnikov, 'Universal Dynamical Response to Slow Driving in Chaotic Systems', arXiv:2606.23810 (2026), https://arxiv.org/abs/2606.23810
  9. Hyeongjin Kim, Souvik Bandyopadhyay, and Anatoli Polkovnikov, 'Temperature and integrability-breaking correspondence via adiabatic transformations', arXiv:2604.01285 (2026), https://arxiv.org/abs/2604.01285
  10. Ysla F. Adans, Marius de Leeuw, and Tristan McLoughlin, 'On deforming and breaking integrability', arXiv:2603.17018 (2026), https://arxiv.org/abs/2603.17018
  11. G. Benettin and A. Ponno, 'Time-Scales to Equipartition in the Fermi-Pasta-Ulam Problem: Finite-Size Effects and Thermodynamic Limit', J. Stat. Phys. 144, 793 (2011), DOI 10.1007/s10955-011-0277-9, https://doi.org/10.1007/s10955-011-0277-9
  12. Yuri V. Lvov and Miguel Onorato, 'Double Scaling in the Relaxation Time in the β-Fermi-Pasta-Ulam-Tsingou Model', Phys. Rev. Lett. 120, 144301 (2018), arXiv:1802.07686, DOI 10.1103/PhysRevLett.120.144301, https://arxiv.org/abs/1802.07686
  13. Nik O. Gjonbalaj, David K. Campbell, and Anatoli Polkovnikov, 'Counterdiabatic driving in the classical β-Fermi-Pasta-Ulam-Tsingou chain', Phys. Rev. E 106, 014131 (2022), arXiv:2112.02422, DOI 10.1103/PhysRevE.106.014131, https://arxiv.org/abs/2112.02422
  14. Alio Issoufou Arzika, Andrea Solfanelli, Harald Schmid, and Stefano Ruffo, 'Quantization of Integrable and Chaotic Three-Particle Fermi-Pasta-Ulam-Tsingou Models', Entropy 25, 538 (2023), DOI 10.3390/e25030538, https://doi.org/10.3390/e25030538
  15. H. Christodoulidi and S. Flach, 'Measuring FPUT thermalization with Toda integrals', Chaos 35, 113127 (2025), DOI 10.1063/5.0285451, https://doi.org/10.1063/5.0285451
  16. Catherine McCarthy, Sarang Gopalakrishnan, and Romain Vasseur, 'Slow crossover from superdiffusion to diffusion in isotropic spin chains', Phys. Rev. B 110, L180301 (2024), DOI 10.1103/PhysRevB.110.L180301, https://doi.org/10.1103/PhysRevB.110.L180301
  17. Adam J. McRoberts and Roderich Moessner, 'Parametrically Long Lifetime of Superdiffusion in Nonintegrable Spin Chains', Phys. Rev. Lett. 133, 256301 (2024), DOI 10.1103/PhysRevLett.133.256301, https://doi.org/10.1103/PhysRevLett.133.256301
  18. Dipankar Roy, Abhishek Dhar, Herbert Spohn, and Manas Kulkarni, 'Robustness of Kardar-Parisi-Zhang scaling in a classical integrable spin chain with broken integrability', Phys. Rev. B 107, L100413 (2023), DOI 10.1103/PhysRevB.107.L100413, https://doi.org/10.1103/PhysRevB.107.L100413