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arXiv:2603.17018

On deforming and breaking integrability

Y. F. Adans, Marius de Leeuw, Tristan Mcloughlin B

Published Mar 17, 2026• 0 citations
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Abstract

In this paper we study nearest-neighbour deformations of integrable models. After expanding in the deformation parameter, we identify four possible types of deformations. First there are deformations that simply break or preserve integrability. Then we find two different subtle cases. The first case is where the deformation is only integrable if all orders of the deformation parameter are taken into account. An example of these are the long-range deformations that appear in holographic models. The second case is when the deformation is perturbatively integrable to some order in the deformation parameter but can not be extended to an integrable model. In this paper we work this out for the XXZ spin chain and discuss the level statistics of each of these cases. We find numerical evidence that the onset of chaos occurs differently in each of these models. For the perturbatively integrable models, we find that the deformation strength at which chaos appears demonstrates a volume-scaling intermediate between strong and weak integrability breaking models.

DOI
10.48550/arxiv.2603.17018

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We consider the spin-12\frac{1}{2}21​ Heisenberg chain (XXX model) weakly perturbed away from integrability by an isotropic next-to-nearest neighbor exchange interaction. Recently, it was conjectured that this model possesses an infinite tower of quasiconserved integrals of motion (charges) [D. Kurlov et al., Phys. Rev. B 105, 104302 (2022)]. In this work we first test this conjecture by investigating how the norm of the adiabatic gauge potential (AGP) scales with the system size, which is known to be a remarkably accurate measure of chaos. We find that for the perturbed XXX chain the behavior of the AGP norm corresponds to neither an integrable nor a chaotic regime, which supports the conjectured quasi-integrability of the model. We then prove the conjecture and explicitly construct the infinite set of quasiconserved charges. Our proof relies on the fact that the XXX chain perturbed by next-to-nearest exchange interaction can be viewed as a truncation of an integrable long-range deformation of the Heisenberg spin chain.
Recently it was suggested that certain perturbations of integrable spin chains lead to a weak breaking of integrability in the sense that integrability is preserved at the first order in the coupling. Here we examine this claim using level spacing distribution. We find that the volume dependent crossover between integrable and chaotic level spacing statistics which marks the onset of quantum chaotic behaviour, is markedly different for weak vs. strong breaking of integrability. In particular, for the gapless case we find that the crossover coupling as a function of the volume LLL scales with a 1/L21/L^21/L2 law for weak breaking as opposed to the 1/L31/L^31/L3 law previously found for the strong case.
A quantum integrable system slightly perturbed away from integrability is typically expected to thermalize on timescales of order $\ensuremath{\tau}\ensuremath{\sim}{\ensuremath{\lambda}}^{\ensuremath{-}2}$, where $\ensuremath{\lambda}$ is the perturbation strength. We here study classes of perturbations that violate this scaling, and exhibit much longer thermalization times $\ensuremath{\tau}\ensuremath{\sim}{\ensuremath{\lambda}}^{\ensuremath{-}2\ensuremath{\ell}}$ where $\ensuremath{\ell}>1$ is an integer. Systems with these ``weak integrability breaking'' perturbations have an extensive number of quasiconserved quantities that commute with the perturbed Hamiltonian up to corrections of order ${\ensuremath{\lambda}}^{\ensuremath{\ell}}$. We demonstrate a systematic construction to obtain families of such weak perturbations of a generic integrable model for arbitrary $\ensuremath{\ell}$. We then apply the construction to various models, including the Heisenberg, XXZ, and XYZ chains, the Hubbard model, models of spinless free fermions, and the quantum Ising chain. Our analytical framework explains the previously observed evidence of weak integrability breaking in the Heisenberg and XXZ chains under certain perturbations.
Quantum spin chains arise naturally from perturbative large- N field theories and matrix models. The Hamiltonian of such a model is a long-range deformation of nearest-neighbour type interactions. Here, we study the most general long-range integrable spin chain with spins transforming in the fundamental representation of . We derive the Hamiltonian and the corresponding asymptotic Bethe ansatz at the leading four perturbative orders with several free parameters. Furthermore, we propose Bethe equations for all orders and identify the moduli of the integrable system. We finally apply our results to plane-wave matrix theory and show that the Hamiltonian in a closed sector is not of this form and therefore not integrable beyond the first perturbative order. This also implies that the complete model is not integrable.
An integrable model perturbed by special ``weak-integrability-breaking'' perturbations thermalizes at timescales much longer than predicted by Fermi's golden rule. Recently, a systematic construction of such perturbations based on the so-called long-range deformations of integrable chains was formulated. These perturbations, obtained as truncations of the long-range deformations in some small parameter expansions, can be viewed as produced by unitary rotations of the short-range integrable models. For infinite systems, several ``generators'' (extensive local, boosted, and bilocal operators) of weak perturbations are known. The main aim of this work is to understand the appropriate generators in finite systems with periodic boundaries since simple counterparts to boosted and bilocal operators are not known in such cases. We approach this by studying the structure of the adiabatic gauge potential (AGP), a proxy for such generators in finite chains, which was originally introduced as a sensitive measure of quantum chaos. We prove an exact relation between the AGPs for the boosted and bilocal classes of generators and note that the counterpart to the boost does not appear in a closed analytic form in finite systems but shows quasilocality nonetheless. We also introduce and study strictly local variants of weak-integrability-breaking perturbations.