Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Weak integrability breaking and level spacing distribution

D. Szász-Schagrin, Pozsgai, Balázs, G. Takács

Published Jan 1, 2021• 24 citations
OpenAlex

Abstract

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 $L$ scales with a $1/L^2$ law for weak breaking as opposed to the $1/L^3$ law previously found for the strong case.

Related papers

Weak integrability breaking and level spacing distribution

Dávid Szász-Schagrin, Balázs Pozsgay, Gábor Takács

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 L L scales with a 1/L^{2} 1 / L 2 law for weak breaking as opposed to the 1/L^{3} 1 / L 3 law previously found for the strong case.

arXiv:2103.06308•SciPost Physics•Aug 23, 2021•28 citations

Weak vs. strong breaking of integrability in interacting scalar quantum field theories

B. Fitos, G. Takács

The recently proposed classification of integrability-breaking perturbations according to their strength is studied in the context of quantum field theories. Using random matrix methods to diagnose the resulting quantum chaotic behaviour, we investigate the \phi^4 ϕ 4 and \phi^6 ϕ 6 interactions of a massive scalar, by considering the crossover between Poissonian and Wigner-Dyson distributions in systems truncated to a finite-dimensional Hilbert space. We find that a naive extension of the scaling of crossover coupling with the volume observed in spin chains does not give satisfactory results for quantum field theory. Instead, we demonstrate that considering the scaling of the crossover coupling with the number of particles yields robust signatures, and is able to distinguish between the strengths of integrability breaking in the \phi^4 ϕ 4 and \phi^6 ϕ 6 quantum field theories.

arXiv:2305.02666•SciPost Physics•Oct 5, 2023•1 citation

On deforming and breaking integrability

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

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.

arXiv:2603.17018•Mar 17, 2026•0 citations

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

David A. Rabson, B. N. Narozhny, Andrew J. Millis

We study numerically the evolution of energy-level statistics as an integrability-breaking term is added to the XXZ Hamiltonian. For finite-length chains, physical properties exhibit a crossover from behavior corresponding to the Poisson level statistics characteristic of integrable models to behavior corresponding to the Wigner-Dyson statistics characteristic of the random-matrix theory used to describe chaotic systems. Different measures of the level statistics are observed to follow different crossover patterns. The range of numerically accessible system sizes is too small to establish with certainty the scaling with system size, but the evidence suggests that in a thermodynamically large system an infinitesimal integrability breaking would lead to Wigner-Dyson behavior.

arXiv:cond-mat/0308089•Physical Review B•Feb 9, 2004•92 citations

Adiabatic gauge potential and integrability breaking with free fermions

Balázs Pozsgay, Rustem Sharipov, Anastasiia Tiutiakina +1 more

We revisit the problem of integrability breaking in free fermionic quantum spin chains. We investigate the so-called adiabatic gauge potential (AGP), which was recently proposed as an accurate probe of quantum chaos. We also study the so-called weak integrability breaking, which occurs if the dynamical effects of the perturbation do not appear at leading order in the perturbing parameter. A recent statement in the literature claimed that integrability breaking should generally lead to an exponential growth of the AGP norm with respect to the volume. However, afterwards it was found that weak integrability breaking is a counter-example, leading to a cross-over between polynomial and exponential growth. Here we show that in free fermionic systems the AGP norm always grows polynomially, if the perturbation is local with respect to the fermions, even if the perturbation strongly breaks integrability. As a by-product of our computations we also find, that in free fermionic spin chains there are operators which weakly break integrability, but which are not associated with known long range deformations.

arXiv:2402.12979•SciPost Physics•Sep 9, 2024•10 citations

Integrability Breaking and Coherent Dynamics in Hermitian and Non-Hermitian Spin Chains with Long-Range Coupling

Y. S. Liu, X. Z. Zhang

Unraveling the mechanisms of ergodicity breaking in complex quantum systems is a central pursuit in nonequilibrium physics. In this work, we investigate a one-dimensional spin model featuring a tunable long-range hopping term, $H_{n}$, which introduces nonlocal interactions and bridges the gap between Hermitian and non-Hermitian regimes. Through a systematic analysis of level-spacing statistics, Krylov complexity, and entanglement entropy, we demonstrate that $H_{n}$ acts as a universal control parameter driving the transition from integrability to quantum chaos. Specifically, increasing the strength of $H_{n}$ induces a crossover from Poissonian to Gaussian Orthogonal Ensemble statistics in the Hermitian limit, and similarly triggers chaotic dynamics in the non-Hermitian case. Most remarkably, despite the onset of global chaos, we identify a tower of exact nonthermal eigenstates that evade thermalization. These states survive as robust quantum many-body scars, retaining low entanglement and coherent dynamics even under strong non-Hermitian perturbations. Our findings reveal a universal mechanism by which long-range and non-Hermitian effects reshape quantum ergodicity, offering new pathways for preserving quantum coherence in complex many-body systems.

quant-pharXiv:2512.14065•Dec 17, 2025•0 citations
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.
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 L L scales with a 1/L^{2} 1 / L 2 law for weak breaking as opposed to the 1/L^{3} 1 / L 3 law previously found for the strong case.
The recently proposed classification of integrability-breaking perturbations according to their strength is studied in the context of quantum field theories. Using random matrix methods to diagnose the resulting quantum chaotic behaviour, we investigate the \phi^4 ϕ 4 and \phi^6 ϕ 6 interactions of a massive scalar, by considering the crossover between Poissonian and Wigner-Dyson distributions in systems truncated to a finite-dimensional Hilbert space. We find that a naive extension of the scaling of crossover coupling with the volume observed in spin chains does not give satisfactory results for quantum field theory. Instead, we demonstrate that considering the scaling of the crossover coupling with the number of particles yields robust signatures, and is able to distinguish between the strengths of integrability breaking in the \phi^4 ϕ 4 and \phi^6 ϕ 6 quantum field theories.
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.
We study numerically the evolution of energy-level statistics as an integrability-breaking term is added to the XXZ Hamiltonian. For finite-length chains, physical properties exhibit a crossover from behavior corresponding to the Poisson level statistics characteristic of integrable models to behavior corresponding to the Wigner-Dyson statistics characteristic of the random-matrix theory used to describe chaotic systems. Different measures of the level statistics are observed to follow different crossover patterns. The range of numerically accessible system sizes is too small to establish with certainty the scaling with system size, but the evidence suggests that in a thermodynamically large system an infinitesimal integrability breaking would lead to Wigner-Dyson behavior.
We revisit the problem of integrability breaking in free fermionic quantum spin chains. We investigate the so-called adiabatic gauge potential (AGP), which was recently proposed as an accurate probe of quantum chaos. We also study the so-called weak integrability breaking, which occurs if the dynamical effects of the perturbation do not appear at leading order in the perturbing parameter. A recent statement in the literature claimed that integrability breaking should generally lead to an exponential growth of the AGP norm with respect to the volume. However, afterwards it was found that weak integrability breaking is a counter-example, leading to a cross-over between polynomial and exponential growth. Here we show that in free fermionic systems the AGP norm always grows polynomially, if the perturbation is local with respect to the fermions, even if the perturbation strongly breaks integrability. As a by-product of our computations we also find, that in free fermionic spin chains there are operators which weakly break integrability, but which are not associated with known long range deformations.
Unraveling the mechanisms of ergodicity breaking in complex quantum systems is a central pursuit in nonequilibrium physics. In this work, we investigate a one-dimensional spin model featuring a tunable long-range hopping term, HnH_{n}Hn​, which introduces nonlocal interactions and bridges the gap between Hermitian and non-Hermitian regimes. Through a systematic analysis of level-spacing statistics, Krylov complexity, and entanglement entropy, we demonstrate that HnH_{n}Hn​ acts as a universal control parameter driving the transition from integrability to quantum chaos. Specifically, increasing the strength of HnH_{n}Hn​ induces a crossover from Poissonian to Gaussian Orthogonal Ensemble statistics in the Hermitian limit, and similarly triggers chaotic dynamics in the non-Hermitian case. Most remarkably, despite the onset of global chaos, we identify a tower of exact nonthermal eigenstates that evade thermalization. These states survive as robust quantum many-body scars, retaining low entanglement and coherent dynamics even under strong non-Hermitian perturbations. Our findings reveal a universal mechanism by which long-range and non-Hermitian effects reshape quantum ergodicity, offering new pathways for preserving quantum coherence in complex many-body systems.