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arXiv:2412.16887 · Physical review. E

Local integrability breaking and exponential localization of leading Lyapunov vectors

Jiaozi Wang, Tomaž Prosen, Giulio Casati

Published Aug 13, 2025• 2 citations
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Abstract

We study integrability breaking and transport in a discrete space-time lattice with a local integrability-breaking perturbation. We find a singular distribution of the Lyapunov spectrum where the majority of Lyapunov exponents vanish in the thermodynamic limit. The subextensive sequence of nonzero exponents, converging in the thermodynamic limit, corresponds to Lyapunov vectors that are exponentially localized with localization lengths proportional to inverse Lyapunov exponents. Moreover, we investigate the transport behavior of the system by considering the spin-spin and current-current spatiotemporal correlation functions. Our results indicate that the overall transport behavior, similar to the purely integrable case, conforms to Kardar-Parisi-Zhang scaling in the thermodynamic limit and at vanishing magnetization. The same dynamical exponent z=3/2 governs the effect of local perturbation spreading in the bulk.

DOI
10.1103/lkjy-xd53

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We study integrability breaking and transport in a discrete space-time lattice with a local integrability-breaking perturbation. We find a singular distribution of the Lyapunov spectrum where the majority of Lyapunov exponents vanish in the thermodynamic limit. The subextensive sequence of nonzero exponents, converging in the thermodynamic limit, corresponds to Lyapunov vectors that are exponentially localized with localization lengths proportional to inverse Lyapunov exponents. Moreover, we investigate the transport behavior of the system by considering the spin-spin and current-current spatiotemporal correlation functions. Our results indicate that the overall transport behavior, similar to the purely integrable case, conforms to Kardar-Parisi-Zhang scaling in the thermodynamic limit and at vanishing magnetization. The same dynamical exponent z=3/2 governs the effect of local perturbation spreading in the bulk.
Recent studies report on anomalous spin transport for the integrable Heisenberg spin chain at its isotropic point. Anomalous scaling is also observed in the time evolution of nonequilibrium initial conditions, the decay of current-current correlations, and nonequilibrium steady state averages. These studies indicate a space-time scaling with x∼t^{2/3} behavior at the isotropic point, in sharp contrast to the ballistic form x∼t generically expected for integrable systems. In our contribution we study the scaling behavior for the integrable lattice Landau-Lifshitz spin chain. We report on equilibrium spatiotemporal correlations and dynamics with step initial conditions. Remarkably, for the case with zero mean magnetization, we find strong evidence that the scaling function is identical to the one obtained from the stationary stochastic Burgers equation, alias Kardar-Parisi-Zhang equation. In addition, we present results for the easy-plane and easy-axis regimes for which, respectively, ballistic and diffusive spin transport is observed, whereas the energy remains ballistic over the entire parameter regime.
Anomalous transport in low dimensional spin chains is an intriguing topic that can offer key insights into the interplay of integrability and symmetry in many-body dynamics. Recent studies have shown that spin-spin correlations in spin chains, where integrability is either perfectly preserved or broken by symmetry-preserving interactions, fall in the Kardar-Parisi-Zhang (KPZ) universality class. Similarly, energy transport can show ballistic or diffusive-like behavior. Although such behavior has been studied under equilibrium conditions, no results on nonequilibrium spin transport in classical spin chains has been reported so far. In this work, we investigate both spin and energy transport in classical spin chains (integrable and nonintegrable) when coupled to two reservoirs at two different temperatures/magnetizations. In both the integrable case and the broken-integrability (but spin-symmetry preserving) case, we report anomalous scaling of spin current with system size (J^{s}∝L^{-μ}), with an exponent value that, within error bars, is close to the KPZ universality class value μ≈2/3. On the other hand, it is noteworthy that the energy current remains ballistic (J^{e}∝L^{-η} with η≈0) in the purely integrable case while there is departure from ballistic behavior (η>0) when integrability is broken regardless of spin-symmetry. We also present results on other interesting observables in the nonequilibrium steady state such as the spatial profiles of magnetization and energy, and spin-spin correlations.
Recent investigations have observed superdiffusion in integrable classical and quantum spin chains. An intriguing connection between these spin chains and the Kardar-Parisi-Zhang (KPZ) universality class has emerged. Theoretical developments (e.g., generalized hydrodynamics) have highlighted the role of integrability as well as spin symmetry in KPZ behavior. However, understanding their precise role on superdiffusive transport still remains a challenging task. The widely used quantum spin chain platform comes with severe numerical limitations. To circumvent this barrier, we focus on a classical integrable spin chain which was shown to have a deep analogy with the quantum spin-12\frac{1}{2}21​ Heisenberg chain. Remarkably, we find that KPZ behavior prevails even when one considers integrability-breaking but spin-symmetry preserving terms, strongly indicating that spin symmetry plays a central role even in the nonperturbative regime. On the other hand, in the nonperturbative regime, we find that energy correlations exhibit clear diffusive behavior. We also study the classical analog of the out-of-time-ordered correlator and Lyapunov exponents. We find a significant presence of chaos for the integrability-broken cases even though KPZ behavior remains robust. The robustness of KPZ behavior is demonstrated for a wide class of spin-symmetry preserving integrability-breaking terms.
We propose a novel framework to characterize the thermalization of many-body dynamical systems close to integrable limits using the scaling properties of the full Lyapunov spectrum. We use a classical unitary map model to investigate macroscopic weakly nonintegrable dynamics beyond the limits set by the KAM regime. We perform our analysis in two fundamentally distinct long-range and short-range integrable limits which stem from the type of nonintegrable perturbations. Long-range limits result in a single parameter scaling of the Lyapunov spectrum, with the inverse largest Lyapunov exponent being the only diverging control parameter and the rescaled spectrum approaching an analytical function. Short-range limits result in a dramatic slowing down of thermalization which manifests through the rescaled Lyapunov spectrum approaching a non-analytic function. An additional diverging length scale controls the exponential suppression of all Lyapunov exponents relative to the largest one.
We investigate the statistical properties of local Lyapunov exponents which characterize magnon localization in the one-dimensional Heisenberg-Mattis spin glass (HMSG) at zero temperature by means of a connection to a suitable version of the Fokker-Planck (FP) equation. We consider the local Lyapunov exponents (LLEs), in particular, the case of instantaneous LLE. We establish a connection between the transfer-matrix recursion relation for the problem and an FP equation governing the evolution of the probability distribution of the instantaneous LLE. The closed-form (stationary) solutions to the FP equation are in excellent accord with numerical simulations for both the unmagnetized and magnetized versions of the HMSG. Scaling properties for nonstationary conditions are derived from the FP equation in a special limit (in which diffusive effects tend to vanish), and also shown to provide a close description to the corresponding numerical-simulation data.
The Heisenberg spin chain is a canonical integrable model. As such, it features stable ballistically propagating quasiparticles, but spin transport is subballistic at any nonzero temperature: An initially localized spin fluctuation spreads in time t to a width t2/3. This exponent as well as the functional form of the dynamical spin correlation function suggest that spin transport is in the Kardar–Parisi–Zhang (KPZ) universality class. However, the full counting statistics of magnetization is manifestly incompatible with KPZ scaling. A simple two-mode hydrodynamic description, derivable from microscopic principles, captures both the KPZ scaling of the correlation function and the coarse features of the full counting statistics, but remains to be numerically validated. These results generalize to any integrable spin chain invariant under a continuous nonabelian symmetry and are surprisingly robust against moderately strong integrability-breaking perturbations that respect the nonabelian symmetry.