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arXiv:2404.04177 · The European Physical Journal B

Diagnosing chaos in a periodically driven Ising model with a ramping field via out-of-time-order correlation saturation

Rohit Kumar Shukla, Gaurav Rudra Malik, S. Aravinda, Sunil Kumar Mishra

Published Aug 1, 2025• 1 citation

Abstract

Abstract The dynamic region of out-of-time-ordered correlators (OTOCs) serves as a powerful indicator of chaos in classical and semiclassical systems, capturing the characteristic exponential growth. However, in spin systems, the dynamic region of OTOCs does not reliably quantify chaos as both integrable and chaotic systems display similar behavior. Instead, we utilize the saturation behavior of OTOCs to differentiate between chaotic and integrable regimes. In integrable systems, the saturation region of OTOCs exhibits oscillatory behavior, while in chaotic systems, it shows a stable saturation. To evaluate this distinction, we investigate a time-dependent Ising spin system subjected to a linearly ramping transverse field, analyzing both integrable (without longitudinal field) and non-integrable (with longitudinal field) scenarios. This setup accelerates system ergodicity due to the introduction of an additional time scale within the system, enhancing chaotic dynamics in the non-integrable regime, and provides a compelling model for studying the interplay between integrability and chaos in quantum systems. To further support our findings, we investigate the level spacing distribution of time-dependent unitary operators, which effectively distinguishes chaotic from regular regions in our system and corroborates the results obtained from the saturation behavior of the OTOC. Additionally, we calculate a metric for the normalized Fourier spectrum of the OTOC which is dependent on the number of frequency components present to gain insights into the observed oscillations and its dependence on the ramping field. Graphical abstract

DOI: 10.1140/epjb/s10051-025-01021-1

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Abstract The dynamic region of out-of-time-ordered correlators (OTOCs) serves as a powerful indicator of chaos in classical and semiclassical systems, capturing the characteristic exponential growth. However, in spin systems, the dynamic region of OTOCs does not reliably quantify chaos as both integrable and chaotic systems display similar behavior. Instead, we utilize the saturation behavior of OTOCs to differentiate between chaotic and integrable regimes. In integrable systems, the saturation region of OTOCs exhibits oscillatory behavior, while in chaotic systems, it shows a stable saturation. To evaluate this distinction, we investigate a time-dependent Ising spin system subjected to a linearly ramping transverse field, analyzing both integrable (without longitudinal field) and non-integrable (with longitudinal field) scenarios. This setup accelerates system ergodicity due to the introduction of an additional time scale within the system, enhancing chaotic dynamics in the non-integrable regime, and provides a compelling model for studying the interplay between integrability and chaos in quantum systems. To further support our findings, we investigate the level spacing distribution of time-dependent unitary operators, which effectively distinguishes chaotic from regular regions in our system and corroborates the results obtained from the saturation behavior of the OTOC. Additionally, we calculate a metric for the normalized Fourier spectrum of the OTOC which is dependent on the number of frequency components present to gain insights into the observed oscillations and its dependence on the ramping field. Graphical abstract

Out-of-time-ordered correlators (OTOCs) have emerged as a diagnostic of information scrambling and quantum chaos in many-body systems. We investigate the imprints of chaos in the dynamics of OTOCs in the Heisenberg spin-

1/2

chain with random fields. The system is parameterized to exhibit a crossover from integrable to chaotic dynamics. We demonstrate numerically that the approach to saturation of the OTOC can distinguish between integrable and chaotic regimes, with a power-law

(1/t)

relaxation for integrable systems and a higher-degree power-law decay

(1/t^α; α\ge 1)

followed by an exponential relaxation for the chaotic regime. We further show that long-range spectral statistics, such as the number variance, are more effective in characterizing quantum chaos in the regime near saturation of OTOC. We also demonstrate that the relaxation and initial scrambling regimes exhibit distinct and universal features, with the former being sensitive and the latter being robust against different realizations of random-fields. The long-time saturation of OTOC also fluctuates with different realizations, and its exact expression is derived through the Eigenstate Thermalization Hypothesis.

Out-of-time-ordered correlators (OTOCs) have been proposed as a probe of chaos in quantum mechanics, on the basis of their short-time exponential growth found in some particular setups. However, it has been seen that this behavior is not universal. Therefore, we query other quantum chaos manifestations arising from the OTOCs, and we thus study their long-time behavior in systems of completely different nature: quantum maps, which are the simplest chaotic one-body system, and spin chains, which are many-body systems without a classical limit. It is shown that studying the long-time regime of the OTOCs it is possible to detect and gauge the transition between integrability and chaos, and we benchmark the transition with other indicators of quantum chaos based on the spectra and the eigenstates of the systems considered. For systems with a classical analog, we show that the proposed OTOC indicators have a very high accuracy that allow us to detect subtle features along the integrability-to-chaos transition.

We investigate signatures of quantum chaos within Ising spin chains subjected to transverse and longitudinal fields, incorporating both local (nearest-neighbor) and non-local (long-range) couplings. While local Ising models may exhibit integrable or chaotic dynamics contingent on interaction strengths and field parameters, systems with non-local interactions generally display a stronger propensity toward chaos, even when the non-local couplings are weak. By examining the distribution of energy level spacings through the level spacing ratio, we delineate the transition from integrable to chaotic regimes and characterize the emergence of quantum chaos in these systems. Our analysis demonstrates that non-local couplings facilitate faster operator spreading and more intricate dynamical behavior, enabling these systems to approach maximal chaos more readily than their local counterparts. Additionally, we analyze Krylov complexity as a dynamical probe of chaos, observing a characteristic peak followed by a plateau at late times in chaotic regimes. This behavior provides a quantitative means to distinguish between integrable and chaotic phases, with the growth rate and saturation level of the complexity serving as effective indicators. Our findings underscore the role of non-local interactions in accelerating the onset of chaos and modifying dynamical complexity in quantum spin chains.

Out-of-time-order correlators (OTOCs) have proven to be a useful tool for studying thermalization in quantum systems. In particular, the exponential growth of OTOCS, or scrambling, is sometimes taken as an indicator of chaos in quantum systems, despite the fact that saddle points in integrable systems can also drive rapid growth in OTOCs. By analyzing the Dicke model and a driven Bose-Hubbard dimer, we demonstrate that the OTOC growth driven by chaos can, nonetheless, be distinguished from that driven by saddle points through the long-term behavior. Besides quantitative differences in the long-term average, the saddle point gives rise to large oscillations not observed in the chaotic case. The differences are also highlighted by entanglement entropy, which in the chaotic-driven dimer matches a Page curve prediction. These results illustrate additional markers that can be used to distinguish chaotic behavior in quantum systems, beyond the initial exponential growth in OTOCs.

Time New Roman 10 We investigate signatures of quantum chaos within Ising spin chains subjected to transverse and longitudinal fields, incorporating both local (nearest-neighbor) and non-local (long-range) couplings. While local Ising models may exhibit integrable or chaotic dynamics contingent on interaction strengths and field parameters, systems with non-local interactions generally display a stronger propensity toward chaos, even when the non-local couplings are weak. By examining the distribution of energy level spacings through the level spacing ratio, we delineate the transition from integrable to chaotic regimes and characterize the emergence of quantum chaos in these systems. Our analysis demonstrates that non-local couplings facilitate faster operator spreading and more intricate dynamical behavior, enabling these systems to approach maximal chaos more readily than their local counterparts. Additionally, we analyze Krylov complexity as a dynamical probe of chaos, observing a characteristic peak followed by a plateau at late times in chaotic regimes. This behavior provides a quantitative means to distinguish between integrable and chaotic phases, with the growth rate and saturation level of the complexity serving as effective indicators. Our findings underscore the role of non-local interactions in accelerating the onset of chaos and modifying dynamical complexity in quantum spin chains.

Fast scrambling, quantified by the exponential initial growth of out-of-time-ordered correlators (OTOCs), is the ability to efficiently spread quantum correlations among the degrees of freedom of interacting systems and constitutes a characteristic signature of local unstable dynamics. As such, it may equally manifest both in systems displaying chaos or in integrable systems around criticality. Here we go beyond these extreme regimes with an exhaustive study of the interplay between local criticality and chaos right at the intricate phase-space region where the integrability-chaos transition first appears. We address systems with a well-defined classical (mean-field) limit, as coupled large spins and Bose-Hubbard chains, thus allowing for semiclassical analysis. Our aim is to investigate the dependence of the exponential growth of the OTOCs, defining the quantum Lyapunov exponent λ_{q} on quantities derived from the classical system with mixed phase space, specifically the local stability exponent of a fixed point λ_{loc} as well as the maximal Lyapunov exponent λ_{L} of the chaotic region around it. By extensive numerical simulations covering a wide range of parameters we give support to a conjectured linear dependence 2λ_{q}=aλ_{L}+bλ_{loc}, providing a simple route to characterize scrambling at the border between chaos and integrability.