Probing chaos and thermalization through out-of-time-ordered correlators in random field spin chains

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.

• The paper demonstrates that OTOCs can differentiate integrable, chaotic, and many-body localized regimes through distinct relaxation behaviors.
• It employs spectral analyses like NNSD and number variance combined with ETH-based saturation formulas to probe operator dynamics in disordered spin chains.
• Key findings include universal power-law scrambling and exponential decay in chaotic regimes, highlighting OTOC's utility for diagnosing thermalization.

Probing Chaos and Thermalization in Random Field Spin Chains via Out-of-Time-Ordered Correlators

Introduction and Motivation

This paper systematically examines the out-of-time-ordered correlator (OTOC) behavior as a probe of quantum chaos and thermalization in Heisenberg spin-$1/2$ chains with random onsite fields. The model permits interpolation between integrable and chaotic regimes by tuning disorder strength, $k$, enabling the exploration of phase transitions from integrability, through chaos, to many-body localization (MBL). OTOCs, which quantify operator spreading and information scrambling, serve as a diagnostic tool for chaos beyond traditional spectral statistics. A central objective is to determine if OTOC dynamics in many-body systems (without a classical counterpart) manifest signatures analogous to those seen in few-body quantum systems with classical limits.

Model and Quantum Chaos Diagnostics

The system under consideration is the XXZ Heisenberg chain, parameterized with uniform random fields in the $x$ and $z$ directions. For zero or weak disorder, the model is integrable. Increasing $k$ breaks integrability, resulting in chaotic spectra characterized by level repulsion and strong eigenvalue correlations. At sufficiently large $k$, the chain transitions into the MBL regime, dominated by localization phenomena.

Traditional chaos indicators are rigorously assessed via:

• Nearest Neighbor Spacing Distribution (NNSD): Its transition from Poisson (integrable) to Wigner-Dyson (chaotic) statistics is mapped precisely as a function of $k$. The crossover is sharp and scales with $1/\sqrt{N}$, where $N$ is the Hilbert space dimension.
• Number Variance, $\Sigma^2(r)$: This characterizes long-range spectral correlations. Integrable systems exhibit $k$0; chaotic systems display logarithmic growth congruent with GOE predictions from Random Matrix Theory (RMT).
• Entanglement Entropy: Bipartition entanglement in mid-spectrum eigenstates quantifies thermalization. Volume-law scaling is observed in the chaotic regime, saturating near Page values. Strong disorder enforces area-law scaling, indicative of MBL.

The study finds that NNSD RMT agreement extends over a wider range of $k$1 than long-range statistics and entanglement. Thus, traditional spectral chaos metrics alone do not offer comprehensive insight, particularly in intermediate “crossover” regimes.

OTOC Dynamics: Scrambling, Relaxation, and Saturation

The OTOC is measured for both local and block (non-local) observables, probing distinct spatial regions of the chain. Its temporal behavior is partitioned into three universal regimes:

Scrambling Regime

• Observations: For both integrable and chaotic dynamics, OTOC grows algebraically (power-law) with time. This regime is insensitive to underlying spectral statistics or random field realizations.
• Universal Feature: Unlike few-body systems with classical analogs (where chaos induces exponential OTOC growth), many-body spin chains exhibit universal power-law growth, irrespective of chaos onset or operator locality.

Relaxation Regime

• Distinguishing Chaos: The relaxation regime provides a clear demarcation between integrable and chaotic behaviors:
  • Integrable Regime: OTOC relaxes via power-law decay ($k$2).
  • Chaotic Regime: OTOC relaxation transitions to a higher-order power law followed by exponential decay. The initial transient may maintain polynomial dependence, but the asymptotic approach is exponential.
  • Sensitivity: Relaxation rates are highly sensitive to random disorder realizations, marking a strong departure from the universal scrambling behavior.
• Correlation with Spectral Statistics: The onset of exponential relaxation correlates with the field strength window where long-range spectral statistics (number variance) align with RMT, rather than with the broader NNSD regime.

Saturation Regime

• OTOC Fluctuations: The long-time saturation value of OTOC fluctuates across disorder realizations.
• Analytical Expression via ETH: Saturation values are derived analytically within the Eigenstate Thermalization Hypothesis (ETH) framework. The derived expressions encompass both diagonal and restricted off-diagonal operator matrix elements in the energy eigenbasis, capturing ensemble-dependent fluctuations absent in RMT predictions.
• Disorder Dependence: As $k$3 increases, saturation value rises, peaking in the strongly chaotic window before decreasing in the MBL phase. This trend inversely mirrors the variance of diagonal operator elements, signifying delocalization and maximal thermalization in the chaotic window.

Numerical Results and Claims

• Clear numerical evidence is established for power-law ($k$4) and exponential relaxation behaviors in integrable and chaotic regimes respectively; strong disorder transitions drive MBL behavior.
• Distinct universality: Scrambling regime is robust to system disorder, but relaxation and saturation are highly sensitive to both the spectral statistics and disorder realization.
• Saturation analysis: ETH provides a precise, realization-sensitive formula for OTOC saturation values, where RMT fails to capture empirical fluctuations.

Implications and Future Directions

The findings demonstrate that OTOC relaxation dynamics—particularly the transition to exponential decay—are a robust probe of quantum chaos in many-body systems, offering finer resolution than NNSD or entanglement entropy alone, especially in the presence of disordered environments. The sensitivity of OTOC to disorder realizations and spectral statistics also highlights the necessity for ensemble-level analysis in practical quantum information scrambling diagnostics.

On a theoretical level, the study advances the application of ETH to dynamical correlators, extending static thermalization predictions to late-time operator spreading. This approach could be generalized to other many-body architectures with tunable disorder and interaction range.

Practically, these results inform the design and analysis of quantum simulators (e.g., trapped ions, cold atoms) where OTOC can be directly measured and chaos/disorder effects are tunable. The clarity in distinguishing regimes via relaxation dynamics makes OTOC a vital metric for benchmarking thermalization, localization, and information leakage in quantum computation platforms.

Future work may extend this framework to larger system sizes, more complex disorder models, and multi-site/multi-operator correlators. Analytical treatment of relaxation rates within ETH remains an open avenue, with implications for the study of quantum ergodicity and dynamical phase transitions.

Conclusion

This paper establishes OTOC as a sensitive and practical probe for distinguishing integrable, chaotic, and localized regimes in random-field Heisenberg spin chains. While scrambling is universal and insensitive to disorder, relaxation and saturation regimes reveal strong dependence on spectral statistics and disorder realizations. The analytical connection of OTOC saturation to ETH is shown to capture ensemble-induced fluctuations, providing a robust framework for diagnosing chaos and thermalization in quantum many-body systems. The practical and theoretical utility of OTOC—especially in the relaxation regime—suggests broader applications in quantum chaos, information theory, and quantum technology diagnostics.