Momentum-Dependent Quantum Ruelle-Pollicott Resonances in the Kicked Ising Model: A Numerical Study

Core Concepts

Analyzing the momentum-dependent spectrum of Ruelle-Pollicott resonances in the kicked Ising model provides a nuanced understanding of quantum chaos and relaxation dynamics in many-body systems, revealing different decay rates for various correlation functions and highlighting the role of almost conserved operators in prethermalization plateaus.

Abstract

This research paper investigates Ruelle-Pollicott (RP) resonances in the context of the kicked Ising (KI) model, a paradigmatic model in quantum chaos. The authors introduce a novel approach by analyzing the momentum-resolved spectrum of a truncated operator propagator on an infinite lattice. This method allows for a detailed study of the decay rates of various correlation functions, revealing a rich interplay between momentum, system parameters, and dynamical regimes.

Bibliographic Information:

Znidaric, M. (2024). Momentum dependent quantum Ruelle-Pollicott resonances in translationally invariant many-body systems. arXiv preprint arXiv:2408.06307v3.

Research Objective:

The study aims to characterize the dynamical regimes of the kicked Ising model by analyzing the momentum-dependent spectrum of RP resonances and their influence on the decay rates of correlation functions.

Methodology:

The authors develop a numerical method based on a truncated operator propagator in momentum space. They analyze the leading eigenvalues of this propagator for different momenta and system parameters, relating them to the decay rates of correlation functions for various observables.

Key Findings:

• The leading RP resonance exhibits a non-trivial dependence on momentum, indicating different decay rates for correlation functions of observables belonging to different momentum sectors.
• The KI model exhibits distinct dynamical regimes, including chaotic regimes with exponential decay of correlations and a mixing regime characterized by power-law decay.
• An almost conserved operator is identified in the mixing regime, leading to a prethermalization plateau in the decay of certain correlation functions.
• The authors conjecture a lower bound for RP resonances based on the singular values of the truncated propagator.

Main Conclusions:

The momentum-resolved spectrum of RP resonances provides a powerful tool for characterizing the dynamics of quantum many-body systems. The study reveals a rich interplay between momentum, system parameters, and dynamical regimes in the KI model, highlighting the importance of considering momentum dependence in analyzing quantum chaos and relaxation dynamics.

Significance:

This research contributes to the understanding of quantum chaos in many-body systems by providing a novel method for analyzing RP resonances and their connection to the decay of correlations. The findings have implications for the study of thermalization, prethermalization, and the emergence of statistical mechanics in closed quantum systems.

Limitations and Future Research:

The study focuses on the KI model as a representative example. Further research could explore the applicability of the method to other many-body systems and investigate the role of disorder and dimensionality. Additionally, a more rigorous mathematical analysis of the conjectured lower bound for RP resonances would be valuable.

Stats

The leading RP resonance for the total magnetization in the z-direction converges to approximately 0.825 at τ = 0.75.
The decay rate observed for the magnetization in the z-direction before reaching the asymptotic regime is approximately 0.76 at τ = 0.75.
The minimum value of the outer radius (rout) of the annular region containing most eigenvalues is approximately 0.64.

Quotes

Key Insights Distilled From

Momentum dependent quantum Ruelle-Pollicott resonances in translationally invariant many-body systems

by Marko Znidar... at arxiv.org 11-06-2024

https://arxiv.org/pdf/2408.06307.pdf

Deeper Inquiries

How does the momentum-dependent spectrum of RP resonances manifest in experimentally realizable quantum many-body systems?

The momentum-dependent spectrum of Ruelle-Pollicott (RP) resonances can be observed in experimentally realizable quantum many-body systems through the decay rates of different correlation functions. Here's how:

Translational Invariance: Systems with translational invariance, like many cold-atom and trapped-ion setups, allow for a well-defined momentum (or quasi-momentum). This means that different collective excitations of the system can be characterized by their momentum.
Correlation Functions as Probes: Correlation functions, which measure how the behavior of one part of the system influences another, are key experimental observables. Examples include density-density correlations in cold atoms or spin-spin correlations in trapped ions.
Distinct Decay Rates: The momentum dependence of RP resonances implies that correlation functions associated with different momenta will generally decay at different rates. This is a direct consequence of the fact that different momentum sectors can exhibit varying degrees of chaoticity or integrability.
Experimental Detection: By measuring the decay of correlation functions for different momenta, one can experimentally reconstruct the momentum-dependent spectrum of RP resonances. This can be achieved by preparing the system in a state with a well-defined momentum and then monitoring the time evolution of the relevant correlation function.
Example: In the context of the kicked Ising model discussed in the paper, the authors show that the decay rates of the total magnetization, magnetization current, and staggered magnetization, all associated with different momenta, are governed by distinct RP resonances. This highlights the experimental relevance of the momentum-dependent spectrum.
Challenges and Considerations:

Finite-Size Effects: Real-world experiments deal with finite systems, while RP resonances are formally defined in the thermodynamic limit. Careful finite-size scaling analysis is crucial to extrapolate to the thermodynamic limit.
Limited Observables: Experiments often have access to a limited set of observables. Choosing the right observables that couple to different momentum sectors is essential for probing the momentum dependence of RP resonances.
Decoherence and Noise: Real systems are subject to decoherence and noise, which can mask the intrinsic decay rates determined by RP resonances. Minimizing these effects is crucial for accurate measurements.

Could there be other factors besides almost conserved quantities contributing to the observed discrepancies between the asymptotic decay rate and the initial faster relaxation in certain parameter regimes of the kicked Ising model?

Yes, besides almost conserved quantities, other factors can contribute to the observed discrepancies between the asymptotic decay rate predicted by the leading RP resonance and the initial faster relaxation in the kicked Ising model:

Non-normality of the Truncated Propagator: The truncated operator propagator used to calculate RP resonances is a non-normal matrix. Non-normal matrices can exhibit transient behavior that deviates significantly from the asymptotic decay determined by their eigenvalues. This transient behavior is often governed by the pseudospectrum of the matrix, which can be much larger than the spectrum.
Finite-Size Effects: Even though the paper focuses on the thermodynamic limit, finite-size effects can still play a role, especially for intermediate timescales. The convergence of the leading RP resonance with increasing system size might be slow, leading to discrepancies for accessible system sizes.
Higher-Order Resonances: While the leading RP resonance determines the asymptotic decay rate, higher-order resonances can contribute to the dynamics at intermediate timescales. These higher-order resonances might be responsible for the initial faster relaxation observed in certain parameter regimes.
Role of Entanglement: The growth and spread of entanglement can influence the relaxation dynamics. It's possible that the initial faster relaxation is related to a rapid initial growth of entanglement, which then slows down as the system approaches a more mixed state.
Further Investigation:

Pseudospectrum Analysis: Calculating the pseudospectrum of the truncated propagator could provide insights into the transient behavior and potential deviations from the asymptotic decay rate.
Systematic Finite-Size Scaling: Performing a systematic finite-size scaling analysis would help disentangle finite-size effects from the intrinsic dynamics governed by RP resonances.
Characterizing Higher-Order Resonances: Identifying and characterizing the contributions of higher-order RP resonances could shed light on the intermediate-timescale dynamics.
Entanglement Dynamics: Studying the entanglement dynamics alongside the decay of correlation functions might reveal connections between entanglement growth and the observed relaxation behavior.

How can the insights gained from studying quantum chaos in the kicked Ising model be applied to understand the dynamics of information scrambling and quantum information processing in complex quantum systems?

The insights gained from studying quantum chaos in the kicked Ising model, particularly through the lens of RP resonances, have significant implications for understanding information scrambling and quantum information processing in complex quantum systems:
Information Scrambling:

Quantifying Scrambling Rates: RP resonances provide a quantitative measure of how quickly local information spreads throughout the system, a hallmark of quantum chaos. A smaller leading RP resonance indicates faster scrambling.
Identifying Chaotic Regimes: The presence of well-defined and isolated RP resonances, as opposed to a continuous spectrum, signals chaotic behavior and efficient scrambling.
Momentum-Dependent Scrambling: The momentum dependence of RP resonances reveals that different types of information, characterized by their momentum, can scramble at different rates. This has implications for understanding the dynamics of spatially structured information.
Quantum Information Processing:

Decoherence and Error Correction: RP resonances can help quantify decoherence rates in quantum information processors. By engineering systems with large RP resonance gaps, one could potentially enhance coherence times.
Quantum Simulation of Chaotic Systems: The kicked Ising model, as a well-studied chaotic system, can serve as a testbed for developing and benchmarking quantum algorithms for simulating chaotic dynamics.
Designing Robust Quantum Gates: Understanding the conditions under which RP resonances become small or large can guide the design of quantum gates that are more robust against noise and imperfections.
Specific Examples:

Spreading of Quantum Information: The decay of correlation functions, governed by RP resonances, directly reflects how initially localized quantum information spreads throughout the system.
Out-of-Time-Order Correlators (OTOCs): OTOCs, a key tool for characterizing information scrambling, are intimately related to RP resonances. The decay rate of OTOCs is often determined by the leading RP resonance.
Many-Body Localization: In contrast to chaotic systems, many-body localized systems exhibit a breakdown of ergodicity and slow information scrambling. RP resonances can be used to distinguish between chaotic and localized phases.
Broader Implications:
The study of quantum chaos in the kicked Ising model provides a valuable framework for understanding the behavior of more complex quantum systems, including:

Black Hole Physics: The fast scrambling of information in black holes has connections to quantum chaos. Insights from the kicked Ising model could contribute to understanding black hole information paradoxes.
High-Energy Physics: The dynamics of strongly interacting quantum field theories, relevant for particle physics, often exhibit chaotic behavior. Techniques from quantum chaos can be applied to study these systems.
Condensed Matter Physics: Many-body quantum systems in condensed matter physics, such as high-temperature superconductors, display complex and often chaotic dynamics. The tools and concepts from quantum chaos can aid in understanding these phenomena.