insight - Scientific Computing - # SYK Model Thermalization

Thermalization Dynamics of a Mixed Sachdev-Ye-Kitaev Model in the Thermodynamic Limit

Q: How might the inclusion of disorder in the coupling constants affect the thermalization dynamics of the mixed SYK model?

Disorder in the coupling constants of the mixed SYK model plays a crucial role in driving the system towards thermalization. Here's how: Breaking of Integrability: The random nature of the couplings $j_q$ and $j_2$ ensures that the mixed SYK Hamiltonian is non-integrable. This means the system lacks an extensive set of conserved quantities, preventing it from being trapped in a non-ergodic subspace and facilitating exploration of the full Hilbert space. Eigenstate Thermalization Hypothesis (ETH): Disorder is believed to be a key ingredient for a system to satisfy the ETH. The ETH posits that in ergodic systems, individual energy eigenstates locally resemble thermal ensembles. The random couplings in the SYK model lead to a complex energy landscape that promotes the fulfillment of ETH, enabling thermalization. Fast Scrambling of Information: The all-to-all nature of the random couplings in the SYK model leads to fast scrambling of quantum information. This means that local perturbations quickly spread throughout the system, effectively destroying local memory and driving the system towards a state of maximal entropy, consistent with thermal equilibrium. Analytical Tractability: While disorder makes the model non-integrable, the specific choice of Gaussian-distributed couplings allows for analytical progress in the large-$N$ and large-$q$ limits. This makes the SYK model a powerful tool for studying the interplay of disorder, chaos, and thermalization. Without disorder, the SYK model would become integrable, and its thermalization dynamics would be significantly altered. The system might exhibit non-ergodic behavior, failing to reach a true thermal equilibrium.

Q: Could the observed finite thermalization times be an artifact of the large-N and large-q limits, and would finite-size effects significantly alter the thermalization behavior?

It's indeed possible that the observed finite thermalization times in the mixed SYK model are influenced by the large-$N$ and large-$q$ limits used in the analysis. Here's why: Large-$N$ Limit: The large-$N$ limit suppresses fluctuations and simplifies the problem by effectively averaging over the disorder. In finite-sized systems, these fluctuations could become important, potentially leading to different thermalization timescales. Large-$q$ Limit: The large-$q$ limit allows for further analytical simplifications, but it also changes the nature of the interactions. In finite-$q$ SYK models, the interactions are less "mean-field-like," and this could affect the thermalization dynamics. Here's how finite-size effects might alter the thermalization behavior: Slower Thermalization: Finite-size systems generally thermalize slower than their thermodynamic limit counterparts. This is because the finite size limits the available phase space for the system to explore. Recurrences: In finite-sized systems, there's a finite probability of the system returning close to its initial state, leading to recurrences in observables. These recurrences would be suppressed in the thermodynamic limit. Non-universal Behavior: The specific details of the thermalization dynamics, such as the precise values of thermalization times, might become dependent on the system size for finite $N$ and $q$. Therefore, while the large-$N$ and large-$q$ limits provide valuable insights into the thermalization of the mixed SYK model, it's crucial to investigate the role of finite-size effects to gain a complete understanding of the system's behavior.

Core Concepts

This study investigates the thermalization process of a closed Sachdev-Ye-Kitaev (SYK) system in the thermodynamic limit, revealing that a mixed SYK model with both interaction and kinetic terms exhibits finite thermalization times and rich dynamics, unlike simpler SYK quench scenarios.

Abstract

Bibliographic Information:

Salazar Jaramillo, S., Jha, R., & Kehrein, S. (2024). Thermalization of a Closed Sachdev-Ye-Kitaev System in the Thermodynamic Limit. arXiv preprint, arXiv:2411.12421v1.

Research Objective:

This study aims to analyze the thermalization dynamics of a closed, large-q Majorana SYK model in the thermodynamic limit when subjected to a quench that introduces a random hopping term alongside the existing interaction term.

Methodology:

The researchers employ the Keldysh formalism to derive the Kadanoff-Baym equations governing the non-equilibrium dynamics of the system. They utilize a large-N limit and a large-q expansion to simplify these equations. Due to the complexity of the equations, they develop a custom numerical integrator based on a predictor-corrector algorithm with causal stepping to solve them. The energy of the system is tracked over time to determine the final temperature and thermalization rate.

Key Findings:

Combining both interaction and kinetic terms in the SYK Hamiltonian leads to finite thermalization times, unlike quenches resulting in a single final term.
The numerical solutions of the Kadanoff-Baym equations demonstrate that the system reaches a stationary state corresponding to thermal equilibrium.
Stronger quenches (larger kinetic term contributions) result in higher final temperatures, approaching the infinite temperature limit observed in pure kinetic quenches.
The thermalization rate exhibits a complex dependence on the final temperature, with faster thermalization for stronger quenches and slower rates for weaker quenches, particularly at lower final temperatures.

Main Conclusions:

This study provides a detailed analysis of the thermalization process in a closed, mixed SYK model in the thermodynamic limit. The results highlight the rich dynamics arising from the interplay between interaction and kinetic terms, contrasting with the instantaneous thermalization observed in simpler SYK quench scenarios. The developed numerical techniques offer a valuable tool for investigating non-equilibrium dynamics in similar complex quantum systems.

Significance:

This research contributes significantly to the understanding of thermalization in closed quantum systems, particularly in the context of the SYK model, which serves as a relatively tractable model for studying quantum chaos and its relation to thermalization.

Limitations and Future Research:

The study focuses on a specific type of quench in the mixed SYK model. Exploring different quench protocols and their impact on thermalization dynamics could provide further insights. Investigating the role of finite-size effects and extending the analysis beyond the large-q limit would be valuable avenues for future research.

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Quotes

"This work addresses the question of thermalization in a closed SYK system in the thermodynamic limit."
"We study the thermalization of a closed quantum system in the thermodynamic limit where analytical equations obtained are exact in nature that we proceed to solve numerically."
"By contrast, the quench to the two-term Hamiltonian described above, referred to as the mixed quench, presents rich relaxation dynamics which are the focus of the present work."
"These limitations and difficulties highlight the importance of our work: we study the thermalization of a closed quantum system in the thermodynamic limit where analytical equations obtained are exact in nature that we proceed to solve numerically. We further quantify finite thermalization rates as well as final temperatures for the quench dynamics."

Key Insights Distilled From

Thermalization of a Closed Sachdev-Ye-Kitaev System in the Thermodynamic Limit

by Santiago Sal... at arxiv.org 11-20-2024

https://arxiv.org/pdf/2411.12421.pdf

Thermalization of a Closed Sachdev-Ye-Kitaev System in the Thermodynamic Limit

Deeper Inquiries

How might the inclusion of disorder in the coupling constants affect the thermalization dynamics of the mixed SYK model?

Disorder in the coupling constants of the mixed SYK model plays a crucial role in driving the system towards thermalization. Here's how:

Breaking of Integrability: The random nature of the couplings $j_q$ and $j_2$ ensures that the mixed SYK Hamiltonian is non-integrable. This means the system lacks an extensive set of conserved quantities, preventing it from being trapped in a non-ergodic subspace and facilitating exploration of the full Hilbert space.
Eigenstate Thermalization Hypothesis (ETH):  Disorder is believed to be a key ingredient for a system to satisfy the ETH. The ETH posits that in ergodic systems, individual energy eigenstates locally resemble thermal ensembles. The random couplings in the SYK model lead to a complex energy landscape that promotes the fulfillment of ETH, enabling thermalization.
Fast Scrambling of Information: The all-to-all nature of the random couplings in the SYK model leads to fast scrambling of quantum information. This means that local perturbations quickly spread throughout the system, effectively destroying local memory and driving the system towards a state of maximal entropy, consistent with thermal equilibrium.
Analytical Tractability: While disorder makes the model non-integrable, the specific choice of Gaussian-distributed couplings allows for analytical progress in the large-$N$ and large-$q$ limits. This makes the SYK model a powerful tool for studying the interplay of disorder, chaos, and thermalization.
Without disorder, the SYK model would become integrable, and its thermalization dynamics would be significantly altered. The system might exhibit non-ergodic behavior, failing to reach a true thermal equilibrium.

Could the observed finite thermalization times be an artifact of the large-N and large-q limits, and would finite-size effects significantly alter the thermalization behavior?

It's indeed possible that the observed finite thermalization times in the mixed SYK model are influenced by the large-$N$ and large-$q$ limits used in the analysis. Here's why:

Large-$N$ Limit: The large-$N$ limit suppresses fluctuations and simplifies the problem by effectively averaging over the disorder. In finite-sized systems, these fluctuations could become important, potentially leading to different thermalization timescales.
Large-$q$ Limit: The large-$q$ limit allows for further analytical simplifications, but it also changes the nature of the interactions. In finite-$q$ SYK models, the interactions are less "mean-field-like," and this could affect the thermalization dynamics.
Here's how finite-size effects might alter the thermalization behavior:

Slower Thermalization: Finite-size systems generally thermalize slower than their thermodynamic limit counterparts. This is because the finite size limits the available phase space for the system to explore.
Recurrences: In finite-sized systems, there's a finite probability of the system returning close to its initial state, leading to recurrences in observables. These recurrences would be suppressed in the thermodynamic limit.
Non-universal Behavior: The specific details of the thermalization dynamics, such as the precise values of thermalization times, might become dependent on the system size for finite $N$ and $q$.
Therefore, while the large-$N$ and large-$q$ limits provide valuable insights into the thermalization of the mixed SYK model, it's crucial to investigate the role of finite-size effects to gain a complete understanding of the system's behavior.

What are the implications of the observed thermalization dynamics in the mixed SYK model for understanding the relationship between quantum chaos and thermalization in more general quantum many-body systems?

The observed thermalization dynamics in the mixed SYK model provide valuable insights into the relationship between quantum chaos and thermalization in more general quantum many-body systems. Here are some key implications:

Quantum Chaos as a Driver of Thermalization: The mixed SYK model, being maximally chaotic, exhibits thermalization, supporting the idea that quantum chaos can be a driving force behind thermalization in closed quantum systems. The random couplings and fast scrambling of information characteristic of chaotic systems promote the exploration of the Hilbert space and the approach to thermal equilibrium.
Role of Non-integrability: The non-integrability of the mixed SYK model, induced by the random couplings, is crucial for its thermalization. This highlights the importance of non-integrability as a necessary condition for thermalization in more general quantum systems. Integrable systems, with their extensive set of conserved quantities, often fail to thermalize.
Testing Ground for ETH: The SYK model serves as a valuable testing ground for the Eigenstate Thermalization Hypothesis (ETH). The observed thermalization in the SYK model, along with its analytical tractability, allows for detailed studies of ETH and its implications for the relationship between individual eigenstates and thermal ensembles.
Insights into Holography: The SYK model's connection to holography suggests that the observed thermalization dynamics might have implications for understanding black hole physics. The fast scrambling and thermalization in the SYK model could provide insights into the process of black hole formation and evaporation.
However, it's important to note that the SYK model, with its specific features like all-to-all couplings and large-$N$ limit, might not be representative of all quantum many-body systems. Nevertheless, its analytical tractability and the insights it provides into the interplay of chaos, disorder, and thermalization make it a valuable tool for advancing our understanding of these fundamental concepts in quantum many-body physics.