Many-body Chaos in The Sachdev-Ye-Kitaev Model

-By Nikita Biliye

The Sachdev-Ye-Kitaev (SYK) Model is an exactly solvable model initially proposed by Subir Sachdev and Jinwu Ye. Later this model was modified by Alexia Kitaev. The SKY model brings insights into the understanding of strongly correlated materials. The Sachdev-Ye-Kitaev (SYK) Model was the first microscopic Quantum system to exhibit many-body chaos.

Over the past few years, many physicists all over the world have conducted researches to investigate the chaos in Quantum systems. These were composed of strongly interacting particles, also referred to as many-body chaos. 

Researchers at the University of California, Berkeley, recently carried out a study examining many-body chaos in the context of the Sachdev-Ye-Kitaev (SKY) model. Many-body chaos came out as a compelling framework to understand Quantum thermalization (the process where Quantum particles reach thermal equilibrium by interacting with each other) in these Quantum systems.

The research disclosed surprising insights into the connection between Quantum chaos and the delocalization or scrambling of quantum information. Theoretical calculations provided support to these ideas. But the verification of validity and observation of Quantum chaos in numerical simulations were proved to be an enduring challenge to the researchers. Kobrin and his research team investigated the chaotic nature of the SKY model by simulating the dynamics of large systems using cutting-edge numerical techniques that they developed. A method based on calculations from Quantum Gravity was used to analyze the collected data.

For a semi-classical chaotic system, OTOCs (out-of-time-order correlators) are expected to exhibit a period of exponential growth, similar to the classical butterfly effect. OTOCs measure the sensitivity of one operator towards a small perturbation influenced by another operator at the earliest. At the intersection of these two perspectives, there arose the discovery of a new form of quantum chaos in strongly interacting systems known as many-body chaos.

The leading order behavior of OTOC is given by eλt/N, where λ Is the Lyapunov exponent and N is the number of degrees of freedom per site. Exceptionally at low temperature, the Lyapunov exponent of the SYK model saturates universal bound λ ≤ 2πt, where T is the temperature of the system.

While studying several other models that show many-body chaos, it was seen that their chaos rate is parametrically slower compared to thermodynamics bound. However, the lack of a reliable numerical toolset created major hurdles in standardizing these experiments and identifying novel models that exhibit many-body chaos.

Researchers gathered numerical evidence of quantum chaos in the Sachdev-Ye-Kitaev (SKY) Model.

To overcome challenges occurring due to lack of reliable numerical toolset, massively parallelized Krylov subspace methods are employed, and new extrapolation tools are developed to characterize many-body chaos. Their findings highlight the valuable interplay between Quantum simulations and theories of Quantum Gravity.

The research team gathered direct numerical evidence of a new dynamic phenomenon referred to as many-body chaos. These pieces of evidence translate chaos from classical mechanics to strongly-interactive Quantum systems.

Presentation of two main results by the researchers:

1. Numerical results for two-point function G(t) = [W(t) W(0)] is demonstrated, showing they agree with analytic predictions of two distinct regimes.

  • At the high-temperature, the mean-field solution of the microscopic model matches with their results.
  • At low temperatures, the results are consistent with full Quantum dynamics of near extremal black holes.

2. Introduction of extrapolation procedure for determining the Lyapunov exponent that explicitly takes into account higher-order terms in the OTOCs.

This new procedure of analyzing results determined the rate of chaos. Also, it precisely showed that at low temperature, it approached the theoretical upper bound. The direct numerical evidence, namely many-body chaos, translates chaos from classical mechanics to the strongly interacting Quantum systems.

A phase diagram showing the behavior of the Sachdev-Ye-Kitaev model for inverse temperatures and the size of the system:

Cr: Kobrin El at
  • In the semiclassical limit (indicated by red, purple), the model is well-described by Schwinger- Dyson equations
  • At low temperatures, the finite size corrections are calculated using Schwarzian action (indicated by blue)
  • At sufficiently small size discreteness of energy-spectrum governs the Dynamics (indicated by Grey)
  • Beyond the semiclassical limit, low-temperature dynamics are captured by Schwarzian theory

Future scope of many-body chaos in Sachdev-Ye-Kitaev Model.

  • Numerical tools created to examine many-body chaos in the SKY model can be used to apply for other models in the future.
  • The recent development in the SKY model can be applied to the models that are difficult to examine using common analysis.
  • These findings could aid the ongoing search for the quantum systems that exhibit behavior similar to that of the black holes.
  • The methods used by this research team could inspire the development and advancements of experimental techniques to simulate Quantum dynamics and controllable Quantum hardware. 



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