[Paper Review] Quantum Chaos, Thermodynamics and Black Hole Microstates in the mass deformed SYK model

Tomoki Nosaka, Tokiro Numasawa|arXiv (Cornell University)|Dec 27, 2019

Black Holes and Theoretical Physics75 references4 citations

TL;DR

This paper investigates the mass-deformed Sachdev-Ye-Kitaev (SYK) model to study black hole microstates, quantum chaos, and thermodynamics. By using SYK boundary states as variational approximations to the ground state, it demonstrates strong agreement with exact results and provides a gravity dual with an IR end-of-the-world brane. The model exhibits no Hawking-Page-like phase transition despite structural similarities to traversable wormhole models.

ABSTRACT

We study various aspects of the mass deformed SYK model which can escape the interiors of pure boundary state black holes. SYK boundary states are given by a simple local boundary condition on the Majorana fermions and then evolved in Euclidean time in the SYK Hamiltonian. We study the ground state of this mass deformed SYK model in detail. We also use SYK boundary states as a variational approximation to the ground state of the mass deformed SYK model. We compare variational approximation with the exact ground state results and they showed a good agreement. We also study the time evolution of the mass deformed ground state under the SYK Hamiltonian. We give a gravity interpretation of the mass deformed ground state and its time evolutions. In gravity side, mass deformation gives a way to prepare black hole microstates that are similar to pure boundary state black holes. Escaping protocol on these ground states simply gives a global AdS2 with an IR end of the world brane. We also study the thermodynamics and quantum chaotic properties of this mass deformed SYK model. Interestingly, we do not observe the Hawking Page like phase transition in this model in spite of similarity of the Hamiltonian with eternal traversable wormhole model where we have the phase transition.

Motivation & Objective

• To investigate the ground state of the mass-deformed SYK model as a candidate for black hole microstates.
• To test the validity of SYK boundary states as a variational approximation to the exact ground state.
• To analyze the thermodynamic and quantum chaotic properties of the mass-deformed SYK model.
• To provide a gravity interpretation of the mass-deformed ground state and its time evolution.
• To explore the absence of Hawking-Page-like phase transitions despite Hamiltonian similarities to eternal traversable wormhole models.

Proposed method

• Constructs the mass-deformed SYK model via a local boundary condition on Majorana fermions.
• Uses exact diagonalization and large-N Schwinger-Dyson equations to compute ground state properties.
• Applies variational approximation using SYK boundary states to compare with exact ground state results.
• Performs finite-N and large-N analysis, including numerical solution of Schwinger-Dyson equations.
• Computes the chaos exponent via the Lyapunov exponent in the large-q limit using a delta-function potential approximation.
• Derives gravity dual interpretation via end-of-the-world brane in global AdS2, with time evolution mapped to bulk dynamics.

Experimental results

Research questions

• RQ1 How do SYK boundary states approximate the exact ground state in the mass-deformed SYK model?
• RQ2 What is the thermodynamic behavior of the mass-deformed SYK model, and does it exhibit a Hawking-Page-like phase transition?
• RQ3 How does the quantum chaotic behavior—measured by the Lyapunov exponent—change under mass deformation?
• RQ4 What is the gravity dual of the mass-deformed ground state and its time evolution?
• RQ5 How do UV effects modify the conformal limit predictions at q=4, particularly in the small-μ regime?

Key findings

• The variational approximation using SYK boundary states shows excellent agreement with the exact ground state, validating its use for studying gapped ground states.
• The model does not exhibit a Hawking-Page-like phase transition, despite structural similarities to the eternal traversable wormhole model.
• In the large-q limit, the Lyapunov exponent is derived as $\lambda\beta = 2\sqrt{\pi}k^{3/2}e^{-k/4}$, showing a non-monotonic dependence on $k$.
• At small $k$, the chaos exponent is corrected as $\frac{\lambda_L\beta}{2\pi} = 1 - \frac{k}{2\pi^2} + \cdots$, with maximal chaos at $k=0$.
• At $q=4$, UV effects cause logarithmic corrections: $E_0(\mu) - E_0 \sim c_{E_0}(\mu/J)^2\log(\mu/J)$ with $c_{E_0} \approx 0.34$, indicating scaling violation.
• Numerical results confirm finite but non-conformal behavior at $q=4$, with fitting coefficients $c_{G_{\text{off}}} \approx -1.2$ and $c_{H_{\text{SYK}}} \approx -0.26$ for off-diagonal and energy observables.