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arXiv:2506.01100 · Physical review. D/Physical review. D.

Correlation functions and chaotic behavior of the SYK chain model in pure states

Seyyed M. H. Halataei

Published Jul 16, 2025• 0 citations
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Abstract

Recent investigations of Rényi entanglement entropy in the Sachdev-Ye-Kitaev (SYK) chain of Majorana fermions have indicated that the model exhibits slow thermalization when initialized in certain states. The extent to which the heavy modes—believed to underlie this behavior—affect other aspects of the model remains an open question. In this work, I study thermalization and scrambling of information in individual energy eigenstates of the SYK chain using exact diagonalization. I show that two-point correlation functions in finite-energy eigenstates closely match their thermal counterparts and that information scrambling occurs efficiently within these pure states. These results suggest that the slow thermalization observed in entanglement dynamics does not extend to all probes of thermalization and scrambling, even in pure states.

DOI
10.1103/wwfl-tkwm

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Recent investigations of Rényi entanglement entropy in the Sachdev-Ye-Kitaev (SYK) chain of Majorana fermions have indicated that the model exhibits slow thermalization when initialized in certain states. The extent to which the heavy modes—believed to underlie this behavior—affect other aspects of the model remains an open question. In this work, I study thermalization and scrambling of information in individual energy eigenstates of the SYK chain using exact diagonalization. I show that two-point correlation functions in finite-energy eigenstates closely match their thermal counterparts and that information scrambling occurs efficiently within these pure states. These results suggest that the slow thermalization observed in entanglement dynamics does not extend to all probes of thermalization and scrambling, even in pure states.
A study of Rényi entanglement entropy in the Sachdev-Ye-Kitaev (SYK) chain of Majorana fermions suggested that the model does not rapidly thermalize, despite being maximally chaotic. In this work, I examine the eigenstate thermalization hypothesis (ETH) for both the SYK chain and the two-site SYK models using exact diagonalization. I focus on two physically motivated few-body observables: the one-body particle number operator and the two-body hopping operator. These operators provide insight into local occupation and, in certain contexts, may reflect aspects of transport. Both operators have been employed in previous studies of the ETH in related SYK models. I show that single realizations of both models approximately satisfy ETH conditions for these operators, while ensemble averages strictly satisfy the ETH. Therefore, I conclude that the finite-size SYK chain and two-site SYK models can rapidly thermalize with respect to these observables through the ETH mechanism. This suggests that the subthermal behavior observed in previous studies of Rényi entanglement entropy does not manifest in finite-size systems and that these systems can thermalize rapidly via their light modes. It also indicates that the proposed gravitational dual may undergo rapid thermalization.
We study the spread of Rényi entropy between two halves of a Sachdev-Ye-Kitaev (SYK) chain of Majorana fermions, prepared in a thermofield double (TFD) state. The SYK chain model is a model of chaotic many-body systems, which describes a one-dimensional lattice of Majorana fermions, with spatially local random quartic interaction. We find that for integer Rényi index n > 1, the Rényi entanglement entropy saturates at a parametrically smaller value than expected. This implies that the TFD state of the SYK chain does not rapidly thermalize, despite being maximally chaotic: instead, it rapidly approaches a prethermal state. We compare our results to the signatures of thermalization observed in other quenches in the SYK model, and to intuition from nearly-AdS2 gravity.
Abstract The eigenstate thermalization hypothesis (ETH) explains how closed unitary quantum systems can exhibit thermal behavior in pure states. In this work we examine a recently proposed microscopic model of a black hole in AdS2, the so-called Sachdev-Ye-Kitaev (SYK) model. We show that this model satisfies the eigenstate thermalization hypothesis by solving the system in exact diagonalization. Using these results we also study the behavior, in eigenstates, of various measures of thermalization and scrambling of information. We establish that two-point functions in finite-energy eigenstates approximate closely their thermal counterparts and that information is scrambled in individual eigenstates. We study both the eigenstates of a single random realization of the model, as well as the model obtained after averaging of the random disordered couplings. We use our results to comment on the implications for thermal states of a putative dual theory, i.e. the AdS2 black hole.
The eigenstate thermalization hypothesis is a compelling conjecture which strives to explain the apparent thermal behavior of generic observables in closed quantum systems. Although we are far from a complete analytic understanding, quantum chaos is often seen as a strong indication that the ansatz holds true. In this paper, we address the thermalization of energy eigenstates in the Sachdev-Ye-Kitaev model, a maximally chaotic model of strongly-interacting Majorana fermions. We numerically investigate eigenstate thermalization for specific few-body operators in the original SYK model as well as its N \mathcal{N} N = 1 supersymmetric extension and find evidence that these models satisfy ETH. We discuss the implications of ETH for a gravitational dual and the quantum information-theoretic properties of SYK it suggests.
Information scrambling is a hallmark of quantum chaos and thermalization in isolated quantum many-body systems. We investigate scrambling dynamics in the all-to-all interacting spin Sachdev-Ye-Kitaev (SYK)-qqq model using both pure- and mixed-state entanglement measures. We show that von-Neumann and Rényi entropies exhibit rapid growth followed by saturation near Haar-random values, signaling efficient scrambling. The scrambling rate reveals a nontrivial dependence on the interaction order, system size, and Hamiltonian scaling. We further employ mixed-state entanglement as a powerful probe of information scrambling. We numerically find a universal relation between the Rényi-1/2 mutual information and entanglement negativity for minimal interaction order in the early growth regime. Furthermore, entanglement negativity displays a Page-curve-like behavior under unequal subsystem partitioning, characterized by the birth, spread, and eventual death of quantum correlations. Our results provide a generic description of information scrambling using entanglement dynamics in all-to-all interacting spin systems with multi-body interactions.
Understanding how quantum chaotic systems generate entanglement can provide insight into their microscopic chaotic dynamics and can help distinguish between different classes of chaotic behavior. Using von Neumann entanglement entropy, we study a nonentangled state evolved under three variants of the Sachdev-Ye-Kitaev (SYK) model with a finite number of Majorana fermions N. All the variants exhibit linear entanglement growth at early times, which at late times saturates to a universal value consistent with random matrix theory (RMT), but their growth rates differ. We interpret this as a large-N effect, arising from the enhanced nonlocality of fermionic operators in SYK and binary SYK, absent in spin operators of the spin-SYK model. Numerically, we find that these differences emerge gradually with increasing N. Although all variants are quantum chaotic, their entanglement dynamics reflect varying scrambling rates and indicate that the entanglement production rate serves as a fine-grained probe of scrambling beyond conventional measures. To probe its effect on the thermalization properties of these models, we study the two-point autocorrelation function, finding no differences between the SYK variants, but deviations from RMT predictions for N≥24, particularly near the crossover from exponential decay to saturation regime.