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arXiv:2606.18529

Tunable Chaos in the Finite Mean SYK Model

Arkaprava Mukherjee, Sumilan Banerjee, Sandip P. Trivedi, Nandini Trivedi

cond-mat.str-elhep-thquant-phSubmitted Jun 16, 2026• Announced Jun 18, 2026• 0 citations
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Abstract

The complex Sachdev-Ye-Kitaev (SYK) model, featuring fermions with all-to-all interactions, serves as a dual paradigm for understanding non-Fermi liquid behavior and the holographic nature of charged black holes. Two defining characteristics of the standard SYK model are its maximal chaos (Lyapunov exponent $λ_{\mathrm{L}}=2πT$ at temperature $T$), and its finite zero-temperature residual entropy. While previous studies have largely focused on couplings drawn from a zero-mean Gaussian distribution, we investigate a generalized model with a finite mean-to-standard-deviation ratio, $g\equiv J_{0}/δJ$ of the coupling distribution in order to get deeper insight into the evolution of chaos. We find that increasing $g$ yields the following effects: (i) The system remains a fast scrambler with $λ_{\mathrm{L}}=A~T$, but with a suppressed coefficient $A<2π$. (ii) In the limit $g\to \infty$, out-of-time-ordered correlators (OTOCs) no longer exhibit exponential growth with $λ_{\mathrm{L}}\simeq 0$. (iii) The spectral correlations indicative of late-time chaos maintain Wigner-Dyson level spacing statistics for all values of $g$. (iv) The system preserves a finite residual entropy, albeit with reduced magnitude, for all $g$ values. We conclude that in this generalized SYK model, there is a chaotic to non-chaotic crossover. Moreover different measures of chaos decouple, demonstrating that the presence of finite residual entropy does not strictly imply maximal chaos.

Comments
23 pages, 12 figures

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The complex Sachdev-Ye-Kitaev (SYK) model, featuring fermions with all-to-all interactions, serves as a dual paradigm for understanding non-Fermi liquid behavior and the holographic nature of charged black holes. Two defining characteristics of the standard SYK model are its maximal chaos (Lyapunov exponent λL=2πTλ_{\mathrm{L}}=2πTλL​=2πT at temperature TTT), and its finite zero-temperature residual entropy. While previous studies have largely focused on couplings drawn from a zero-mean Gaussian distribution, we investigate a generalized model with a finite mean-to-standard-deviation ratio, g≡J0/δJg\equiv J_{0}/δJg≡J0​/δJ of the coupling distribution in order to get deeper insight into the evolution of chaos. We find that increasing ggg yields the following effects: (i) The system remains a fast scrambler with λL=A Tλ_{\mathrm{L}}=A~TλL​=A T, but with a suppressed coefficient A<2πA<2πA<2π. (ii) In the limit g→∞g\to \inftyg→∞, out-of-time-ordered correlators (OTOCs) no longer exhibit exponential growth with λL≃0λ_{\mathrm{L}}\simeq 0λL​≃0. (iii) The spectral correlations indicative of late-time chaos maintain Wigner-Dyson level spacing statistics for all values of ggg. (iv) The system preserves a finite residual entropy, albeit with reduced magnitude, for all ggg values. We conclude that in this generalized SYK model, there is a chaotic to non-chaotic crossover. Moreover different measures of chaos decouple, demonstrating that the presence of finite residual entropy does not strictly imply maximal chaos.
Abstract It has been known that the large-q complex Sachdev-Ye-Kitaev (SYK) model falls under the same universality class as that of van der Waals (mean-field) and saturates the Maldacena-Shenker-Stanford (MSS) bound, both features shared by various black holes. This makes the SYK model a useful tool in probing the fundamental nature of quantum chaos and holographic duality. This work establishes the robustness of this shared universality class and chaotic properties for SYK-like models by extending to a system of coupled large-q complex SYK models of different orders. We provide a detailed derivation of thermodynamic properties, specifically the critical exponents for an observed phase transition, as well as dynamical properties, in particular the Lyapunov exponent, via the out-of-time correlator calculations. Our analysis reveals that, despite the introduction of an additional scaling parameter through interaction strength ratios, the system undergoes a continuous phase transition at low temperatures, similar to that of the single SYK model. The critical exponents align with the Landau–Ginzburg (mean-field) universality class, shared with van der Waals gases and various AdS black holes. Furthermore, we demonstrate that the coupled SYK system remains maximally chaotic in the large-q limit at low temperatures, adhering to the MSS bound, a feature consistent with the single SYK model. These findings establish robustness and open avenues for broader inquiries into the universality and chaos in complex quantum systems. We provide a detailed outlook for future work by considering the ‘very’ low-temperature regime, where we discuss relations with the Hawking–Page phase transition observed in the holographic dual black holes. We present preliminary calculations and discuss the possible follow-ups that might be taken to make the connection robust.
We present an exact solution in the large-NNN limit of the Lévy Sachdev-Ye-Kitaev (LSYK) model introduced in Ref. [1], wherein the couplings are drawn from a Lévy Stable distribution parameterized by a tail exponent μ∈[0,2]μ\in [0, 2]μ∈[0,2]. Starting from the Hamiltonian and its associated partition function, we highlight the key differences from the standard Gaussian SYK model and derive the large-NNN Schwinger-Dyson equations via a bosonic oscillator representation of the action. These equations are solved both numerically and analytically in the large-qqq and infrared limits. We subsequently analyze the chaotic properties of the model by computing the Krylov exponent from the large-qqq Green's function and extracting the Lyapunov exponent from the 444-point function. The parameter μμμ continuously interpolates between a free theory at μ=0μ= 0μ=0 and the conventional, maximally chaotic Gaussian SYK model at μ=2μ= 2μ=2, with non-maximal chaos persisting throughout the intermediate regime 0<μ<20 < μ< 20<μ<2. Thermodynamic quantities, including the entropy, free energy, average energy, and specific heat capacity, are computed and compared with their Gaussian SYK counterparts. The interpretations of the thermodynamics are discussed with respect to the holographic dual and non-Fermi liquid theory. Finally, we discuss an alternative representation of the LSYK model based on a distinct decomposition of the Lévy Stable distribution, which establishes a non-trivial connection to Gaussian SYK, and provide supporting analytical and numerical results in the appendices.
We introduce two disorder-free variants of the Sachdev-Ye-Kitaev (SYK) model, demonstrate their integrability, and study their static and dynamical properties. Unlike diagrammatic techniques, the integrability of these models allows us to obtain dynamical correlation functions even when the number of Majorana fermions is finite. From the solutions, we find that out-of-time-order correlators (OTOCs) in these models exhibit exponential growth at early times, resembling that of many-body quantum chaotic systems, such as those with disorder or external kick terms, despite their large N behavior differing from that of typical chaotic systems. Conversely, our analysis shows no evidence of random-matrix behavior in level spacing statistics or the spectral form factor. Our findings illustrate that the clean versions of the SYK models represent simple but nontrivial examples of disorder-free quantum many-body systems displaying chaoslike behavior of OTOCs.
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.
We analyze the relation between quantum chaotic behavior and phase transitions of the Yukawa--Sachdev-Ye-Kitaev model as a function of filling and temperature, which describes random Yukawa interactions between NNN complex fermions and MMM bosons in zero spatial dimensions for both the non-Fermi liquid and insulating states at finite temperature and chemical potential. We solve the ladder equations for the out-of-time-order correlator (OTOC) for both bosons and fermions. Despite the appearance of the chemical potential in the Hamiltonian, which explicitly introduces an additional energy scale, the OTOCs for the fermions and bosons in the non-Fermi liquid state turn out to be unaffected, and the Lyapunov exponents that diagnose chaos remain maximal. As the chemical potential increases, the system is known to experience a first-order transition from a critical phase to a gapped insulating phase. We postulate that the boundary of the region in parameter space where each phase is (meta)stable coincides with the curve on which the Lyapunov exponent is maximal. By calculating the exponent in the insulating phase and comparing to numerical results on the boundaries of stability, we show that this is plausible.
Quantum chaos remains a challenging concept because classical signatures of chaos do not directly transfer to quantum systems. This thesis investigates which indicators of chaos are shared between classical and quantum mechanics and how they can be used to characterise non-integrable many-body systems. We examine ergodicity and thermalisation by means of the Eigenstate Thermalisation Hypothesis (ETH) as signatures of non-integrability, and study the Out-of-Time-Ordered Correlator (OTOC) as a probe of information scrambling and as a measure of sensitivity to initial conditions. We briefly discuss the appearance of chaotic wavefunctions and observations such as quantum scars and quantum ergodicity in individual eigenmodes of the Bunimovich stadium potential. Using the OTOC, we discuss extensions of the Lyapunov exponent to quantum systems and apply these ideas to the Sachdev–Ye–Kitaev (SYK) model. In the large-N limit, and using the AdS/CFT correspondence, we find that the model might potentially be the dual of a black hole and that it is maximally chaotic. We conclude by discussing additional proposed signatures of quantum chaos and prospects for experimental realisations of the SYK model.