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arXiv:2211.11385 · Journal of High Energy Physics

Non-maximal chaos in some Sachdev-Ye-Kitaev-like models

Ma Chen, Chushun Tian

Published May 2, 2023• 1 citation
arXivHTMLPDF

Abstract

Abstract We study the chaos exponent of some variants of the Sachdev-Ye-Kitaev (SYK) model, namely, the $$ \mathcal{N} $$ N = 1 supersymmetry (SUSY)-SYK model and its sibling, the (N|M)-SYK model which is not supersymmetric, for arbitrary interaction strength. We find that for large q the chaos exponent of these variants, as well as the SYK and the $$ \mathcal{N} $$ N = 2 SUSY-SYK model, all follow a single-parameter scaling law. By quantitative arguments we further make a conjecture, i.e. that the found scaling law might hold for general one-dimensional (1D) SYK-like models with large q. This points out a universal route from maximal chaos towards completely regular or integrable motion in the SYK model and its 1D variants.

DOI
10.1007/jhep05(2023)009

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cond-mat.str-elCross-listedarXiv:2606.18529•Jun 18, 2026•0 citations
Abstract We study the chaos exponent of some variants of the Sachdev-Ye-Kitaev (SYK) model, namely, the N\mathcal{N}N N = 1 supersymmetry (SUSY)-SYK model and its sibling, the (N|M)-SYK model which is not supersymmetric, for arbitrary interaction strength. We find that for large q the chaos exponent of these variants, as well as the SYK and the N\mathcal{N}N N = 2 SUSY-SYK model, all follow a single-parameter scaling law. By quantitative arguments we further make a conjecture, i.e. that the found scaling law might hold for general one-dimensional (1D) SYK-like models with large q. This points out a universal route from maximal chaos towards completely regular or integrable motion in the SYK model and its 1D variants.
Abstract We investigate a model closely related to both the original Sachdev-Ye-Kitaev (SYK) model and the N \mathcal{N} N N = 1 supersymmetric SYK model. It consists of N real Majorana fermions and M auxiliary bosons with Yukawa interactions. We consider the large N and M limit and keep the ratio M/N fixed. The model has two branches characterized by the conformal dimensions of fields, which we compute as a function of the ratio M/N. One of the branches contains the supersymmetric saddle for M = N. As we take the limit M/N → ∞ both branches coincide and we obtain the same conformal dimensions as SYK. Furthermore, we determine the Lyapunov exponent of the model and find maximal chaos independent of M/N.
Abstract We study the generalization of the Sachdev-Ye-Kitaev (SYK) model to a 1 + 1 dimensional chiral SYK model of N flavors of right-moving chiral Majorana fermions with all-to-all random 4-fermion interactions. The interactions in this model are exactly marginal, leading to an exact scaling symmetry. We show the Schwinger-Dyson equation of this model in the large N limit is exactly solvable. In addition, we show this model is integrable for small N ≤ 6 by bosonization. Surprisingly, the two point function in the large N limit has exactly the same form as that for N = 4, although the four point functions of the two cases are quite different. The ground state entropy in the large N limit is the same as that of N free chiral Majorana fermions, leading to a zero ground state entropy density. The OTOC of the model in the large N limit exhibits a non-trivial spacetime structure reminscent of that found by Gu and Kitaev [1] for generic SYK-like models. Specifically we find a Lyapunov regime inside an asymmetric butterfly cone, which are signatures of quantum chaos, and that the maximal velocity dependent Lyapunov exponent approaches the chaos bound 2π/β as the interaction strength approaches its physical upper bound. Finally, the model is integrable for (at least) N ≤ 6 but chaotic in the large N limit, leading us to conjecture that there is a transition from integrability to chaos as N increases past a critical value.
We investigate a model closely related to both the original Sachdev-Ye-Kitaev (SYK) model and the mathcalN=1\\mathcal{N}=1mathcalN=1 supersymmetric SYK model. It consists of NNN real Majorana fermions and MMM auxiliary bosons with Yukawa interactions. We consider the large NNN and MMM limit and keep the ratio M/NM/NM/N fixed. The model has two branches characterized by the conformal dimensions of fields, which we compute as a function of the ratio M/NM/NM/N. One of the branches contains the supersymmetric saddle for M=NM=NM=N. Furthermore, we determine the Lyapunov exponent of the model and find maximal chaos independent of M/NM/NM/N.
The SYK model of Sachdev, Ye and Kitaev is a recent model of N Majorana fermions coupled via quartic random interaction. It has the extraordinary properties of being exactly solvable at large N , having emergent conformal symmetry in the IR, and exhibiting maximal quantum chaos. The confluence of these properties make SYK a highly attractive candidate for a solvable AdS/CFT correspondence. What’s more, it was found to capture elements of non-Fermi-liquid behavior in strange metals. This thesis introduces the SYK model and discusses a recent variant, the SYK∗ model of Marcus and Vandoren. In the latter, the effective action is derived and branches of conformal solutions are discussed. The parameter space of SYK∗ is investigated for large and small M . The former reveals a phase separation and a N critical point in the T−M plane is claimed to exist.
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.