Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2004.05313 · Physical Review Letters

Comment on “Chaotic-Integrable Transition in the Sachdev-Ye-Kitaev Model”

Jaewon Kim, Xiangyu Cao

Published Mar 11, 2021• 15 citations
arXivHTMLPDF

Abstract

In arXiv:1707.02197, the authors considered the Sachdev-Ye-Kitaev model with quadratic perturbation (also known as mass-deformed SYK), and claimed that the quantum Lyapunov exponent would vanish in the regime of low temperature and small quartic coupling. We calculate the exponent exactly in that regime and show that it is nonzero.
DOI
10.1103/physrevlett.126.109101

Related papers

Chaotic-Integrable Transition in the Sachdev-Ye-Kitaev Model

Antonio M. Garcı́a-Garcı́a, Bruno Loureiro, Aurelio Romero-Bermúdez +1 more

Quantum chaos is one of the distinctive features of the Sachdev-Ye-Kitaev (SYK) model, N Majorana fermions in 0+1 dimensions with infinite-range two-body interactions, which is attracting a lot of interest as a toy model for holography. Here we show analytically and numerically that a generalized SYK model with an additional one-body infinite-range random interaction, which is a relevant perturbation in the infrared, is still quantum chaotic and retains most of its holographic features for a fixed value of the perturbation and sufficiently high temperature. However, a chaotic-integrable transition, characterized by the vanishing of the Lyapunov exponent and spectral correlations given by Poisson statistics, occurs at a temperature that depends on the strength of the perturbation. We speculate about the gravity dual of this transition.

arXiv:1707.02197•Physical Review Letters•Jun 15, 2018•136 citations

Tunable quantum chaos in the Sachdev-Ye-Kitaev model coupled to a thermal bath

Yiming Chen, Hui Zhai, Pengfei Zhang

The Sachdev-Ye-Kitaev (SYK) model describes Majorana fermions with random interaction, which displays many interesting properties such as non-Fermi liquid behavior, quantum chaos, emergent conformal symmetry and holographic duality. Here we consider a SYK model or a chain of SYK models with N Majorana fermion modes coupled to another SYK model with N 2 Majorana fermion modes, in which the latter has many more degrees of freedom and plays the role as a thermal bath. For a single SYK model coupled to the thermal bath, we show that although the Lyapunov exponent is still proportional to temperature, it monotonically decreases from 2π/β (β = 1/(k B T), T is temperature) to zero as the coupling strength to the thermal bath increases. For a chain of SYK models, when they are uniformly coupled to the thermal bath, we show that the butterfly velocity displays a crossover from a $$ \sqrt{T} $$ -dependence at relatively high temperature to a linear T-dependence at low temperature, with the crossover temperature also controlled by the coupling strength to the thermal bath. If only the end of the SYK chain is coupled to the thermal bath, the model can introduce a spatial dependence of both the Lyapunov exponent and the butterfly velocity. Our models provide canonical examples for the study of thermalization within chaotic models.

arXiv:1705.09818•Journal of High Energy Physics•Jul 1, 2017•81 citations

A sharp transition in quantum chaos and thermodynamics of mass deformed SYK model

Tomoki Nosaka

We review our recent work [arXiv:2009.10759] where we studied the chaotic property of the two coupled Sachdev-Ye-Kitaev systems exhibiting a Hawking-Page like phase transition. By computing the out-of-time-ordered correlator in the large N limit by using the bilocal field formalism, we found that the chaos exponent of this model shows a discontinuous fall-off at the phase transition temperature. Hence in this model the Hawking-Page like transition is correlated with a transition in chaoticity, as expected from the relation between a black hole geometry and the chaotic behavior in the dual field theory.

arXiv:2012.10628•Dec 19, 2020•0 citations

Chaos in a classical limit of the Sachdev-Ye-Kitaev model

Thomas Scaffidi, Ehud Altman

We study chaos in a classical limit of the Sachdev-Ye-Kitaev (SYK) model obtained in a suitably defined large-$S$ limit. The low-temperature Lyapunov exponent is found to depend linearly on temperature, with a slope that is parametrically different than in the quantum case: It is proportional to $N/S$. The classical dynamics can be understood as the rotation of an $N$-dimensional body with a random inertia tensor, corresponding to the random couplings of the SYK Hamiltonian. This allows us to find an extensive number of fixed points, corresponding to the body's principal axes of rotation. The thermodynamics is mapped to the $p$-spin model with $p=2$, which exhibits a spin glass phase at low temperature whose presence does not preclude the existence of chaos.

arXiv:1711.04768•Physical review. B./Physical review. B•Oct 15, 2019•40 citations

Competition between Chaotic and Nonchaotic Phases in a Quadratically Coupled Sachdev-Ye-Kitaev Model

Xin Chen, Ruihua Fan, Yiming Chen +2 more

The Sachdev-Ye-Kitaev (SYK) model is a concrete solvable model to study non-Fermi liquid properties, holographic duality, and maximally chaotic behavior. In this work, we consider a generalization of the SYK model that contains two SYK models with a different number of Majorana modes coupled by quadratic terms. This model is also solvable, and the solution shows a zero-temperature quantum phase transition between two non-Fermi liquid chaotic phases. This phase transition is driven by tuning the ratio of two mode numbers, and a nonchaotic Fermi liquid sits at the critical point with an equal number of modes. At a finite temperature, the Fermi liquid phase expands to a finite regime. More intriguingly, a different non-Fermi liquid phase emerges at a finite temperature. We characterize the phase diagram in terms of the spectral function, the Lyapunov exponent, and the entropy. Our results illustrate a concrete example of the quantum phase transition and critical behavior between two non-Fermi liquid phases.

arXiv:1705.03406•Physical Review Letters•Nov 15, 2017•89 citations

Lyapunov exponents in a Sachdev-Ye-Kitaev-type model with population imbalance in the conformal limit and beyond

A. S. Shankar, Mikael Fremling, Stephan Plugge +1 more

The Sachdev-Ye-Kitaev (SYK) model shows chaotic behavior with a maximal Lyapunov exponent. In this paper, we investigate the four-point function of a SYK-type model numerically, which gives us access to its Lyapunov exponent. The model consists of two sets of Majorana fermions, called A and B, and the interactions are restricted to being exclusively pairwise between the two sets, not within the sets. We find that the Lyapunov exponent is still maximal at strong coupling. Furthermore, we show that even though the conformal dimensions of the A and B fermions change with the population ratio, the Lyapunov exponent remains constant, not just in the conformal limit where it is maximal, but also in the intermediate and weak coupling regimes.

arXiv:2302.08876•Physical review. D/Physical review. D.•Nov 28, 2023•1 citation
Quantum chaos is one of the distinctive features of the Sachdev-Ye-Kitaev (SYK) model, N Majorana fermions in 0+1 dimensions with infinite-range two-body interactions, which is attracting a lot of interest as a toy model for holography. Here we show analytically and numerically that a generalized SYK model with an additional one-body infinite-range random interaction, which is a relevant perturbation in the infrared, is still quantum chaotic and retains most of its holographic features for a fixed value of the perturbation and sufficiently high temperature. However, a chaotic-integrable transition, characterized by the vanishing of the Lyapunov exponent and spectral correlations given by Poisson statistics, occurs at a temperature that depends on the strength of the perturbation. We speculate about the gravity dual of this transition.
The Sachdev-Ye-Kitaev (SYK) model describes Majorana fermions with random interaction, which displays many interesting properties such as non-Fermi liquid behavior, quantum chaos, emergent conformal symmetry and holographic duality. Here we consider a SYK model or a chain of SYK models with N Majorana fermion modes coupled to another SYK model with N 2 Majorana fermion modes, in which the latter has many more degrees of freedom and plays the role as a thermal bath. For a single SYK model coupled to the thermal bath, we show that although the Lyapunov exponent is still proportional to temperature, it monotonically decreases from 2π/β (β = 1/(k B T), T is temperature) to zero as the coupling strength to the thermal bath increases. For a chain of SYK models, when they are uniformly coupled to the thermal bath, we show that the butterfly velocity displays a crossover from a T \sqrt{T} T​ -dependence at relatively high temperature to a linear T-dependence at low temperature, with the crossover temperature also controlled by the coupling strength to the thermal bath. If only the end of the SYK chain is coupled to the thermal bath, the model can introduce a spatial dependence of both the Lyapunov exponent and the butterfly velocity. Our models provide canonical examples for the study of thermalization within chaotic models.
We review our recent work [arXiv:2009.10759] where we studied the chaotic property of the two coupled Sachdev-Ye-Kitaev systems exhibiting a Hawking-Page like phase transition. By computing the out-of-time-ordered correlator in the large N limit by using the bilocal field formalism, we found that the chaos exponent of this model shows a discontinuous fall-off at the phase transition temperature. Hence in this model the Hawking-Page like transition is correlated with a transition in chaoticity, as expected from the relation between a black hole geometry and the chaotic behavior in the dual field theory.
We study chaos in a classical limit of the Sachdev-Ye-Kitaev (SYK) model obtained in a suitably defined large-SSS limit. The low-temperature Lyapunov exponent is found to depend linearly on temperature, with a slope that is parametrically different than in the quantum case: It is proportional to N/SN/SN/S. The classical dynamics can be understood as the rotation of an NNN-dimensional body with a random inertia tensor, corresponding to the random couplings of the SYK Hamiltonian. This allows us to find an extensive number of fixed points, corresponding to the body's principal axes of rotation. The thermodynamics is mapped to the ppp-spin model with p=2p=2p=2, which exhibits a spin glass phase at low temperature whose presence does not preclude the existence of chaos.
The Sachdev-Ye-Kitaev (SYK) model is a concrete solvable model to study non-Fermi liquid properties, holographic duality, and maximally chaotic behavior. In this work, we consider a generalization of the SYK model that contains two SYK models with a different number of Majorana modes coupled by quadratic terms. This model is also solvable, and the solution shows a zero-temperature quantum phase transition between two non-Fermi liquid chaotic phases. This phase transition is driven by tuning the ratio of two mode numbers, and a nonchaotic Fermi liquid sits at the critical point with an equal number of modes. At a finite temperature, the Fermi liquid phase expands to a finite regime. More intriguingly, a different non-Fermi liquid phase emerges at a finite temperature. We characterize the phase diagram in terms of the spectral function, the Lyapunov exponent, and the entropy. Our results illustrate a concrete example of the quantum phase transition and critical behavior between two non-Fermi liquid phases.
The Sachdev-Ye-Kitaev (SYK) model shows chaotic behavior with a maximal Lyapunov exponent. In this paper, we investigate the four-point function of a SYK-type model numerically, which gives us access to its Lyapunov exponent. The model consists of two sets of Majorana fermions, called A and B, and the interactions are restricted to being exclusively pairwise between the two sets, not within the sets. We find that the Lyapunov exponent is still maximal at strong coupling. Furthermore, we show that even though the conformal dimensions of the A and B fermions change with the population ratio, the Lyapunov exponent remains constant, not just in the conformal limit where it is maximal, but also in the intermediate and weak coupling regimes.