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:2204.07194 · Journal of High Energy Physics

Spectral form factor in sparse SYK models

Elena Cáceres, Anderson Seigo Misobuchi, Amir Raz

Published Aug 23, 2022• 19 citations
arXivHTMLPDF

Abstract

Abstract We investigate the spectral form factor of the sparse Sachdev-Ye-Kitaev model. We use numerical methods to establish that at intermediate times the connected part of the spectral form factor is the dominant one. These connected contributions arise from fluctuations around the disconnected geometry, not from a new saddle point. A similar effect was previously conjectured in SYK but required a value of N out of reach of current numerical simulations.

DOI
10.1007/jhep08(2022)236

Related papers

Spectral fluctuations in the Sachdev-Ye-Kitaev model

Yiyang Jia, J. J. M. Verbaarschot

Abstract We present a detailed quantitative analysis of spectral correlations in the Sachdev-Ye-Kitaev (SYK) model. We find that the deviations from universal Random Matrix Theory (RMT) behavior are due to a small number of long-wavelength fluctuations (of the order of the number of Majorana fermions N) from one realization of the ensemble to the next one. These modes can be parameterized effectively in terms of Q-Hermite orthogonal polynomials, the main contribution being due to scale fluctuations for which we give a simple analytical estimate. Our numerical results for N = 32 show that only the lowest eight polynomials are needed to eliminate the nonuniversal part of the spectral fluctuations. The covariance matrix of the coefficients of this expansion can be obtained analytically from low-order double-trace moments. We evaluate the covariance matrix of the first six moments and find that it agrees with the numerics. We also analyze the spectral correlations in terms of a nonlinear σ-model, which is derived through a Fierz transformation, and evaluate the one and two-point spectral correlation functions to two-loop order. The wide correlator is given by the sum of the universal RMT result and corrections whose lowest-order term corresponds to scale fluctuations. However, the loop expansion of the σ-model results in an ill-behaved expansion of the resolvent, and it gives universal RMT fluctuations not only for q = 4 or higher even q-body interactions, but also for the q = 2 SYK model albeit with a much smaller Thouless energy while the correct result in this case should have been Poisson statistics. In our numerical studies we analyze the number variance and spectral form factor for N = 32 and q = 4. We show that the quadratic deviation of the number variance for large energies appears as a peak for small times in the spectral form factor. After eliminating the long-wavelength fluctuations, we find quantitative agreement with RMT up to an exponentially large number of level spacings for the number variance or exponentially short times in the case of the spectral form factor.

arXiv:1912.11923•Journal of High Energy Physics•Jul 1, 2020•35 citations

Spectral form factor of quadratic R-para-particle SYK model with random matrix coupling

Tingfei Li

A bstract This paper investigates the spectral form factor (SFF) of the quadratic R -para-particle Sachdev-Ye-Kitaev ( R -PSYK 2 ) model with various random matrix ensemble couplings. We generalize previous work on Gaussian Unitary Ensemble (GUE) couplings to all three Gaussian ensembles (GUE, GOE, GSE) and three circular ensembles (CUE, COE, CSE). Through analytical and numerical methods, we establish precise correspondences between GUE and CUE results, demonstrating their SFFs satisfy $$ {\mathcal{K}}_{\textrm{GUE}}(2t) $$ K GUE 2 t ≈ $$ {\mathcal{K}}_{\textrm{CUE}}(t) $$ K CUE t in the time regime 1 ≪ t ≪ N . For the symplectic ensembles, we observe similar behavior with appropriate time rescaling, while we only provide the calculation method for the orthogonal ensembles.

arXiv:2504.19159•Journal of High Energy Physics•Oct 28, 2025•2 citations

Quantum chaos in the sparse SYK model

Patrick Orman, Hrant Gharibyan, John P. Preskill

A bstract The Sachdev-Ye-Kitaev (SYK) model is a system of N Majorana fermions with random interactions and strongly chaotic dynamics, which at low energy admits a holographically dual description as two-dimensional Jackiw-Teitelboim gravity. Hence the SYK model provides a toy model of quantum gravity that might be feasible to simulate with near-term quantum hardware. Motivated by the goal of reducing the resources needed for such a simulation, we study a sparsified version of the SYK model, in which interaction terms are deleted with probability 1 −p . Specifically, we compute numerically the spectral form factor (SFF, the Fourier transform of the Hamiltonian’s eigenvalue pair correlation function) and the nearest-neighbor eigenvalue gap ratio r (characterizing the distribution of gaps between consecutive eigenvalues). We find that when p is greater than a transition value p 1 , which scales as 1/ N 3 , both the SFF and r match the values attained by the full unsparsified model and with expectations from random matrix theory (RMT). But for p < p 1 , deviations from unsparsified SYK and RMT occur, indicating a breakdown of holography in the highly sparsified regime. Below an even smaller value p 2 , which also scales as 1/ N 3 , even the spacing of consecutive eigenvalues differs from RMT values, signaling a complete breakdown of spectral rigidity. Our results cast doubt on the holographic interpretation of very highly sparsified SYK models obtained via machine learning using teleportation infidelity as a loss function.

arXiv:2403.13884•Journal of High Energy Physics•Feb 26, 2025•21 citations

Many-body spectral transitions through the lens of a variable-range quadratic Sachdev-Ye-Kitaev model

Andrea Legramandi, Soumik Bandyopadhyay, Philipp Hauke

The Sachdev-Ye-Kitaev (SYK) model is a cornerstone in the study of quantum chaos and holographic quantum matter. Real-world implementations, however, deviate from the idealized all-to-all connectivity, raising questions about the robustness of its chaotic properties. In this work, we investigate a quadratic SYK model with distance-dependent interactions governed by a power-law decay. By analytically and numerically studying the spectral form factor (SFF), we uncover how transitions present in the single-particle limit carry over to the many-body system. Nontrivial cancellations in the one-loop contributions lead to a robustness of the SFF under a considerable reduction of the interaction range. Further suppression leads to a breakdown of perturbation theory around the infinite-range path-integral saddle and the appearance of spectral regimes, marked by a higher dip and the emergence of a secondary plateau. Our results highlight the interplay between single-particle criticality and many-body dynamics, offering insights into the quantum chaos-to-localization transition and its reflection in spectral statistics.

arXiv:2412.14280•Physical review. B./Physical review. B•Mar 30, 2026•1 citation

A derivation of the soft modes effective action for the SYK spectral form factor

Matteo A. Cardella

The SYK model spectral form factor exhibits absence of information loss in the form of a ramp and a plateau that are typical of random matrix theory. In a large N collective fields description, the ramp was reproduced by Saad, Shenker and Stanford by a connected component of the analytically continued thermal partition function two point correlator. The existence of an \emph{off shell} connected component for the real time partition function is a consequence of disorder averaging over a statistical ensemble of two replica systems. In particular, the ramp arises from a family of vanishing action replica-non-diagonal conformal saddles. We provide a derivation of the soft modes effective action for fluctuations around the ramp conformal saddles, by adapting to the two replica system a method used by Kitaev and Suh in regular SYK.

•Jul 22, 2019•0 citations

Exponential Ramp in the Quadratic Sachdev-Ye-Kitaev Model

Michael Winer, Shao-Kai Jian, Brian G. Swingle

A long period of linear growth in the spectral form factor provides a universal diagnostic of quantum chaos at intermediate times. By contrast, the behavior of the spectral form factor in disordered integrable many-body models is not well understood. Here we study the two-body Sachdev-Ye-Kitaev model and show that the spectral form factor features an exponential ramp, in sharp contrast to the linear ramp in chaotic models. We find a novel mechanism for this exponential ramp in terms of a high-dimensional manifold of saddle points in the path integral formulation of the spectral form factor. This manifold arises because the theory enjoys a large symmetry group. With finite nonintegrable interaction strength, these delicate symmetries reduce to a relative time translation, causing the exponential ramp to give way to a linear ramp.

arXiv:2006.15152•Physical Review Letters•Dec 18, 2020•71 citations
Abstract We investigate the spectral form factor of the sparse Sachdev-Ye-Kitaev model. We use numerical methods to establish that at intermediate times the connected part of the spectral form factor is the dominant one. These connected contributions arise from fluctuations around the disconnected geometry, not from a new saddle point. A similar effect was previously conjectured in SYK but required a value of N out of reach of current numerical simulations.
Abstract We present a detailed quantitative analysis of spectral correlations in the Sachdev-Ye-Kitaev (SYK) model. We find that the deviations from universal Random Matrix Theory (RMT) behavior are due to a small number of long-wavelength fluctuations (of the order of the number of Majorana fermions N) from one realization of the ensemble to the next one. These modes can be parameterized effectively in terms of Q-Hermite orthogonal polynomials, the main contribution being due to scale fluctuations for which we give a simple analytical estimate. Our numerical results for N = 32 show that only the lowest eight polynomials are needed to eliminate the nonuniversal part of the spectral fluctuations. The covariance matrix of the coefficients of this expansion can be obtained analytically from low-order double-trace moments. We evaluate the covariance matrix of the first six moments and find that it agrees with the numerics. We also analyze the spectral correlations in terms of a nonlinear σ-model, which is derived through a Fierz transformation, and evaluate the one and two-point spectral correlation functions to two-loop order. The wide correlator is given by the sum of the universal RMT result and corrections whose lowest-order term corresponds to scale fluctuations. However, the loop expansion of the σ-model results in an ill-behaved expansion of the resolvent, and it gives universal RMT fluctuations not only for q = 4 or higher even q-body interactions, but also for the q = 2 SYK model albeit with a much smaller Thouless energy while the correct result in this case should have been Poisson statistics. In our numerical studies we analyze the number variance and spectral form factor for N = 32 and q = 4. We show that the quadratic deviation of the number variance for large energies appears as a peak for small times in the spectral form factor. After eliminating the long-wavelength fluctuations, we find quantitative agreement with RMT up to an exponentially large number of level spacings for the number variance or exponentially short times in the case of the spectral form factor.
A bstract This paper investigates the spectral form factor (SFF) of the quadratic R -para-particle Sachdev-Ye-Kitaev ( R -PSYK 2 ) model with various random matrix ensemble couplings. We generalize previous work on Gaussian Unitary Ensemble (GUE) couplings to all three Gaussian ensembles (GUE, GOE, GSE) and three circular ensembles (CUE, COE, CSE). Through analytical and numerical methods, we establish precise correspondences between GUE and CUE results, demonstrating their SFFs satisfy KGUE(2t) {\mathcal{K}}_{\textrm{GUE}}(2t) KGUE​(2t) K GUE 2 t ≈ KCUE(t) {\mathcal{K}}_{\textrm{CUE}}(t) KCUE​(t) K CUE t in the time regime 1 ≪ t ≪ N . For the symplectic ensembles, we observe similar behavior with appropriate time rescaling, while we only provide the calculation method for the orthogonal ensembles.
A bstract The Sachdev-Ye-Kitaev (SYK) model is a system of N Majorana fermions with random interactions and strongly chaotic dynamics, which at low energy admits a holographically dual description as two-dimensional Jackiw-Teitelboim gravity. Hence the SYK model provides a toy model of quantum gravity that might be feasible to simulate with near-term quantum hardware. Motivated by the goal of reducing the resources needed for such a simulation, we study a sparsified version of the SYK model, in which interaction terms are deleted with probability 1 −p . Specifically, we compute numerically the spectral form factor (SFF, the Fourier transform of the Hamiltonian’s eigenvalue pair correlation function) and the nearest-neighbor eigenvalue gap ratio r (characterizing the distribution of gaps between consecutive eigenvalues). We find that when p is greater than a transition value p 1 , which scales as 1/ N 3 , both the SFF and r match the values attained by the full unsparsified model and with expectations from random matrix theory (RMT). But for p < p 1 , deviations from unsparsified SYK and RMT occur, indicating a breakdown of holography in the highly sparsified regime. Below an even smaller value p 2 , which also scales as 1/ N 3 , even the spacing of consecutive eigenvalues differs from RMT values, signaling a complete breakdown of spectral rigidity. Our results cast doubt on the holographic interpretation of very highly sparsified SYK models obtained via machine learning using teleportation infidelity as a loss function.
The Sachdev-Ye-Kitaev (SYK) model is a cornerstone in the study of quantum chaos and holographic quantum matter. Real-world implementations, however, deviate from the idealized all-to-all connectivity, raising questions about the robustness of its chaotic properties. In this work, we investigate a quadratic SYK model with distance-dependent interactions governed by a power-law decay. By analytically and numerically studying the spectral form factor (SFF), we uncover how transitions present in the single-particle limit carry over to the many-body system. Nontrivial cancellations in the one-loop contributions lead to a robustness of the SFF under a considerable reduction of the interaction range. Further suppression leads to a breakdown of perturbation theory around the infinite-range path-integral saddle and the appearance of spectral regimes, marked by a higher dip and the emergence of a secondary plateau. Our results highlight the interplay between single-particle criticality and many-body dynamics, offering insights into the quantum chaos-to-localization transition and its reflection in spectral statistics.
The SYK model spectral form factor exhibits absence of information loss in the form of a ramp and a plateau that are typical of random matrix theory. In a large N collective fields description, the ramp was reproduced by Saad, Shenker and Stanford by a connected component of the analytically continued thermal partition function two point correlator. The existence of an \emph{off shell} connected component for the real time partition function is a consequence of disorder averaging over a statistical ensemble of two replica systems. In particular, the ramp arises from a family of vanishing action replica-non-diagonal conformal saddles. We provide a derivation of the soft modes effective action for fluctuations around the ramp conformal saddles, by adapting to the two replica system a method used by Kitaev and Suh in regular SYK.
A long period of linear growth in the spectral form factor provides a universal diagnostic of quantum chaos at intermediate times. By contrast, the behavior of the spectral form factor in disordered integrable many-body models is not well understood. Here we study the two-body Sachdev-Ye-Kitaev model and show that the spectral form factor features an exponential ramp, in sharp contrast to the linear ramp in chaotic models. We find a novel mechanism for this exponential ramp in terms of a high-dimensional manifold of saddle points in the path integral formulation of the spectral form factor. This manifold arises because the theory enjoys a large symmetry group. With finite nonintegrable interaction strength, these delicate symmetries reduce to a relative time translation, causing the exponential ramp to give way to a linear ramp.