The moments of the spectral form factor in SYK

Year: 2025

Authors: Legramandi A., Talwar N.

Autors Affiliation: Univ Trento, CNR, INO, Pitaevskii BEC Ctr, I-I38123 Trento, Italy; Univ Trento, Dipartimento Fis, I-I38123 Trento, Italy; Trento Inst Fundamental Phys & Applicat, INFN, TIFPA, Trento, Italy; Swansea Univ, Dept Phys, Swansea SA2 8PP, Wales; SISSA, Via Bonomea 265, Trieste 34136, Italy.

Abstract: In chaotic quantum systems the spectral form factor exhibits a universal linear ramp and plateau structure with superimposed erratic oscillations. The mean signal and the statistics of the noise can be probed by the moments of the spectral form factor, also known as higher-point spectral form factors. We identify saddle points in the SYK model that describe the moments during the ramp region. Perturbative corrections around the saddle point indicate that SYK mimics random matrix statistics for the low order moments, while large deviations for the high order moments arise from fluctuations near the edge of the spectrum. The leading correction scales inversely with the number of random parameters in the SYK Hamiltonian and is amplified in a sparsified version of the SYK model, which we study numerically, even in regimes where a linear ramp persists. Finally, we study the q = 2 SYK model, whose spectral form factor exhibits an exponential ramp with increased noise. These findings reveal how deviations from random matrix universality arise in disordered systems and motivate their interpretation from a bulk gravitational perspective.

Journal/Review: JOURNAL OF HIGH ENERGY PHYSICS

Volume: (8)      Pages from: 108-1  to: 108-58

More Information: We would like to thank Timothy J. Hollowood and S. Prem Kumar for many helpful discussions throughout this work, and S. Prem Kumar for originally suggesting to study the moments of the spectral form factor. We also would like to thank Alex Windey for useful discussions. AL acknowledges support from Provincia Autonoma di Trento and by Q@TN, the joint lab between University of Trento. This work was supported by the Swiss State Secretariat for Education, Research and Innovation (SERI) under contract number UeMO19-5.1. NT acknowledges support from STFC grant ST/T506473/1 and ERC-COG grant NP-QFT No. 864583 Non-perturbative dynamics of quantum fields: from new deconfined phases of matter to quantum black holes, by the MUR-FARE2020 grant No. R20E8NR3HX The Emergence of Quantum Gravity from Strong Coupling Dynamics.
KeyWords: AdS-CFT Correspondence; Random Systems; Field Theories in Lower Dimensions; 1/N Expansion
DOI: 10.1007/JHEP08(2025)108