Abstract
In this note we study the spectral form factor in the SYK model in large q limit at infinite temperature. We construct analytic solutions for the saddle point equations that describe the slope and the ramp regions of the spectral form factor time dependence. These saddle points are obtained by taking different approaches to the large q limit: the slope region is described by a replica-diagonal solution and the ramp region is described by a replica-nondiagonal solution. We find that the onset of the ramp behavior happens at the Thouless time of order q log q. We also evaluate the one-loop corrections to the slope and ramp solutions for late times, and study the transition from the slope to the ramp. We show this transition is accompanied by the breakdown of the perturbative 1/q expansion, and that the Thouless time is defined by the consistency of extrapolation of this expansion to late times.
Highlights
The interplay between the ensemble averaging and semiclassical nature of the large N expansion raises questions about its structure, in particular about the replica trick and the difference between quenched and annealed averaging [48,49,50,51]
We construct analytic solutions for the saddle point equations that describe the slope and the ramp regions of the spectral form factor time dependence. These saddle points are obtained by taking different approaches to the large q limit: the slope region is described by a replica-diagonal solution and the ramp region is described by a replica-nondiagonal solution
The main result of the present work is the analytic description of the slope and ramp regimes of the spectral form factor in the double scaled SYK model
Summary
After averaging over the disorder, one can rewrite S(T ) as a path integral over collective bilocal fields G and Σ [13], analogously to the replica partition function for the Euclidean SYK at finite temperature [2,3,4,5, 49]. We sketch this derivation in appendix A. The goal of the present work is to find and study the main solutions of these saddle point equations which determine the dynamics of the spectral form factor, at late times
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