Spectral form factor in the double-scaled SYK model

TL;DR

This paper analytically studies the spectral form factor in the large q limit of the SYK model at infinite temperature, identifying the onset of ramp behavior and the transition from slope to ramp regions.

Contribution

It provides explicit saddle point solutions for the spectral form factor in the SYK model, revealing the nature of the slope and ramp regions and the Thouless time scale.

Findings

Ramp onset occurs at Thouless time ~ q log q

Transition from slope to ramp involves breakdown of 1/q expansion

One-loop corrections refine late-time behavior analysis

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.