Exact four point function for large $q$ SYK from Regge theory

TL;DR

This work solves for the exact leading large-$N$ four-point function of the fermions in the large-$q$ SYK model at arbitrary temperature by a Regge-theory based Sommerfeld-Watson resummation. The analysis shows the four-point function is governed by three Regge poles at $m=0,\pm v$, yielding TOC and OTOC expressions; the OTOC growth rate is $\lambda_L=2\pi T\,v$, interpolating between non-chaotic and maximally chaotic regimes as temperature varies. The TOC result is independent of the center-of-mass time variable and the OTOC result encodes the full Regge-pole structure, with results in exact agreement with Streicher. This Regge-theory perspective provides a transparent, analytic handle on scrambling in a solvable quantum chaotic system and connects weakly coupled and maximally chaotic dynamics in the large-$q$ SYK model.

Abstract

Motivated by the goal of understanding quantum systems away from maximal chaos, in this note we derive a simple closed form expression for the fermion four point function of the large $q$ SYK model valid at arbitrary temperatures and to leading order in $1/N$. The result captures both the large temperature, weakly coupled regime, and the low temperature, nearly conformal, maximally chaotic regime of the model. The derivation proceeds by the Sommerfeld-Watson resummation of an infinite series that recasts the four point function as a sum of three Regge poles. The location of these poles determines the Lyapunov exponent that interpolates between zero and the maximal value as the temperature is decreased. Our results are in complete agreement with the ones by Streicher arxiv:1911.10171 obtained using a different method.

Figures (1)

• Figure 1: Illustration of the contour manipulation for the Sommerfeld-Watson transformation. We first construct a meromorphic function that decays exponentially at infinity and that has poles at the values of $m$ that we want to sum over. The contribution of these poles are picked up by the green contour, which we deform into the purple dashed contour, and finally into the red contour picking up the contribution of the Regge poles. On these contour plots we plot $\abs{f_+(m)S_+(m)}^{1/4}$ for $v=3/5$, on the left for the OTOC configuration $x=x'=\pi/2,\, y=\pi/4$, while on the right for the TOC configuration $x=x'=\pi/4,\, y=\pi/2$. These figures demonstrate that for the TOC configuration the $m=\pm v$ Regge poles disappear, that the arc contributions at infinity can be dropped and that there are no additional poles or cuts on the complex $m$ plane that our analysis would have missed.