A semiclassical ramp in SYK and in gravity

TL;DR

The paper investigates how a non-decaying, ramp-like late-time behavior in finite-entropy quantum systems arises from semiclassical dynamics. By constructing two-replica saddles in the SYK model (Brownian and regular) and identifying corresponding gravitational geometries (notably a time-identified two-sided black hole or double cone in JT gravity), the authors show a mechanism for linear-in-T ramp contributions with zero action and a symmetry-breaking zero mode. They connect these saddles to random-matrix expectations for the spectral form factor and discuss the plateau's origin as a nonperturbative effect, while highlighting unresolved issues such as factorization, wiggles, and the precise plateau mechanism. The work advances the understanding of how ensemble-averaged bulk descriptions can reproduce universal late-time spectral statistics and informs the black hole information problem by tying ramp phenomena to semiclassical gravity and replica-based field theories.

Abstract

In finite entropy systems, real-time partition functions do not decay to zero at late time. Instead, assuming random matrix universality, suitable averages exhibit a growing "ramp" and "plateau" structure. Deriving this non-decaying behavior in a large $N$ collective field description is a challenge related to one version of the black hole information problem. We describe a candidate semiclassical explanation of the ramp for the SYK model and for black holes. In SYK, this is a two-replica nonperturbative saddle point for the large $N$ collective fields, with zero action and a compact zero mode that leads to a linearly growing ramp. In the black hole context, the solution is a two-sided black hole that is periodically identified under a Killing time translation. We discuss but do not resolve some puzzles that arise.

Figures (8)

• Figure 1: A log-log plot of the spectral form factor in SYK for $q=4, ~N=34, ~\beta J = 5$Cotler:2016fpe. A single sample (red, erratic) is plotted together with an average of 90 samples (black, smoother). The ramp is approximately linear $\propto T$ in standard variables, not just in the log-log variables.
• Figure 2: In the Brownian model, the $O(1)$ late time value arises from a nontrivial saddle point in the collective field description. In regular SYK, we propose a similar explanation for the ramp. Both plots are simply sketches. (In regular SYK at $\beta = 0$ there are regular oscillations in the slope region. We are sketching the envelope.)
• Figure 3: We show a path-integral repesentation of the quantity $\text{Tr}(e^{-\frac{\beta_{\text{aux}}}{2}H}e^{-iHT}e^{-\frac{\beta_{\text{aux}}}{2}H}e^{iHT})$. The open and filled circles should be identified. In SYK one could study this quantity by path $G,\Sigma$ path integral. The leading saddle point is simply the analytic continuation to this contour of the standard thermal saddle point for $Z(\beta_{\text{aux}})$. The idea is that the equations for $G,\Sigma$ in the shaded region are similar to our equations, with $L$ and $R$ referring two the two sides of the contour, (and with a factor of $i^q$ that arises from a different convention for $G_{LR}$).
• Figure 4: Example numerical solutions to (\ref{['saddlePt']}). Left: the disconnected saddle point with $G_{LR} = 0$. Solid blue is $\text{Re}(G_{LL}) = \text{Re}(G_{RR})$, dotted red is $\text{Im}(G_{LL}) = -\text{Im}(G_{RR})$, and dashed yellow is $\text{Im}(G_{LR})$. Middle: a connected saddle point. Right: the same solution but with nonzero $\Delta$. For the connected solutions, increasing $\beta_{\text{aux}}$ would broaden the features and increase the magnitude of the imaginary part of $G_{LL}$. Increasing $T$ would extend the middle part of the plots where the solutions are very small.
• Figure 5: The double cone. At left we indicate two identification surfaces in AdS${}_2$ (red and green, indicated by arrows). The blue curved lines represent the regulated boundaries. At right we have folded the geometry into a double cone. We also made the regulated boundary wiggly to represent the boundary graviton degree of freedom.