[Paper Review] A semiclassical ramp in SYK and in gravity

Phil Saad, Stephen H. Shenker+1 more|arXiv (Cornell University)|Jun 18, 2018

Quantum many-body systemsPhysics and Astronomy73 references220 citations

TL;DR

The paper offers a semiclassical, two-replica saddle-point explanation for the late-time ramp in SYK and discusses a gravity interpretation via periodically identified two-sided black holes, connecting to random-matrix universality.

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.

Motivation & Objective

• Motivate the search for a semiclassical origin of the ramp/plateau in finite-entropy systems.
• Develop a large-N G,Σ (and variants) formalism to identify non-decaying contributions
• Explain Brownian SYK ramp via zero-action, LR-coupled saddles and extend insights to regular SYK
• Connect SYK saddles to gravity through JT/Schwarzian descriptions and Lorentzian wormhole configurations
• Discuss unresolved issues about factorization and the plateau in the collective-field framework

Proposed method

• Study Brownian SYK with local in-time couplings to identify zero-action, LR-coupled saddles
• Formulate a G,Σ path integral for Tr[U(T)] Tr[U(T)]*, derive zero-action saddles G_LR=±i/2, Σ_LR=∓iJ/2^{q-2}
• Generalize to Tr[U(T)^k] Tr[U(T)^k]* and show k cyclic saddles yielding a k-linear ramp
• Extend to regular SYK by introducing matrix-valued G_ij, Σ_ij and solving saddle-point equations in frequency space
• Relate SYK saddles to bulk JT gravity via Schwarzian theory and discuss a Lorentzian double-cone/periodic Killing-time identification
• Address potential issues with fluctuations and factorization, and outline the emergence of a plateau problem

Experimental results

Research questions

• RQ1 How can a semiclassical large-N description reproduce the ramp and plateau observed in spectral form factors?
• RQ2 What are the saddle points of the G,Σ effective action that correlate the two replicas and yield non-decaying late-time behavior?
• RQ3 Can Brownian and regular SYK saddles be unified or related to gravitational configurations in JT gravity?
• RQ4 What is the interpretation of the ramp/plateau in gravity, and how does periodic time identification of a two-sided black hole contribute?
• RQ5 What puzzles remain about factorization and the plateau within this semiclassical framework?

Key findings

• In Brownian SYK, non-decaying late-time behavior arises from saddles with zero action and a compact LR zero mode, producing an O(1) ramp.
• In regular SYK, a family of zero-action saddles correlates the replicas and explains a linear ramp in ⟨|Z(iT)|^2⟩, though the plateau remains not fully resolved in this framework.
• For unitary-k replicas, saddles corresponding to cyclic permutations yield a k-linear ramp, mirroring random-matrix expectations.
• The gravity interpretation uses a two-sided black hole with periodic Killing-time identification, viewed as a Lorentzian double cone, offering a semiclassical bridge to ramp-like behavior.
• The paper identifies puzzles about fluctuations, factorization, and plateau in the gravitational/collective-field picture and does not fully resolve them.