Effective description of sub-maximal chaos: stringy effects for SYK scrambling

TL;DR

This paper develops a formalism for describing sub-maximal chaos in quantum systems, specifically analyzing stringy effects in SYK models and connecting these to holographic string corrections.

Contribution

It introduces a general path integral approach to sub-maximal chaos, identifying relevant modes and effective actions in SYK models at finite coupling, and links these to stringy holographic effects.

Findings

Identified scramblon modes and derived their effective actions in SYK models.

Matched chaos descriptions to stringy corrections in holographic CFTs.

Provided analytical treatment of finite coupling effects in SYK quantum dot and chain.

Abstract

It has been proposed that the exponential decay and subsequent power law saturation of out-of-time-order correlation functions can be universally described by collective 'scramblon' modes. We develop this idea from a path integral perspective in several examples, thereby establishing a general formalism. After reformulating previous work on the Schwarzian theory and identity conformal blocks in two-dimensional CFTs relevant for systems in the infinite coupling limit with maximal quantum Lyapunov exponent, we focus on theories with sub-maximal chaos: we study the large-q limit of the SYK quantum dot and chain, both of which are amenable to analytical treatment at finite coupling. In both cases we identify the relevant scramblon modes, derive their effective action, and find bilocal vertex functions, thus constructing an effective description of chaos. The final results can be matched in detail to stringy corrections to the gravitational eikonal S-matrix in holographic CFTs, including a stringy Regge trajectory, bulk to boundary propagators, and multi-string effects that are unexplored holographically.