Semiclassical geometry in double-scaled SYK

TL;DR

This paper explores the semiclassical geometry of double-scaled SYK at finite energies, showing how bulk curvature and fluctuations depend on coupling strength, and analyzing entanglement entropy in this regime.

Contribution

It introduces a semiclassical approximation for double-scaled SYK with a Liouville description, linking correlation functions to a bulk geometry and studying curvature corrections.

Findings

Bulk curvature increases with depth and coupling strength.

Fluctuations are suppressed at small coupling, leading to a rigid (A)dS spacetime.

Entanglement entropy of thermal states is computed in the semiclassical limit.

Abstract

We argue that at finite energies, double-scaled SYK has a semiclassical approximation controlled by a coupling $\lambda $ in which all observables are governed by a non-trivial saddle point. The Liouville description of double-scaled SYK suggests that the correlation functions define a geometry in a two-dimensional bulk, with the 2-point function describing the metric. For small coupling, the fluctuations are highly suppressed, and the bulk describes a rigid (A)dS spacetime. As the coupling increases, the fluctuations become stronger. We study the correction to the curvature of the bulk geometry induced by these fluctuations. We find that as we go deeper into the bulk the curvature increases and that the theory eventually becomes strongly coupled. In general, the curvature is related to energy fluctuations in light operators. We also compute the entanglement entropy of partially entangled thermal states in the semiclassical limit.