Semiclassical geometry in double-scaled SYK

TL;DR

This work develops a λ-controlled semiclassical expansion for double-scaled SYK, casting the model in a Liouville GΣ framework that yields an emergent two-dimensional bulk geometry. At small λ the geometry is rigid AdS2 (or dS2, depending on sign choices), with curvature corrections arising from 1-loop fluctuations that intensify deeper into the bulk and encode energy fluctuations of light operators. The authors compute the leading classical partition function and 2-point function, extend to 1-loop corrections, and analyze the induced bulk metric and its curvature, linking geometric deformations to operator energy fluctuations. They also analyze partially entangled thermal states (PETS), deriving Renyi entropies for light and heavy operators, thereby connecting bulk geometric notions to boundary entanglement structures and providing a platform to explore connections with JT gravity and kinematic-space ideas.

Abstract

We argue that at finite energies, double-scaled SYK has a semiclassical approximation controlled by a coupling $λ$ 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.

Figures (7)

• Figure 1: The correction to the curvature at small temperatures.
• Figure 2: Numerical plot of $\Delta R$ at $v=0.99$ for $\tau \in (0,\beta )$.
• Figure 3: Numerical plot of $\Delta R$ at $v=0.6$.
• Figure 4: Plot of the curvature $R/\mathbb{J}^2$ as a function of $\tau \mathbb{J}$ for $v=0.6$, and two values of $q$: $q=0.36$ (left) and $q=0.8$ (right).
• Figure 5: Plot of the curvature $R/\mathbb{J}^2$ as a function of $\tau \mathbb{J}$ for $v=0.9$, and two values of $q$: $q=0.36$ (left) and $q=0.8$ (right).