Modular Witten Diagrams and Quantum Extremality

TL;DR

This paper develops a perturbative framework to compute entanglement entropy in excited holographic CFT states, incorporating quantum extremality and backreaction effects via Witten diagrams and modular flow.

Contribution

It introduces a modular Witten diagram approach to evaluate quantum extremal surfaces and entanglement entropy perturbatively in excited states of holographic CFTs.

Findings

Explicit calculation of entanglement entropy to second order in source amplitude.

Reformulation of a graviton-exchange diagram to match quantum Ryu-Takayanagi terms.

Demonstration of shape deformation effects on entanglement wedge due to quantum backreaction.

Abstract

We study entanglement entropy for ball-shaped regions in excited states of holographic conformal field theories. The excited states are prepared by the Euclidean path integral in the CFT with a source turned on for some double-trace operator, with a small, $O(1)$ amplitude $\lambda$. On the gravity side, the double-trace operator deforms the bulk geometry as well as the entanglement structure of the state of bulk matter fields. By the quantum extremal surface formula, this leads to a deformation of the shape of the entanglement wedge, an effect which becomes manifest in the entanglement entropy at $O(\lambda^2 G_N)$. On the CFT side, we explicitly calculate the entanglement entropy perturbatively in the source amplitude to $O(\lambda^2)$, in terms of modular-flowed correlation functions of double-trace operators. We then evaluate these modular-flowed correlation functions using Witten diagrams. This calculation involves a Schwinger-Keldysh contour ordering prescription in the bulk, which we motivate using analytic continuation from Euclidean replica correlators. Focusing on a particular graviton-exchange diagram, we rewrite it in a form where it manifestly reproduces the canonical energy term present in the quantum Ryu-Takayanagi formula, including the shape deformation of the entanglement wedge due to backreaction and quantum effects.