Entanglement Entropy in Jackiw-Teitelboim Gravity

TL;DR

This work provides a coherent Lorentzian framework for gravitational entanglement entropy in pure JT gravity by factorizing the Hilbert space with a brick wall boundary and introducing a universal defect operator. The central construction is the isometric map J that relates JT gravity to two Schwarzian theories via a half defect, yielding a Lorentzian interpretation of the Euclidean replica trick and a modified Renyi trace involving D. The entanglement entropy naturally splits into a bulk term S_bulk and a quantum area term A, with the defect operator accounting for topological and winding data, and the results reproduce the known disk entropy for the Hartle-Hawking state. The approach generalizes the FLM/HRT paradigm to a fully Lorentzian setting in JT gravity and offers a path toward extending these ideas to matter couplings and higher dimensions.

Abstract

We compute the entanglement entropy and Renyi entropies of arbitrary pure states in pure Jackiw-Teitelboim gravity in Lorentz signature. We apply the quantum Hubeny-Rangamani-Ryu-Takayanagi formula by computing the quantum corrected area term and the bulk entropy term. The sum of these two terms for the Hartle-Hawking state agrees with the black hole entropy above extremality computed from the Euclidean disk path integral. We interpret the area term as the universal contribution of a defect operator that plays a crucial role in our Lorentzian interpretation of the Euclidean replica trick in gravity.

Figures (3)

• Figure 1: We illustrate how to use a local boundary condition to define a cutting map from an unfactorized Hilbert space to a factorized Hilbert space. Given a state in the unfactorized Hilbert space, we can evolve for a small time $\delta$ with the Euclidean path integral with an additional boundary inserted, shown in blue. The boundary condition at the blue boundary defines each of the Hilbert space factors. It also defines the cutting map. The boundary conditions on the dashed red boundaries correspond to defining states in the Hilbert space. In quantum field theory, we additionally need to specify the size and shape of the blue boundary. In quantum gravity, the geometry is dynamical, so it is enough to only specify the boundary condition. The time parameter $\delta$ should be taken to zero. We leave $\delta$ implicit in the remainder of this paper. The blue boundary is often referred to as a brick wall.
• Figure 2: We illustrate the brick wall method of computing entanglement entropy in quantum field theory. In a), we show how a brick wall can be used to factorize an arbitrary state, prepared with the Euclidean path integral with the possible insertion of some local operators. The condition that the factorization map is an isometry in the limit $\epsilon \rightarrow 0$ is visualized in b). In c), we show how the brick wall may be used to construct the (unnormalized) reduced density matrix of a state. The Renyi partition functions are then given by $\text{Tr } \rho^n$. On the dashed lines, we fix the values of the fields that appear in the path integral. The brick wall is colored blue.
• Figure 3: We illustrate the process of using the Euclidean path integral to compute the Hartle-Hawking wavefunction in a factorized Hilbert space. If the integral of the spin connection on the brick wall boundary (given in blue) satisfies $\int \omega = \theta$, then the geometry will have an opening angle of $\theta$ between the two boundaries that represent the two Hilbert space factors. For $\theta \neq \pi$, there is a kink. An isometric factorization map cannot have a kink, or else the norm of the Hartle-Hawking wavefunction in the factorized Hilbert space would be incorrect.