Local quenches, bulk entanglement entropy and a unitary Page curve

TL;DR

This work tests the Faulkner–Faulkner–Faulkner–Faulkner (FLM) prescription for entanglement entropy in a fully time-dependent AdS3/CFT2 setup, using a local quantum quench in the boundary theory and its bulk dual. The authors compute both the area correction and the bulk entanglement entropy correction at order unity, showing that the bulk and boundary calculations match and thereby validating FLM in a dynamical context. They decompose the entanglement entropy into universal (kinematic) and dynamical parts in the CFT, and reproduce these features holographically through geometric and bulk quantum corrections, including a bulk-based Page curve signaling unitary evolution in this toy model of black hole evaporation. The results illuminate how unitarity can persist in semiclassical gravity for simple excitations, connect bulk modular Hamiltonians to CFT data, and provide a concrete framework for exploring quantum corrections to holographic entanglement in time-dependent spacetimes with potential extensions to higher dimensions and more general quench protocols.

Abstract

Quantum corrections to the entanglement entropy of matter fields interacting with dynamical gravity have proven to be very important in the study of the black hole information problem. We consider a one-particle excited state of a massive scalar field infalling in a pure AdS$_3$ geometry and compute these corrections for bulk subregions anchored on the AdS boundary. In the dual CFT$_2$, the state is given by the insertion of a local primary operator and its evolution thereafter. We calculate the area and bulk entanglement entropy corrections at order $\mathcal{O}(N^0)$, both in AdS and its CFT dual. The two calculations match, thus providing a non-trivial check of the FLM formula in a dynamical setting. Further, we observe that the bulk entanglement entropy follows a Page curve. We explain the precise sense in which our setup can be interpreted as a simple model of black hole evaporation and comment on the implications for the information problem.

Figures (13)

• Figure 1: Schematic representation of the uniformization map (\ref{['eq:zofw']}). After this transformation is implemented, each sheet of the original Riemann surface $\Sigma_n^A$ maps to a wedge in the complex plane $\mathbb{C}$ with $2\pi (k-1)/n \leq \theta \leq 2\pi k/n$. The insertion points are mapped according to (\ref{['eq:oddz']}) and (\ref{['eq:z1z2map']}) which means that $i)$ only two of these points are inserted in each wedge and $ii)$ points with $k\geq1$ differ only by a phase to those with $k=0$. For this plot we have set $n=12$ as an illustration, but for our particular calculation we are interested in smaller values of $n$, specifically, in the vicinity of $n\approx 1$.
• Figure 2: Small interval limit of the $2n$-correlator in the complex plane $\mathbb{C}$. Assuming $R\to0$ all pairs of insertion points with equal $k$ (i.e., in the same sheet of the original Riemann surface $\Sigma_n^A$) approach to each other and one can carry out an OPE expansion between the two. This is possible provided that $n$ is not too large, so points with different $k$ are still a finite distance away from each other as $R\to0$. For this plot we have set $n=12$ as an illustration, but for our particular calculation we are interested in smaller values of $n$, specifically, in the vicinity of $n\approx 1$.
• Figure 3: Pictorial representation of the universal contribution to the spread of entanglement entropy after a local quantum quench in 2d CFTs. The state at $t=0$ is the vacuum state perturbed by a local operator smeared over a region of compact support $\sim \alpha$. The state evolves under the CFT Hamiltonian $H$ for $t > 0$, generating an entangled pair of wave packets that move in opposite directions at the speed of light. The wave packets eventually increase the entanglement entropy of region $A=\{x|x\in[x_L,x_R]\}$ ($x_L\equiv x_c-R$, $x_R\equiv x_c+R$), and then decrease it as they disperse to infinity.
• Figure 4: Schematic representation of the holographic dual of a local quantum quench. The model consists of a small perturbation that arises by acting locally with an operator $\mathcal{O}_\Delta$ on the vacuum state. The perturbation falls into the AdS interior and excites the metric and other bulk fields. The asymptotic values of the metric and scalar field determine the one point function of the stress-energy tensor $T_{\mu\nu}$ and the scalar operator $\mathcal{O}_\Delta$ in the boundary CFT. For finite $\alpha$, the state at $t=0$ can be prepared by smearing the operator over a region with finite support $\sim\alpha$. This is consistent with the standard notion of UV/IR connection.
• Figure 5: Two global coordinate systems and their Poincaré patches. In the left figure we have plotted the original global frame, represented with coordinates $(\tau,r,\theta)$, where the particle lies at the origin $r=0$. Specializing to a Poincaré patch maps this geodesic to $z^2=L^2+t^2$, so the minimum approach to the boundary is $z=L$ at $t=0$. This surface is depicted in brown. In the right figure we have plotted the boosted global frame $(\tau',r',\theta')$ where the particle oscillates according to (\ref{['eq:osscil']}). Specializing to a Poincaré patch of this boosted frame maps the particle's trajectory to $z^2=\alpha^2+t^2$, with $\alpha\equiv L\, e^{\beta}$, so it can now get arbitrarily close to the boundary. The minimum approach is now $z=\alpha$ at $t=0$. This surface is also depicted in brown in the corresponding cylinder.