Time evolution of entanglement entropy from a pulse

TL;DR

This work analyzes the time evolution of entanglement entropy in a 1{+}1 CFT with a holographic dual when a localized left-moving energy pulse passes through. It builds a finite diffeomorphism from $AdS_3$ in Poincaré coordinates to a general solution with arbitrary $T_{\pm\pm}$, enabling exact holographic entanglement calculations via geodesic lengths and Schwarzian connections to the boundary stress tensor. The authors derive a transformation law for entanglement entropy under conformal maps, confirm it against a CFT computation using twist operators, and explore the shockwave limit where the pulse becomes delta-like, yielding a simplified, local-in-time entanglement response. The results validate the covariant holographic entanglement entropy framework in a time-dependent setting and provide analytic tools for studying nonperturbative time-dependent backgrounds in $AdS_3$ with potential extensions to higher dimensions and Rényi entropies.

Abstract

We calculate the time evolution of the entanglement entropy in a 1+1 CFT with a holographic dual when there is a localized left-moving packet of energy density. We find the gravity result agrees with a field theory result derived from the transformation properties of Rényi entropy. We are able to reproduce behavior which qualitatively agrees with CFT results of entanglement entropy of a system subjected to a local quench. In doing so we construct a finite diffeomorphism which tales three-dimensional anti-de Sitter space in the Poincaré patch to a general solution, generalizing the diffeomorphism that takes the Poincaré patch a BTZ black hole. We briefly discuss the calculation of correlation functions in these backgrounds and give results at large operator dimension.

Figures (6)

• Figure 1: A representation of the system we wish to construct. The red shaded diagonal area is the region of nonzero left-moving energy density with a (lightcone coordinate) width $\Delta$, and the green vertical region is the time-swept path of the spacial regions we wish to calculate entanglement entropy for.
• Figure 2: The map $f_+$ for $\Delta =1/4,~ \tau=2.$ Note that the map is the identity for $x_+<0$. On the left we also show the region with nonzero $T_{++}$ and on the right we shade the region of $x$ not covered by the $y$ coordinates.
• Figure 3: Geodesics connecting two points at equal $x$ time, with $\tau=2,~\Delta=1,~d=3$. On the left we have the projection to the $(x,t)$ plane, with dashed red lines denoting where the region of nonzero $L_+$ is. On the right we have the projection onto the $(x,z)$ plane, with $z$ shifted by time for clarity. Note that while the curves look timelike, this is due to the frame-dragging from $L_+$ and they are indeed spacelike. Note that when $\Delta > d$ the curves are very similar, and when they are entirely within the pulse they are simply extermal BTZ geodesics.
• Figure 4: The six regions for the piecewise smooth function (\ref{['D_regions']}), labelled I - V. The region of interest is always between $x_+-d$ and $x_+$. Note that IIIa only occurs for $d<\Delta$ and IIIb for $\Delta<d$.
• Figure 5: Plots of $\Delta S= S-\frac{c}{6}\log\left[ 2d^2/\delta^2\right]$ at fixed $\tau$ and $\Delta$ as we increase $d$. The lightest curve has $d=1<\Delta$ and the darkest curve has $d=3>\Delta$.