Holographic Entanglement Entropy of Local Quenches in AdS$_4$/CFT$_3$: A Finite-Element Approach

TL;DR

This work investigates the holographic entanglement entropy evolution after local quenches in AdS$_4$/CFT$_3$ using a finite-element numerical scheme to extremize spacelike surfaces. It demonstrates that early-time growth is consistent with the first law for small operator dimensions $Δ$, while large $Δ$ and late times reveal distinct, dimension-dependent dynamics compared to the AdS$_3$/CFT$_2$ case, including sub-linear growth and the absence of a clear logarithmic regime within reachable times. The study provides quantitative bounds and scaling relations, such as $ΔS_A(t=0) ≤ (π Δ)/2$ for small $Δ$ and a horizon-dominated large-$Δ$ limit, and highlights substantial deviations from lower-dimensional behavior at late times. These findings underscore the richer entanglement structure in higher dimensions and motivate further numerical and analytical explorations, including backreacted bulk dynamics and extensions to more complex boundary regions.

Abstract

Understanding quantum entanglement in interacting higher-dimensional conformal field theories is a challenging task, as direct analytical calculations are often impossible to perform. With holographic entanglement entropy, calculations of entanglement entropy turn into a problem of finding extremal surfaces in a curved spacetime, which we tackle with a numerical finite-element approach. In this paper, we compute the entanglement entropy between two half-spaces resulting from a local quench, triggered by a local operator insertion in a CFT$_3$. We find that the growth of entanglement entropy at early time agrees with the prediction from the first law, as long as the conformal dimension $Δ$ of the local operator is small. Within the limited time region that we can probe numerically, we observe deviations from the first law and a transition to sub-linear growth at later time. In particular, the time dependence at large $Δ$ shows qualitative differences to the simple logarithmic time dependence familiar from the CFT$_2$ case. We hope that our work will motivate further studies, both numerical and analytical, on entanglement entropy in higher dimensions.

Figures (15)

• Figure 1: Absolute value of the metric component $g_{t t}$ in Poincaré coordinates (top) and corresponding boundary energy density $E=T_{t t}$ (bottom) for time $t=0,2,4$ in the AdS$_4$/CFT$_3$ local quench model. Horizontal axis shows radius $\rho=\sqrt{x_1^2 + x_2^2}$, boundary is at $z=0$. Units correspond to $M=R=\alpha=1$.
• Figure 2: Schematic boundary cutoff for an extremal surface arond a horizon region. Curve represents full numerical solution, dashed line corresponds to static integration region (extends to infinity in both directions). The geodesic lengths $l_1$ and $l_2$ between the horizon and the boundary of the full solution serve as an effective cutoff distance.
• Figure 3: Example of a cutoff at proper distances $l_1=1.5$ (left) and $l_2=2.0$ (right) from the horizon, in global (top) and Poincaré coordinates (bottom). Global coordinate labels are $(X,Y,Z)=(r \sin\theta \cos\phi,r \sin\theta \sin\phi,r \cos\theta)$. The center of the coordinate horizon is shown as a black dot. Surfaces were computed for Poincaré time $t=3$ and mass parameter $M=1$. The time coordinates are omitted. Units in $R=\alpha=1$.
• Figure 4: Numerically minimized surfaces in global coordinates at quench time $t=0$ for mass parameters $M=0.05$ (top), $M=1$ (middle), and $M=40$ (bottom). The black spheres show the respective coordinate horizons of the metric. Coordinate labels are $(X,Y,Z)=(r \sin\theta \cos\phi,r \sin\theta \sin\phi,r \cos\theta)$. Units in $R=\alpha=1$.
• Figure 5: Half-space entanglement entropy $\Delta S_A$ of a local quench at time $t=0$. Data points show numerical computations in AdS$_4$, dashed line is the analytical AdS$_3$ result. The solid lines correspond to functions $\Delta S_A(M) = {\pi\over 4} M$ (red) and ${\pi\over 2} M^{2/3}$ (orange). Units in $G_N = R = 1$.