Interpolating between Space-like and Time-like Entanglement via Holography

TL;DR

This work develops a holographic framework to interpolate entanglement entropy between space-like and time-like separations by studying boosted slab regions. It reveals two classes of extremal surfaces, Type I (real turning point) and Type II (complex turning point), enabling a unified treatment that connects the Ryu–Takayanagi prescription to its time-like generalizations; complex saddles naturally arise near the light cone and are necessary for a consistent analytic continuation. In conformal backgrounds, EE smoothly interpolates across causal boundaries, while in confining geometries it exhibits phase transitions and confinement-regulated timelike entanglement, with real Type I surfaces possible even for time-like separations. The results emphasize the importance of complex extremal surfaces in extending holographic EE beyond purely spacelike settings and hint at a broader framework where causality is encoded through analytic structures in the bulk, with future work exploring more general backgrounds and deeper field-theoretic interpretations of the complex saddles.

Abstract

We study entanglement entropy for slab like regions in quantum field theories, using their holographic duals. We focus on the transition between space like and time like separations. By considering boosted subsystems in conformal and confining holographic backgrounds, we identify two classes of extremal surfaces: real ones (Type I) and complex surfaces (Type II). These interpolate between the usual Ryu Takayanagi prescription and its time like generalisations. We derive explicit expressions for the entanglement entropy in both conformal and confining cases. We discuss their behaviour across phase transitions and null limits. The interpolation between Type I and Type II surfaces reveals an analytic continuation of the extremal surface across the light cone. Our analysis also finds the existence of a Ryu Takayanagi surface (Type I) even for time like separations in the confining field theory case.

Figures (7)

• Figure 1: The red hyper-plane ($c_y>c_t$) corresponds to space-like separation in a boosted frame of reference. The black dotted hyper-plane ($c_y<c_t$) represents time-like separation in a boosted coordinate system.
• Figure 2: Plot of the interval $\Delta$ and the entanglement in terms of $u_0$, with $u_\Lambda=Q =1$.
• Figure 3: Parametric plot of $S_{tEE}$ vs. $\Delta$. We fix $u_\Lambda=Q=1$.
• Figure 4: Extremal surface for $u_0=0.1$ and $d=4$.
• Figure 5: Plot of (46) and (42) with $c_y=0.05$, $u_\Lambda=Q =1$.