Time-like Entanglement Entropy: a top-down approach

TL;DR

This work introduces a top-down holographic framework for time-like entanglement entropy (tEE) that avoids Euclidean-to-Lorentzian analytic continuation by incorporating a sign parameter $\lambda=\pm 1$ in the bulk metric. It provides exact and analytic approximate expressions for $S_{tEE}$ and the time separation $T$ for slab and spherical entangling regions in generic CFTs and confining backgrounds, along with a bulk-embedding stability criterion $Z(u_0)<0$ and a direct route to Liu–Mezei central charges across dimensions. The authors validate the approach on a 4d $\mathcal{N}=2$ SCFT, showing that tEE encodes the dual CFT central charge via quantities ${\cal N}$ and $\widehat{\cal N}$, and extend the analysis to confining models, where phase transitions in tEE correlate with confinement. Analytic approximations for $S_{tEE}$ and $T$ facilitate practical calculations, and the framework sets the stage for exploring infinite families of CFTs and confining theories (to be detailed in NRtoappear).

Abstract

We investigate the concept of time-like entanglement entropy (tEE) within the framework of holography. We introduce a robust top-down prescription for computing tEE in higher-dimensional QFTs, both conformal and confining, eliminating the ambiguities typically associated with analytic continuation from Euclidean to Lorentzian signatures. We present accurate analytic approximations for tEE and time-like separations in slab geometries. We establish a clear stability criterion for bulk embeddings and demonstrate that tEE serves as a powerful tool for computing CFT central charges, extending and strengthening previous results. Finally, we apply our framework to holographic confining backgrounds, revealing distinctive behaviours like phase transitions.

Figures (4)

• Figure 1: On the left panel, the exact time separation $|T|$ in eq.(\ref{['TEED4']}) in terms of the turning point $u_0$. On the right, the approximate $|T_{app}|$ in eq.(\ref{['TEEappD4']}) in terms of $u_0$.
• Figure 2: The exact time like entanglement entropy in terms of $u_0$, on the left. The right panel displays the approximate time-like entanglement in terms of $u_0$. Note that an integration constant can take account of the shift of both plots.
• Figure 3: The exact time-like entanglement entropy in terms $|T|$ on the left, both conveniently normalised. On the right, the approximate entanglement in terms of the approximate separation (both conveniently normalised). These display the signs of a phase transition.
• Figure 4: The exact time-like entanglement entropy for the Anabalón-Ross model in terms $|T|$ on the left, both conveniently normalised. On the right, the approximate entanglement in terms of the approximate separation (both conveniently normalised). These display the signs of a phase transition.