Holographic timelike entanglement and subregion complexity with scalar hair

TL;DR

This paper explores holographic timelike entanglement entropy and subregion complexity in deformed AdS black holes, revealing new dependence on time intervals and interior contributions, with analytical and numerical results across dimensions.

Contribution

It introduces a novel approach to compute holographic timelike entanglement and complexity in hairy black holes, highlighting differences from pure AdS cases and extending to higher dimensions.

Findings

Deformation breaks invariance of imaginary HTEE component in pure AdS3.

Analytical expressions match numerical results at small time intervals.

Interior region dominates the UV-finite term of subregion complexity in BTZ black holes.

Abstract

We investigate the holographic timelike entanglement entropy (HTEE) and timelike subregion complexity of a thermal CFT$_d$ deformed by a relevant scalar operator $\phi_0$, dual to a hairy black hole in AdS$_{d+1}$. We employ the prescription of merging spacelike and timelike surfaces at the interior, constructing an extremal surface homologous to a boundary timelike subsystem with a time interval $\Delta t$. Consequently, this deformation breaks the invariance of the imaginary component of HTEE observed in pure AdS$_3$ and BTZ geometry, introducing a nontrivial dependence on $\Delta t$. At small $\Delta t$, we derive analytical expressions that are in agreement with numerical results, and observe partial consistency with analytic continuation to temporal or spacelike entanglement entropy at the level of the near-boundary expansion. However, analytic continuation of CFT temporal entanglement entropy fails to reproduce the HTEE calculations under boundary deformation, even in $d=2$. Furthermore, we extend the numerical calculations to higher dimensions ($d=3$). In addition, we study holographic timelike subregion complexity within the complexity=volume conjecture and find that it remains real-valued, providing a complementary geometric probe of the black hole interior. In particular, for the BTZ black hole, we analytically show that the UV-finite term of the subregion complexity receives its entire contribution from the interior region alone.