Holographic timelike entanglement and subregion complexity with scalar hair

Abstract

We investigate the holographic timelike entanglement entropy (HTEE) and timelike subregion complexity of a thermal CFT${}_d$ deformed by a relevant scalar operator ${\varphi }_0$, 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.

arXiv comment42 pages, 11 figures, v3 matches the published version

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