Holographic timelike entanglement and subregion complexity with scalar hair

TL;DR

This work investigates holographic timelike entanglement entropy (HTEE) and timelike subregion complexity in a thermal CFT$d$ deformed by a relevant scalar source $\phi_0$, dual to a hairy AdS$_{d+1}$ black hole. Using the surface-merging prescription, extremal spacelike and timelike surfaces are glued inside the Kasner-like interior, revealing that the deformation induces a nontrivial $\Delta t$-dependence in the imaginary part of HTEE and enhances the real part, with IR contributions altering analytic continuation from spacelike/temporal entanglement. Timelike complexity, computed via the complexity=volume conjecture, remains real and shows interior-dominated, linear-in-$\Delta t$ growth at late times, with hair amplifying the finite volume change $\Delta V$. In BTZ, the leading contribution to subregion complexity arises from the interior, and the scalar hair shifts interior–exterior balance, indicating HTEE and timelike CV probe complementary aspects of the black hole interior. Altogether, the results demonstrate that relevant boundary deformations break the prior analytic equivalence between spacelike and timelike entanglement, and that HTEE provides a sharper geometric probe of bulk interior structure than spacelike observables alone.

Abstract

We investigate the holographic timelike entanglement entropy (HTEE) and timelike subregion complexity of a thermal CFT$_d$ deformed by a relevant scalar operator $φ_0$, 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 $Δt$. 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 $Δt$. At small $Δt$, 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.

Figures (11)

• Figure 1: Illustration of the AdS$_{d+1}$ black brane setup where the CFT$_d$ is deformed by $\phi_0$ (higher dimensions are suppressed). The timelike subregion $\mathcal{T}$ is expressed as a strip with time interval $\Delta t$ localized at $x=0$. Here, $r$ is the holographic direction, with the asymptotic boundary at $r=0$ and the horizon at $r=r_H$. The blue curves denote spacelike extremal surfaces (real area), while the red curves denote timelike ones (imaginary area). The two surfaces merge at $r\rightarrow\infty$.
• Figure 2: Illustration of the $s=+1$ surfaces (blue) and $s=-1$ surfaces (red) for $\tilde{\phi}_0=0$ (left) and $\tilde{\phi}_0=2.1533$ (right). Here we choose $d = 2$, $\Delta =1.5$, $r_H=1$, and $r_0=0.5$.
• Figure 3: Relations between boundary time interval $\Delta t$ and the turning point $r_0$ for various dimensions and $\tilde{\phi}_0$. Magenta solid line represents the analytical solution for $d=2$, which is given by Eq. \ref{['dtanalytical']}. The colored dashed lines are the fitting function in Eq. \ref{['fittinginterval']}.
• Figure 4: Real part of the total area versus the boundary time interval $\Delta t$ for various boundary deformation parameter $\tilde{\phi}_0=T^{-d+\Delta}\phi_0$. The solid magenta line represents analytical result from Eq. \ref{['analyticalarea']} for $d=2$ and solid colored lines represent fitted functions from Eqs. \ref{['fitd2D32']} and \ref{['fitd2D1']}. In $d=3$ cases, the solid magenta line represents a linear function of $\Delta t$, fitted with the $\tilde{\phi}_0$ results while colored lines represent fitted functions from Eqs. \ref{['fitd3D2']} and \ref{['fitd3D32']}. However, our numerical fitting breaks down at large $\tilde{\phi}_0$ for $d=3$ due to the breakdown of the near-boundary approximation.
• Figure 5: Imaginary part of the total area versus the boundary time interval $\Delta t$ for various boundary deformation parameter $\tilde{\phi}_0=T^{-d+\Delta}\phi_0$. For $d=2$, the solid magenta line represents analytical result from Eq. \ref{['analyticalarea']} and solid colored lines represent analytical fitting from Eqs. \ref{['fitd2D32']} and \ref{['fitd2D1']} for small $\Delta t$. In $d=3$, we fit the functions presented in Eqs \ref{['ImaginaryFit1']} and \ref{['ImaginaryFit2']} to the $\tilde{\phi}_0=0$ results.