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We propose a new model of the spherical symmetric quantum black hole in the reduced phase space formulation. We deparametrize gravity by coupling to the Gaussian dust which provides the material coordinates. The foliation by dust coordinates covers both the interior and exterior of the black hole. After the spherical symmetry reduction, our model is a 1 + 1 dimensional field theory containing infinitely many degrees of freedom. The effective dynamics of the quantum black hole is generated by an improved physical HamiltonianH_Δ. The holonomy correction inH_Δis implemented by the$\overline{\mu }$-scheme regularization with a Planckian area scale Δ (which often chosen as the minimal area gap in loop quantum gravity). The effective dynamics recovers the semiclassical Schwarzschild geometry at low curvature regime and resolves the black hole singularity with Planckian curvature, e.g.R_μνρσR^μνρσ∼ 1/Δ². Our model predicts that the evolution of the black hole at late time reaches the charged Nariai geometry dS₂×S²with Planckian radii$\sim \sqrt{\mathrm{\Delta }}$. The Nariai geometry is stable under linear perturbations but may be unstable by nonperturbative quantum effects. Our model suggests the existence of quantum tunneling of the Nariai geometry and a scenario of black-hole-to-white-hole transition.

 

Entanglement entropy satisfies a first law-like relation, whichequates the first order perturbation of the entanglement entropy for theregion A A to the first order perturbation of the expectation value of the modularHamiltonian, \delta S_{A}=\delta \langle K_A \rangle δ S A = δ ⟨ K A ⟩ .We propose that this relation has a finer version which states that, thefirst order perturbation of the entanglement contour equals to the firstorder perturbation of the contour of the modular Hamiltonian,i.e.  \delta s_{A}(\textbf{x})=\delta \langle k_{A}(\textbf{x})\rangle δ s A ( 𝐱 ) = δ ⟨ k A ( 𝐱 ) ⟩ .Here the contour functions s_{A}(\textbf{x}) s A ( 𝐱 ) and k_{A}(\textbf{x}) k A ( 𝐱 ) capture the contribution from the degrees of freedom at \textbf{x} 𝐱 to S_{A} S A and K_A K A respectively. In some simple cases k_{A}(\textbf{x}) k A ( 𝐱 ) is determined by the stress tensor. We also evaluate the quantumcorrection to the entanglement contour using the fine structure of theentanglement wedge and the additive linear combination (ALC) proposalfor partial entanglement entropy (PEE) respectively. The fine structurepicture shows that, the quantum correction to the boundary PEE can beidentified as a bulk PEE of certain bulk region. While the shows thatthe quantum correction to the boundary PEE comes from the linearcombination of bulk entanglement entropy. We focus on holographictheories with local modular Hamiltonian and configurations of quantumfield theories where the applies.