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.
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Entanglement entropy from entanglement contour: higher dimensions
We study the entanglement contour and partial entanglement entropy (PEE) in quantum field theories in 3 and higher dimensions. The entanglement entropy is evaluated from a certain limit of the PEE with a geometric regulator. In the context of the entanglement contour, we classify the geometric regulators, study their difference from the UV regulators. Furthermore, for spherical regions in conformal field theories (CFTs) we find the exact relation between the UV and geometric cutoff, which clarifies some subtle points in the previous literature. We clarify a subtle point of the additive linear combination (ALC) proposal for PEE in higher dimensions. The subset entanglement entropies in the ALC proposal should all be evaluated as a limit of the PEE while excluding a fixed class of local-short-distance correlation. Unlike the 2-dimensional configurations, naively plugging the entanglement entropy calculated with a UV cutoff will spoil the validity of the ALC proposal. We derive the entanglement contour function for spherical regions, annuli and spherical shells in the vacuum state of general-dimensional CFTs on a hyperplane.
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- Award ID(s):
- 2207763
- NSF-PAR ID:
- 10395790
- Date Published:
- Journal Name:
- SciPost Physics Core
- Volume:
- 5
- Issue:
- 2
- ISSN:
- 2666-9366
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.21468/SciPostPhysCore.5.2.020
