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arXiv:1703.01759

Entanglement entropy and kinematic space in BCFT

Samrat Bhowmick, Suchetan Das, Bobby Ezhuthachan

Published Mar 6, 2017• 3 citations
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Abstract

The relation between kinematic space metric and entanglement entropy provides us with a differential equation for entanglement entropy. For BCFT on upper half plane we solve this equation to obtain an expression for entanglement entropy consistent with known results in the literature. We also discuss how this relation can be used to recast the RG flow, under relevant deformations of a CFT, as a flow in the space of kinematic space metrics.

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The relation between kinematic space metric and entanglement entropy provides us with a differential equation for entanglement entropy. For BCFT on upper half plane we solve this equation to obtain an expression for entanglement entropy consistent with known results in the literature. We also discuss how this relation can be used to recast the RG flow, under relevant deformations of a CFT, as a flow in the space of kinematic space metrics.
Entanglement entropy between tangent balls in CFTD_DD​
We apply the universal method developed in \cite{Jiang:2025jnk} to compute the entanglement entropy between two tangent balls in CFTD_DD​. When taking the radius of one ball to infinity, it gives the entanglement entropy between a ball and its tangent half plane. In two-dimensional case, this configuration is equivalent to the entanglement in boundary conformal field theory (BCFT) between the negative half-axis and an interval ending on the boundary.
We derive differential equations for the flow of entanglement entropy as a function of the metric and the couplings of the theory. The variation of the universal part of entanglement entropy under a local Weyl transformation is related to the variation under a local change in the couplings. We show that this relation is, in fact, equivalent to the trace Ward identity. As a concrete application of our formalism, we express the entanglement entropy for massive free fields as a two-point function of the energy-momentum tensor.
We explore entanglement entropy of a cap-like region for a generic quantum field theory residing in the Bunch-Davies vacuum on de Sitter space. Entanglement entropy in our setup is identical with the thermal entropy in the static patch of de Sitter, and we derive a simple relation between the vacuum expectation value of the energy-momentum tensor trace and the RG flow of entanglement entropy. In particular, renormalization of the bare couplings and logarithmic divergence of the entanglement entropy are interrelated in our setup. We confirm our findings by recovering known universal contributions for a free field theory deformed by a mass operator as well as obtain correct universal behaviour at the fixed points. Simple examples of entanglement entropy flows are elaborated in d=2,3,4. Inthreedimensionswefindthatwhiletherenormalizedentanglemententropy is stationary at the fixed points, it is not monotonic. We provide a computational evidence that the universal ‘area law’ for a conformally coupled scalar is different from the known result in the literature, and argue that this difference survives in the limit of flat space. Finally, we carry out the spectral decomposition of entanglement entropy flow and discuss its application to the F-theorem.
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The of the entanglement entropies between disjoint subsystems enables us to show that the dynamical equation of the entanglement entropy in CFT 2 is precisely the three-dimensional Einstein equation. We establish a profound relation between the cosmological constant and CFT 2 entanglement entropy. Thus, entanglement entropies induce internal gravitational geometries in CFT 2 . Extracting the dual metric from an entanglement entropy becomes a straightforward procedure. Remarkably, we discover that the renormalization group equation is a geometric identity.
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