Abstract
We study the holographic mutual information in AdS4 of disjoint spatial domains in the boundary which are delimited by smooth closed curves. A numerical method which approximates a local minimum of the area functional through triangulated surfaces is employed. After some checks of the method against existing analytic results for the holographic entanglement entropy, we compute the holographic mutual information of equal domains delimited by ellipses, superellipses or the boundaries of two dimensional spherocylinders, finding also the corresponding transition curves along which the holographic mutual information vanishes.
Highlights
With ρA, namely SA ≡ −TrA(ρA log ρA), and it measures the entanglement between A and B
Since γA reaches the boundary of the asymptotically AdSD+2 spacetime, its area AA is divergent and it must be regularized through the introduction of a cutoff ε in the holographic direction, which corresponds to the ultraviolet cutoff of the dual conformal field theory
In this paper we focus on D = 2 and we study the shape dependence of the holographic entanglement entropy and of the holographic mutual information (1.1) in AdS4, which is dual to the zero temperature vacuum state of the three dimensional conformal field theory on the boundary
Summary
Finding the minimal area surface spanning a curve is a classic problem in geometry and physics. A physical realization of the problem is obtained by dipping a stiff wire frame of some given shape in soapy water and removing it: as the energy of the film is proportional to the area of the water/air interface, the lowest energy configuration consists of a surface of minimal area In this mundane setting, the requirement of minimal area results into a well known equation. According to the prescription of [9, 10], to compute the holographic entanglement entropy, first we have to restrict ourselves to a t = const slice and we have to find, among all the surfaces γA spanning the curve ∂A, the one minimizing the area functional. As a consequence of the latter property, the boundary is a geodesic of γA (see section A)
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