On shape dependence of holographic mutual information in AdS4
IIFT-UAM/CSIC-14-118
On shape dependence of holographic mutual information in AdS Piermarco Fonda a, , Luca Giomi b, , Alberto Salvio c d, and Erik Tonni a, a SISSA and INFN, via Bonomea 265, 34136, Trieste, Italy b Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands c Instituto de Física Teórica IFT-UAM/CSIC, Universidad Autónoma de Madrid, Madrid 28049, Spain d Departamento de Física Teórica, Universidad Autónoma de Madrid, Madrid, Spain
Abstract
We study the holographic mutual information in AdS of disjoint spatial domains in the boundary whichare delimited by smooth closed curves. A numerical method which approximates a local minimum of thearea functional through triangulated surfaces is employed. After some checks of the method against existinganalytic results for the holographic entanglement entropy, we compute the holographic mutual informationof 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. [email protected] [email protected] [email protected] [email protected] a r X i v : . [ h e p - t h ] N ov ontents
33 Simply connected regions 4 H
29B Numerical Method 30C Superellipse: a lower bound for F A
32D Some generalizations to AdS D +2 D.1 Sections of the infinite strip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33D.2 Annular domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
E Elliptic integrals 39
Entanglement entropy has been extensively studied during the last decade and its important role in quantumgravity, quantum field theory and condensed matter physics is widely recognized.Given a quantum system in its ground state | ψ i and assuming that its Hilbert space can be decomposedas H = H A ⊗ H B , one can introduce the reduced density matrix ρ A ≡ Tr B ρ by tracing over H B the densitymatrix ρ = | ψ ih ψ | of the whole system. Here we focus on a bipartition of the Hilbert space associated with aseparation of a spatial slice into two complementary regions. The entanglement entropy is the Von Neumannentropy associated with ρ A , namely S A ≡ − Tr A ( ρ A log ρ A ), and it measures the entanglement between A and B . In the same way, one can introduce ρ B ≡ Tr A ρ and S B . Since ρ is a pure state, we have that S B = S A . Understanding the dependence of S A on the geometry of the region A is an important task.Let us consider a conformal field theory in D + 1 dimensions at zero temperature in its ground state. Theentanglement entropy S A between a D dimensional spatial region A and its complement B can be writtenas an expansion in the ultraviolet cutoff ε , where the leading divergence is S A ∝ Area( ∂A ) /ε D − + . . . [1, 2].This behaviour is known as the area law for the entanglement entropy and ∂A is sometimes called entanglingsurface. When D = 1 and the domain A is an interval, ∂A is made by its two endpoints and the area law isviolated because the leading divergence is logarithmic. In particular, S A = ( c/
3) log( ‘/ε ) + const, where c isthe central charge of the model [3, 4].By virtue of the holographic correspondence [5–7] (see [8] for a review), the entanglement entropy S A ofa conformal field theory in a D + 1 dimensional Minkowski spacetime can be also calculated from its dualgravitational model defined in a D + 2 dimensional asymptotically anti-De Sitter (AdS) spacetime whose1oundary is the spacetime of the original conformal field theory. In the regime where it is enough to consideronly classical gravity, the holographic prescription to compute the entanglement entropy is [9, 10] S A = A A G N , (1.1)where G N is the D + 2 dimensional Newton constant and A A is the area of the codimension two minimalarea spacelike surface ˜ γ A at some fixed time slice such that ∂ ˜ γ A = ∂A . Since ˜ γ A reaches the boundary of theasymptotically AdS D +2 spacetime, its area A A is divergent and therefore it must be regularized through theintroduction of a cutoff ε in the holographic direction, which corresponds to the ultraviolet cutoff of the dualconformal field theory. The leading divergence of (1.1) as ε → D +2 occur alsoin the holographic dual of the expectation values of the Wilson loops [15, 16]. Nevertheless, while the bulksurfaces for the Wilson loops are always two dimensional, for the holographic entanglement entropy they havecodimension two. Thus, when D = 2 the minimal surfaces to compute for the holographic entanglemententropy (1.1) are the same ones occurring in the gravitational counterpart of the correlators of spacelikeWilson loops.As for the dependence of A A on the geometry of ∂A , analytic results have been found for the infinitestrip and for the sphere when D is generic [9, 10]. Spherical domains play a particular role because theirreduced density matrix can be related to a thermal one [17]. When D = 2, the O (1) term in the expansionof S A as ε → F , which decreases along any renormalizationgroup flow [18–20]. Some interesting results have been found about A A for an entangling surface ∂A with ageneric shape [21–28], but a complete understanding is still lacking.When A = A ∪ A is made by two disjoint spatial regions, an important quantity to study is the mutualinformation I A ,A ≡ S A + S A − S A ∪ A . (1.2)It is worth remarking that S A ∪ A provides the entanglement between A ∪ A and the remaining part ofthe spatial slice. In particular, it does not quantify the entanglement between A and A , which is measuredby other quantities, such as the logarithmic negativity [29–32]. In the combination (1.2), the area lawdivergent terms cancel and the subadditivity of the entanglement entropy guarantees that I A ,A (cid:62)
0. Fortwo dimensional conformal field theories, the mutual information depends on the full operator content of themodel [33–36]. When D (cid:62)
2, the computation of (1.2) is more difficult because non local operators must beintroduced along ∂A [37–39].The holographic mutual information is (1.2) with S A given by (1.1). The crucial term to evaluate is S A ∪ A , which depends on the geometric features of the entangling surface ∂A = ∂A ∪ ∂A , including alsothe distance between A and A and their relative orientation, being ∂A made by two disjoint components.It is well known that, keeping the geometry of A and A fixed while their distance increases, the holographicmutual information has a kind of phase transition with discontinuous first derivative, such that I A ,A = 0when the two regions are distant enough. This is due to the competition between two minima correspondingto a connected configuration and to a disconnected one. While the former is minimal at small distances, thelatter is favoured for large distances, where the holographic mutual information therefore vanishes [40–42].This phenomenon has been also studied much earlier in the context of the gravitational counterpart of theexpectation values of circular spacelike Wilson loops [43–46]. The transition of the holographic mutualinformation is a peculiar prediction of (1.1) and it does not occur if the quantum corrections are taken intoaccount [47]. A similar transition due to the competition of two local minima of the area functional occursalso for the holographic entanglement entropy of a single region at finite temperature [48–50].In this paper we focus on D = 2 and we study the shape dependence of the holographic entanglemententropy and of the holographic mutual information (1.1) in AdS , which is dual to the zero temperaturevacuum state of the three dimensional conformal field theory on the boundary. This reduces to finding the2inimal area surface ˜ γ A spanning a given boundary curve ∂A (the entangling curve) defined in some spatialslice of the boundary of AdS . The entangling curve ∂A could be made by many disconnected components.When ∂A consists of one or two circles, the problem is analytically tractable [9, 10, 15, 51–55]. However,for an entangling curve having a generic shape (and possibly many components), finding analytic solutionsbecomes a formidable task. In order to make some progress, we tackle the problem numerically with thehelp of Surface Evolver [56, 57], a widely used open source software for the modelling of liquid surfacesshaped by various forces and constraints. A section at constant time of AdS gives the Euclidean hyperbolicspace H . Once the curve embedded in H is chosen, this software constructs a triangular mesh whichapproximates the surface spanning such curve which is a local minimum of the area functional, computingalso the corresponding finite area. The number of vertices V , edges E and faces E of the mesh are relatedvia the Euler formula, namely V − E + F = χ , being χ = 2 − g − b the Euler characteristic of the surface,where g is its genus and b the number of its boundaries. In this paper we deal with surfaces of genus g = 0with one or more boundaries.The paper is organized as follows. In §2 we state the problem, introduce the basic notation and reviewsome properties of the minimal surfaces occurring in our computations. In §3 we address the case of surfacesspanning simply connected curves. First we review two analytically tractable examples, the circle andthe infinite strip; then we address the case of some elongated curves (i.e. ellipse, superellipse and theboundary of the two dimensional spherocylinder) and polygons. Star shaped and non convex domains arealso briefly discussed. In §4 we consider ∂A made by two disjoint curves. The minimal surface spanningsuch disconnected curve can be either connected or disconnected, depending on the geometrical features ofthe boundary, including the distance between them and their relative orientation. The cases of surfacesspanning two disjoint circles, ellipses, superellipses and the boundaries of two dimensional spherocylindersare quantitatively investigated for a particular relative orientation. Further discussions and technical detailsare reported in the appendices. R thisis known as Plateau’s problem. A physical realization of the problem is obtained by dipping a stiff wireframe of some given shape in soapy water and then removing it: as the energy of the film is proportional tothe area of the water/air interface, the lowest energy configuration consists of a surface of minimal area. Inthis mundane setting, the requirement of minimal area results into a well known equation H = 0 , (2.1)where H = k ii / k ij = e i,j · N , with N the surface normal vector, e i a generic tangent vector, such that the surface metric tensor is h ij = e i · e j ,and ( · ) ,i = ∂ i ( · ).The metric of AdS in Poincaré coordinates reads ds = − dt + dx + dy + dz z , (2.2)where the AdS radius has been set to one for simplicity. The spatial slice t = const provides the Euclideanhyperbolic space H and the region A is defined in the z = 0 plane. According to the prescription of [9, 10],to compute the holographic entanglement entropy, first we have to restrict ourselves to a t = const sliceand then we have to find, among all the surfaces γ A spanning the curve ∂A , the one minimizing the areafunctional A [ γ A ] = ˆ γ A d A = ˆ U A √ h du du z , (2.3)where U A is a coordinate patch associated with the coordinates ( u , u ) and h = det( h ij ). We denote by ˜ γ A the area minimizing surface, so that A [˜ γ A ] ≡ A A provides the holographic entanglement entropy through the3yu-Takayanagi formula (1.1). Since all the surfaces γ A reach the boundary of AdS , their area is divergentand therefore one needs to introduce a cutoff in the holographic direction to regularize it, namely z (cid:62) ε > ε is an infinitesimal parameter. The holographic dictionary tells us that this cutoff corresponds to theultraviolet cutoff in the dual three dimensional conformal field theory. Considering z (cid:62) ε >
0, the area A [ γ A ]and therefore A A as well become ε dependent quantities which diverge when ε →
0. Important insights canbe found by writing A A as an expansion for ε →
0. When ∂A is a smooth curve, this expansion reads A A = P A ε − F A + o (1) , (2.4)where P A = length( ∂A ) is the perimeter of the entangling curve and o (1) indicates vanishing terms when ε →
0. When the entangling curve curve ∂A contains a finite number of vertices, also a logarithmic divergenceoccurs, namely A A = P A ε − B A log( P A /ε ) − W A + o (1) . (2.5)The functions F A , B A and W A are defined through (2.4) and (2.5). They depend on the geometry of ∂A ina very non trivial way. We remark that the section of ˜ γ A at z = ε provides a curve which does not coincidewith ∂A because of the non trivial profile of ˜ γ A in the bulk.As the area element in AdS is factorized in the form d A = du du √ h/z , a surface in AdS is equivalentto a surface in R endowed with a potential energy density of the form 1 /z . By using the standard machineryof surface geometry (see §A), one can find an analog of (2.1) in the form H + ˆ z · N z = 0 , (2.6)where ˆ z is a unit vector in the z direction. The relation (2.6) implies that, in order for the mean curvatureto be finite, the surface must be orthogonal to the ( x, y ) plane at z = 0: i.e. ˆ z · N = 0 at z = 0. As aconsequence of the latter property, the boundary is also a geodesic of ˜ γ A (see §A). In this section we consider cases in which the region A is a simply connected domain. We first review thesimple examples of the disk and of the infinite strip, which can be solved analytically [9, 10]. In §3.1 wenumerically analyze the case in which A is an elongated region delimited by either an ellipse, a superellipseor the boundary of a two dimensional spherocylinder, while in §3.2 we address the case in which ∂A is aregular polygon. In §3.3, star shaped and non convex domains are briefly discussed.If A is a disk of radius R , the minimal area surface ˜ γ A is a hemisphere, as it can be easily proved from adirect substitution in (2.6). Taking N = r / | r | , with r = ( x, y, z ) and | r | = R , one finds ˆ z · N = z/R , hence H = − /R , which is the mean curvature of a sphere whose normal is outward directed. The area of the partof the hemisphere such that ε (cid:54) z (cid:54) R is A A = 2 πRε − π . (3.1)Comparing this expression with (2.4), one finds that F A = 2 π in this case. It is worth remarking , as peculiarfeature of the disk, that in (3.1) o (1) terms do not occur.A special case of (2.6) is obtained when the surface is fully described by a function z = z ( x, y ) representingthe height of the surface above the ( x, y ) plane at z = 0. In this case A [ γ A ] = ˆ γ A z q z ,x + z ,y dxdy , (3.2)and (2.6) becomes the following second order non linear partial differential equation for z (see §A for somedetails on this derivation) z ,xx (1 + z ,y ) + z ,yy (1 + z ,x ) − z ,xy z ,x z ,y + 2 z (1 + z ,x + z ,y ) = 0 , (3.3)4ith the boundary condition that z = 0 when ( x, y ) ∈ ∂A . The partial differential equation (3.3) is verydifficult to solve analytically for a generic curve ∂A ; but for some domains A it reduces to an ordinarydifferential equation. Apart from the simple hemispherical case previously discussed, this happens also foran infinite strip A = { ( x, y ) ∈ R , | y | (cid:54) R } , whose width is 2 R . The corresponding minimal surface isinvariant along the x axis and therefore it is fully characterized by the profile z = z ( y ) for | y | (cid:54) R . Taking z ,x = 0 in (3.3) yields z ,yy + 2 z (1 + z ,y ) = 0 . (3.4)Equivalently, the infinite strip case can be studied by considering the one dimensional problem obtainedsubstituting z = z ( y ) directly in (3.2) [8–10]. Since the resulting effective Lagrangian does not depend on y explicitly, one easily finds that z q z ,y is independent of y . Taking the derivative with respect to y of this conservation law, (3.4) is recovered, as expected. The constant value can be found by considering y = 0, where z (0) ≡ z ∗ and z ,y (0) = 0. Notice that z ∗ is the maximal height attained by the curve along the z direction. Integrating the conservation law, one gets y ( z ) = √ π Γ(3 / / z ∗ − z z ∗ F (cid:18) ,
34 ; 74 ; z z ∗ (cid:19) , z ∗ = Γ(1 / √ π Γ(3 / R , (3.5)where Γ is the Euler gamma and F is the hypergeometric function. Thus, the minimal surface ˜ γ A consistsof a tunnel of infinite length along the x direction, finite width R along the y direction and whose shape inthe ( y, z ) plane is described by (3.5). Considering a finite piece of this surface which extends for R (cid:29) R in the x direction, whose projection on the ( x, y ) plane is delimited by the dashed lines in the bottom panelof Fig. 1, its area is given by [9, 10, 15, 16] A A = 4 R ε − R s ∞ R + o (1) , s ∞ ≡ π Γ(1 / , (3.6)where ε (cid:54) z (cid:54) z ∗ . Comparing (2.4) with P A = 4 R and (3.6), one concludes that F A = s ∞ R /R .In order to compare (3.6) with our numerical results, we find it useful to construct an auxiliary surfaceby closing this long tunnel segment with two planar “caps” placed at x = ± R , whose profile is described inthe ( y, z ) plane by (3.5), with a cutoff at z = ε . These regions are identical by construction and their area(see §D.1) is given by A cap = 2 R /ε − π/ o (1). Thus, the total area of the auxiliary surface reads A A + 2 A cap = 4( R + R ) ε − R s ∞ R − π + o (1) , (3.7)where the coefficient of the leading divergence is the perimeter of the rectangle in the boundary (dashedcurve in Fig. 1). It is worth remarking that this surface is not the minimal area surface anchored on thedashed rectangle in Fig. 1. Indeed, in this case an additional logarithmic divergence occurs (see §3.2).Since in the following we will compute numerically A A for various domains keeping ε fixed, let us introduce e F A ≡ − (cid:18) A A − P A ε (cid:19) . (3.8)From (2.4) one easily observes that e F A = F A + o (1) when ε →
0. Notice that for the disk we have e F A = F A .In Fig. 2 the values of e F A for the surfaces discussed above are represented together with other ones comingfrom different curves that will be introduced in §3.1: the black dot corresponds to the disk (see (3.1)), thedotted horizontal line is obtained from (3.6) for the infinite strip, while the dashed line is found from thearea (3.7) of the auxiliary surface. 5 y z x y R R Figure 1: Top panel: Minimal surfaces constructed by using Surface Evolver where the entangling curve ∂A is a circle with radius R = 1 (red), an ellipse (orange), a superellipse (3.9) with n = 8 (purple) and theboundary of a spherocylinder (green) with R = 3 R . The cutoff is ε = 0 .
03 and only the y (cid:62) y = 0. Bottom panel:In the ( x, y ) plane, we show the superellipses with R = 3 R with n = 2 (orange), n = 4 (blue), n = 6(magenta) and n = 8 (purple), the circle with radius R (red curve) and the rectangle circumscribing thesuperellipses (dashed lines). The green curve is the boundary of the two dimensional spherocylinder with R = 3 R . The first examples of entangling curves ∂A we consider for which analytic expressions of the correspondingminimal surfaces are not known are the superellipse and the boundary of the two dimensional spherocylinder,whose geometries depend on two parameters. 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" (cid:1)(cid:2)(cid:3)(cid:4)(cid:1)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:6)(cid:10) % !* !% "* "% ellipsespherocylinder superellipse n = 4superellipse n = 6superellipse n = 8superellipse n = 3 R R F A R /R R = 2 R = 1 circle Figure 2: Numerical data for e F A , defined in (3.8), corresponding to domains A which are two dimensionalspherocylinders or delimited by superellipses. Here ε = 0 .
03. In the main plot R = 1, while in the inset,which shows a zoom of the initial part of the main plot in logarithmic scale on both the axes, we have alsoreported data with R = 2. The horizontal dotted black line corresponds to the infinite strip (3.6) and thedashed one to the auxiliary surface where the sections at x = ± R have been added (see (3.7)). The redand blue dotted horizontal lines come from the asymptotic result (C.10) evaluated for n = 2 and n = 3respectively.is described by the equation | x | n R n + | y | n R n = 1 , R (cid:62) R > , n (cid:62) , (3.9)where R , R and n are real and positive parameters. The curve (3.9) is also known as Lamé curve and herewe consider only integers n (cid:62) n = 2 in (3.9) is the ellipse with semi-majorand semi-minor axes given by R and R respectively. As the positive integer n increases, the superellipseapproximates the rectangle with sides 2 R and 2 R . When R = R , the curves (3.9) for various n areknown as squircles because they have intermediate properties between the ones of a circle ( n = 2) and theones of a square ( n → ∞ ). In the bottom panel of Fig. 1, we show some superellipses with R = 3 R , thecircle with radius R included in all the superellipses and the rectangle circumscribing them.In order to study the interpolation between the circle and the infinite strip, a useful domain to consider isthe two dimensional spherocylinder. The spherocylinder (also called capsule) is a three dimensional volumeconsisting of a cylinder with hemispherical ends. Here we are interested in its two dimensional version, whichis a rectangle with semicircular caps. In particular, the two dimensional spherocylinder circumscribed bythe rectangle with sides 2 R and 2 R is defined as the set S ≡ D ∪ C + ∪ C − , where the rectangle D and thedisks C ± are D ≡ (cid:8) ( x, y ) , | y | (cid:54) R , | x | (cid:54) R − R (cid:9) , C ± ≡ (cid:8) ( x, y ) , (cid:2) x ± ( R − R ) (cid:3) + y (cid:54) R (cid:9) . (3.10)7he perimeter of this domain is P A = 2 πR + 4( R − R ) and an explicit example of ∂ S with R = 3 R is given by the green curve in the bottom panel of Fig. 1. When R = R , the curve ∂ S becomes a circle,while for R (cid:29) R it provides a kind of regularization of the infinite strip. Indeed, when R → ∞ at fixed R the two dimensional spherocylinder S becomes the infinite strip with width 2 R . Let us remark thatthe curvature of ∂ S is discontinuous while the curvature of the superellipse (3.9) is continuous. Moreover,the choice to regularize the infinite strip through the circles C ± in (3.10) is arbitrary; other domains canbe chosen (e.g. regions bounded by superellipses) without introducing vertices in the entangling curve. Astraightforward numerical analysis allows to observe that a superellipses with n > ∂ S in the first quadrant outside the Cartesian axes.In Fig. 2 we show the numerical data for e F A , defined in (3.8), when A is given by the domains discussedabove: disk, infinite strip, two dimensional spherocylinder and two dimensional regions delimited by superel-lipses. In particular, referring to the bottom panel of Fig. 1, we fixed R and increased R . For the twodimensional spherocylinder, this provides an interpolation between the circle and the infinite strip. SurfaceEvolver has been employed to compute the area A A and for the cutoff in the holographic direction we choose ε = 0 .
03. Below this value, the convergence of the local minimization algorithm employed by Surface Evolverbecomes problematic, as well as for too large domains A , as discussed in §B.When R = R , we observe that e F A for the squircles with different n > n . For large R /R , the limits of e F A / ( R /R ) for the domains we address are finite and positive. The values of theselimits associated with the superellipses are ordered in the opposite way in n with respect to the startingpoint at R = R and therefore they cross each other as R /R increases. We remark that the curvecorresponding to the two dimensional spherocylinder stays below the ones associated with the superellipsesfor the whole range of R /R that we considered. In Fig. 2 the horizontal black dotted line corresponds to theinfinite strip (see (3.6)) while the dashed curve is obtained from the auxiliary surface described above (see(3.7)). The latter one is our best analytic approximation of the data corresponding to the two dimensionalspherocylinder.Focussing on the regime of large R /R , from Fig. 2 we observe that the asymptotic value of e F A / ( R /R )for the two dimensional spherocylinder is very close to the one of the auxiliary surface obtained from (3.7)and therefore it is our best approximation of the result corresponding to the infinite strip. This is reasonablebecause the two dimensional spherocylinder is a way to regularize the infinite strip without introducing ver-tices in the entangling curve, as already remarked above. As for the minimal surfaces spanning a superellipsewith a given n (cid:62)
2, in §C an asymptotic lower bound is obtained (see (C.10)), generalizing the constructionof [28]. In Fig. 2 this bound is shown explicitly for n = 2 and n = 3 (red and blue dotted horizontal linesrespectively). Since this value is strictly larger than the corresponding one associated with the infinite strip(see (3.6)), we can conclude that e F A / ( R /R ) for the superellipse at fixed n does not converge to the value s ∞ associated with the infinite strip. In this section we consider the minimal area surfaces associated with simply connected regions A whoseboundary is a convex polygon with N sides. These are prototypical examples of minimal surfaces spanningentangling curves with geometric singularities. For quantum field theory results about the entanglemententropy of domains delimited by such curves, see e.g. [58–60].The main feature to observe about the area A A of the minimal surface is the occurrence of a logarithmicdivergence, besides the leading one associated with the area law, in its expansion as ε →
0. We find itconvenient to introduce e B A ≡ ε/P A ) (cid:18) A A − P A ε (cid:19) . (3.11)Since (2.5) holds in this case, we have that e B A = B A + o (1).When ∂A is a convex polygon with N sides, denoting by α i < π its internal angle at the i -th vertex, for8igure 3: Minimal area surfaces constructed with Surface Evolver whose ∂A is a polygon with three (left),four (middle) and eight (right) sides. The red polygons ∂A lie in the plane at z = 0 and the z axis pointsdownward but, according to our regularization, the triangulated surfaces are anchored to the same polygonsat z = ε . 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(cid:1) (cid:1)(cid:1) (cid:1)(cid:1) (cid:1) (cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1) (cid:1)(cid:1) (cid:1) (cid:1)(cid:1) (cid:1)(cid:1) (cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1) (cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1) (cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:2) (cid:2)(cid:2) (cid:2)(cid:2) (cid:2)(cid:2)(cid:2) (cid:2)(cid:2) (cid:2)(cid:2) (cid:2) (cid:2)(cid:2)(cid:2)(cid:2) (cid:2) (cid:2)(cid:2)(cid:2)(cid:2) (cid:2) (cid:2)(cid:2) (cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3) (cid:3) (cid:4) (cid:4)(cid:4) (cid:4)(cid:4) (cid:4) (cid:4)(cid:4) (cid:4) (cid:4)(cid:4) (cid:4) (cid:4) (cid:4)(cid:4) (cid:4) (cid:4)(cid:4)(cid:4) (cid:4) (cid:4)(cid:4) (cid:4)(cid:4)(cid:4) (cid:4) (cid:4)(cid:4) (cid:4)(cid:4)(cid:4)(cid:4) (cid:4)(cid:4)(cid:4) (cid:4)(cid:4)(cid:4) (cid:4)(cid:4)(cid:4) (cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:5) (cid:5) (cid:5)(cid:5)(cid:5) (cid:5)(cid:5) (cid:5)(cid:5) (cid:5)(cid:5)(cid:5) (cid:5) (cid:5)(cid:5)(cid:5) (cid:5)(cid:5)(cid:5)(cid:5) (cid:5)(cid:5) (cid:5) (cid:5) (cid:5)(cid:5)(cid:5) (cid:5) (cid:5)(cid:5) (cid:5) (cid:5)(cid:5) (cid:5)(cid:5) (cid:5)(cid:5) (cid:5)(cid:5)(cid:5) (cid:5)(cid:5)(cid:5)(cid:5) (cid:5) (cid:5) (cid:5)(cid:5)(cid:5)(cid:5) (cid:5)(cid:5)(cid:5) (cid:5) (cid:5)(cid:5) (cid:5) (cid:5)(cid:5) (cid:5) (cid:3) (cid:2) (cid:1) (cid:4) (cid:5) ! !"!& !"! zz squaretriangleoctagon Figure 4: Left: Section of the minimal surfaces anchored to an equilateral triangle (red, magenta and purplepoints), a square (blue points) or an octagon (green points) inscribed in a circle, as indicated in the insetby the black line. The continuos lines are z = ρ/f ( α N ), where f ( α ) is found from (3.15) with N = 3(red), N = 4 (blue) or N = 8 (green). The dashed black curve is the hemisphere corresponding to the circlecircumscribing the polygons at z = 0 (dashed in the inset), while the dashed grey horizontal line correspondsto the cutoff ε = 0 .
03. Right: A zoom of the left panel around the origin, placed in the common vertex of thepolygons. For the triangle, three different values of ε ∈ { . , . , . } has been considered to highlighthow the agreement with the analytic result improves as ε → B A ≡ N X i =1 b ( α i ) . (3.12)The function b ( α ) has been first found in [61], where the holographic duals of the correlators of Wilsonloops with cusps have been studied, by considering the minimal surface near a cusp whose opening angleis α . Notice that (3.12) does not depend on the lengths of the edges but only on the convex angles ofthe polygon. Further interesting results have been obtained in the context of the holographic entanglemententropy [52, 62].Introducing the polar coordinates ( ρ, φ ) in the z = 0 plane, one considers the domain {| φ | (cid:54) α/ , ρ < L } ,9 (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3)(cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) ! "! (cid:1) (cid:5) ! !"! ! & (cid:2) (cid:5) ! !"! ! ' (cid:3) (cid:5) ! !"! ! ( (cid:4) (cid:5) ! !"! ! &! (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3)(cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) ! "! (cid:1) (cid:5) ! !"! ! & (cid:2) (cid:5) ! !"! ! ' (cid:3) (cid:5) ! !"! ! ( (cid:4) (cid:5) ! !"! ! &! α α = 1 = 2 = 4 = 10 (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3)(cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) ! "! (cid:1) (cid:5) ! !"! ! & (cid:2) (cid:5) ! !"! ! ' (cid:3) (cid:5) ! !"! ! ( (cid:4) (cid:5) ! !"! ! &! (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3)(cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) ! "! (cid:1) (cid:5) ! !"! ! & (cid:2) (cid:5) ! !"! ! ' (cid:3) (cid:5) ! !"! ! ( (cid:4) (cid:5) ! !"! ! &! α α α = 1 = 2 = 4 = 10 B A B A Figure 5: The quantity e B A in (3.11) with A A evaluated with Surface Evolver when the entangling curve ∂A is either an isosceles triangle whose basis has length ‘ (top panel) or a rhombus whose side length is ‘ (bottom panel). Here ε = 0 .
03. The black continuous curves are obtained from (3.12) and (3.16).where L (cid:29)
1. By employing scale invariance, one introduces the following ansatz [61] z = ρf ( φ ) , (3.13)in terms of a positive function f ( φ ), which is even in the domain | φ | (cid:54) α/ f → + ∞ for | φ | → α/ (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3)(cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4)(cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5)(cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:6) ! !" (cid:1) ! ! " (cid:2) ! ! (cid:3) ! ! $ (cid:4) ! ! "% (cid:5) ! ! "$ (cid:6) ! ! (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3)(cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4)(cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5)(cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:6) ! !" (cid:1) ! ! " (cid:2) ! ! (cid:3) ! ! $ (cid:4) ! ! "% (cid:5) ! ! "$ (cid:6) ! ! α α R = 2 R = 1 R = 5 R = 10 R = 15 R = 20 R B A Figure 6: The quantity e B A in (3.11) corresponding to ∂A given by polygons with N equal sides circumscribedby a circle with radius R . The cutoff is ε = 0 .
03 and the values of N are indicated above the correspondingseries of data points. The black curve is given by (3.12) and (3.16).Plugging (3.13) into the area functional, the problem becomes one dimensional, similarly to the case of theinfinite strip slightly discussed in §3. Since the resulting integrand does not depend explicitly on φ , thecorresponding conservation law tells us that ( f + f ) / p ( f ) + f + f is independent of φ . Thus, theprofile for 0 (cid:54) φ < α/ − α/ < φ (cid:54) φ = ˆ ff ζ (cid:20) ( ζ + 1) (cid:18) ζ ( ζ + 1) f ( f + 1) − (cid:19)(cid:21) − dζ , (3.14)being f ≡ f (0). When f → ∞ , we require that the l.h.s. of (3.14) becomes α/ f = f ( α ). In this limit the integral in (3.14) can be evaluated analytically interms of elliptic integrals Π and K (see §E for their definitions) as follows α ( f ) = 2 ˜ f s − f − ˜ f h Π (cid:0) − ˜ f , ˜ f (cid:1) − K (cid:0) ˜ f (cid:1)i , ˜ f ≡ f f ∈ [0 , / . (3.15)Notice that when f → α → π , which means absence of the corner, while α → f → ∞ .As for the area of the minimal surface given by (3.13), one finds that b ( α ) ≡ ˆ ∞ − s ζ + f + 1 ζ + 2 f + 1 ! dζ = E (cid:0) ˜ f (cid:1) − (cid:0) − ˜ f (cid:1) K (cid:0) ˜ f (cid:1)q − f , (3.16)where f = f ( α ) can be found by inverting numerically (3.15). The function (3.16) has a pole when α → b ( α ) = Γ( ) / ( πα ) + . . . ) while b ( π ) = 0, which is expected because α = π means no cuspand the logarithmic divergence does not occur for smooth entangling curves.An interesting family of curves to study is the one made by the convex regular polygons. They areequilateral, equiangular and all vertices lie on a circle. For instance, a rhombus does not belong to this11amily. Denoting by R the radius of the circumscribed circle and by N the number of sides, the length ofeach side is ‘ = 2 R sin( π/N ) and all the internal angles are α N ≡ N − N π . When N → ∞ we have that α N → π and the polygon becomes a circle. Thus, the area of the minimal surface spanning these regularpolygons is (2.5) with P A = N ‘ and B A = 2 N b ( α N ).It is interesting to compare the analytic results presented above with the corresponding numerical onesobtained with Surface Evolver. Some examples of minimal surfaces anchored on curves ∂A given by a polygonare given in Fig. 3, where the triangulations are explicitly shown. In Fig. 4 we take as ∂A an equilateraltriangle, a square and an octagon which share a vertex and consider the section of the corresponding minimalsurfaces through a vertical plane which bisects the angles associated with the common vertex, as shown inthe inset of the left panel. Focussing on the part of the curves near the common vertex, we find thatthe numerical results are in good agreement with the analytic expression z = ρ/f , where f = f ( α N ) isobtained from (3.14). It would be interesting to find analytic results for the profiles shown in the left panelof Fig. 4.By employing Surface Evolver, we can also consider entangling curves given by polygons which are notregular, as done in Fig. 5, where we have reported the data for e B A (defined in (3.11)) corresponding to thearea of the minimal surfaces ˜ γ A when ∂A is either an isosceles triangle (top panel) or a rhombus with side ‘ (bottom panel). These examples allow us to consider also cusps with small opening angles. The size ofthe isosceles triangles has been changed by varying the angles α adjacent to the basis. Thus, the limitingregimes are the segment ( α = 0) and the semi infinite strip ( α = π ). As for the rhombus, denoting by α theangle indicated in the inset, its limiting regimes are the segment ( α = 0) and the square ( α = π ). The cutoffin the holographic direction has been fixed to ε = 0 .
03 (see the discussion in §B). Increasing the size of thepolygon improves the agreement with the curve given by (3.12) and (3.16), as expected, because ε/P A getscloser to zero. Moreover, the agreement between the numerical data and the analytic curve gets worse as α becomes very small.In Fig. 6 we report the data for e B A found with Surface Evolver for regular polygons with various number N of edges. The agreement with the curve given by (3.12) and (3.16) is quite good and it improves for largerdomains.It is worth emphasizing that, for entangling surfaces ∂A containing corners, the way we have employedto construct the minimal surfaces with Surface Evolver (i.e. by defining ∂A at z = ε ) influences the term W A in the expansion (2.5) for the area, as already remarked in [61].It could be helpful to compute the length P ε of the curve defined as the section at z = ε of the minimalsurface anchored on the long segments of a large wedge with opening angle α , which has been introducedabove. From (3.13) we find that, in terms of polar coordinates whose center is the projection of the vertexat z = ε , this curve is given by ρ = εf ( φ ). Being L (cid:29)
1, we find that P ε reads P ε = 2 ˆ α ε / q ρ + ( ∂ φ ρ ) dφ = 2 ε ˆ α ε / q f + ( ∂ φ f ) dφ = 2 ε ˆ L/εf q f ( ∂ f φ ) df = 2 L − f ε + . . . , (3.17)where α ε ’ α is defined by the relation L = εf ( α ε /
2) and in the last step a change of variable has beenperformed. It is easy to observe that α ε < α . Considering the integral in the intermediate step of (3.17),one notices that it diverges because of its upper limit of integration (see the text below (3.13)), while thelower limit of integration gives a finite result, providing a contribution O ( ε ) to P ε . The expression of ∂ f φ can be read from the integrand of (3.14), finding that f ( ∂ f φ ) = O (1 /f ) when f → + ∞ . Since L/ε (cid:29) f , we obtain that this integral diverges like L/ε − f + . . . ,where the finite term has been found numerically. As a cross check of the finite term, we observe that f = 0when α = π (see below (3.15)), as expected. Thus, we can conclude that P ε = 2 L + O ( ε ), being P A = 2 L the length of the boundary of the wedge at z = 0. Notice that, performing this computation for the minimalsurface anchored on a circle of radius R , which is a hemisphere, one finds that P ε = 2 πR + O ( ε ).Let us remark that P ε is not related to the regularization we adopt in our numerical analysis, as it canbe realized from the right panel of Fig. 4. Indeed, in order to analytically the profiles given by the numericaldata in the right panel of Fig. 4 the ansatz (3.13) cannot be employed and a partial differential equationmust be solved. 12 Figure 7: Minimal surface constructed with Surface Evolver corresponding to a star convex domain delimitedby the red curve given by r ( φ ) = R + a cos( kφ ) in polar coordinates in the z = 0 plane, with R = 1, a = 0 . k = 4. Here the cutoff is ε = 0 .
03 and (
V, F ) = (6145 , The crucial assumption throughout the above discussions is that the minimal surface ˜ γ A can be fully describedby z = z ( x, y ), where ( x, y ) ∈ A . Nevertheless, there are many domains A for which this parameterizationcannot be employed because there are pairs of different points belonging to the minimal surfaces ˜ γ A withthe same projection ( x, y ) / ∈ A in the z = 0 plane. In these cases, being the analytic approach quite difficultin general, one can employ our numerical method to find the minimal surfaces and to compute their area.The numerical data obtained with Surface Evolver would be an important benchmark for analytic resultsthat could be found in the future.An interesting class of two dimensional regions to consider is given by the star shaped domains. Aregion A at z = 0 belongs to this set of domains if a point P ∈ A exists such that the segment connectingany other point of the region to P entirely belongs to A . As for the minimal surface anchored on astar shaped domain A , by introducing a spherical polar coordinates system ( r, φ, θ ) centered in P (theangular ranges are φ ∈ [0 , π ) and θ ∈ [0 , π/ ρ = r sin θ and z = r cos θ , being ( ρ, φ ) the polar coordinates of the z = 0 plane. Some interestinganalytic results about these domains have been already found. In particular, [22] considered minimal surfacesobtained as smooth perturbations around the hemisphere and in [23] the behaviour in the IR regime forgapped backgrounds [63] has been studied. Our numerical method allows a more complete analysis because,within our approximations, we can find (numerically) the area of the corresponding minimal surface withoutrestrictions.In Fig. 7 we show a star convex domain A delimited by the red curve at z = 0, which does not containvertices, and the corresponding minimal surface ˜ γ A anchored on it. Notice that there are pairs of pointsbelonging to ˜ γ A having the same projection ( x, y ) / ∈ A on the z = 0 plane. It is worth recalling that inour regularization the numerical construction of the minimal surface with Surface Evolver has been done bydefining the entangling curve ∂A at z = ε .In order to give a further check of our numerical method, we find it useful to compare our numerical13 (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) 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! ! " " ! ! ! "$% !$% ! !$% z Figure 8: Minimal surfaces corresponding to entangling curves ∂A at z = 0 given by (3.21) with R = 3, k = 4, µ = 0 and for different values of the parameter a , which delimit star shaped domains (red curves inthe inset). In the inset, where the z direction points downward, we show the minimal surfaces constructedthrough Surface Evolver with ε = 0 .
03. In the main plot, the solid curves are their sections of the minimalsurfaces of the inset at φ = π/ a ∈ { . , . , . , . } (red, green, blue and black respectively), while in the inset a increases startingfrom the top left surface and going to the top right, bottom left and bottom right ones.results against the analytic ones obtained in [22], where the equation of motion coming from (2.3) writtenin polar coordinates ( r, φ, θ ) has been linearized to second order around the hemisphere solution with radius R , finding r ( θ, φ ) = R + a r ( θ, φ ) + a r ( θ, φ ) + O ( a ) , (3.18)where the r ( θ, φ ) and r ( θ, φ ) are given by [22] r ( θ, φ ) = [tan( θ/ k (1 + k cos θ ) cos( kφ ) , (3.19) r ( θ, φ ) = [tan( θ/ k R n (1 + k cos θ ) + (cid:2) µ (1 + 2 k cos θ ) + k cos θ (cid:3) cos(2 kφ ) o , (3.20)being k ∈ N and µ ∈ R two parameters of the linearized solution. The minimal surface equation comingfrom (2.3) is satisfied by (3.18) at O ( a ). Notice that r ( θ = 0 , φ ) = r ( θ = 0 , φ ) = 0, which means thatthe maximum value reached by the linearized solution along the z direction is R , like for the hemisphere.Neglecting the O ( a ) terms in (3.18), one has a surface spanning the curve r ( π/ , φ ) ≡ R ( φ ) at z = 0,14 Figure 9: Minimal surfaces constructed with Surface Evolver corresponding to non convex domains at z = 0delimited by the red and blue curves, which are made by arcs of circle centered either in the origin or inthe points identified by the black dots. The green and magenta curves are sections of the minimal surfacesanchored on the red and the blue curves respectively.which reads R ( φ ) ≡ R + a cos( kφ ) + a R (cid:2) µ cos(2 kφ ) (cid:3) . (3.21)In Fig. 8 we construct the minimal surfaces providing the holographic entanglement entropy of some examplesof star shaped regions A delimited by (3.21) where R and µ are kept fixed while a takes different values,taking the φ = π/ a/R and it gets worse as a/R increases, as expected.Our numerical method is interesting because it does not rely on any particular parameterization of thesurface and this allows us to study the most generic non convex domain. In Fig. 9 we show two examples ofnon convex domains A which are not star shaped: one is delimited by the red curve and the other one by theblue curve. We could see these domains as two two dimensional spherocylinders which have been bended ina particular way. Constructing the minimal surfaces ˜ γ A anchored on their boundaries and considering theirsections given by the green and magenta curves, one can clearly observe that some pair of points belongingto the minimal surfaces have the same projection ( x, y ) / ∈ A on the z = 0 plane, as already remarked above.An analytic description of these surfaces is more difficult with respect to the minimal surfaces anchored onthe boundary of star shaped domains because it would require more patches.15 Figure 10: Minimal surface constructed with Surface Evolver for a domain A = A ∪ A delimited bytwo disjoint and equal ellipses at z = 0 (blue curves). Here ε = 0 .
03 and the minimal surface is anchoredon ∂A defined at z = ε , according to our regularization prescription. The minimal surface has ( V, F ) =(18936 , E can be found from the Euler formula with vanishing genus and twoboundaries). Only half surface is shown in order to highlight the curves given by the two sections suggestedby the symmetry of the surface. In this section we discuss the main result of this paper, which is the numerical study of the holographic mutualinformation of disjoint equal domains delimited by some of the smooth curves introduced in §3.1. For twoequal disjoint ellipses, an explicit example of the minimal surface whose area determines the correspondingholographic mutual information is shown in Fig. 10.Let us consider two dimensional domains A = A ∪ A made by two disjoint components A and A , whereeach component is a simply connected domain delimited by a smooth curve. The boundary is ∂A = ∂A ∪ ∂A and the shapes of ∂A and ∂A could be arbitrary, but we will focus on the geometries discussed in §3. Sincethe area law holds also for S A ∪ A and P A = P A + P A , the leading divergence O (1 /ε ) cancels in thecombination (1.2), which is therefore finite when ε → I A ,A as follows I A ,A ≡ I A ,A G N , (4.1)where G N is the four dimensional Newton constant. Since ∂A and ∂A are smooth curves, from (2.4) and(3.8) we have I A ,A = e F A ∪ A − e F A − e F A = F A ∪ A − F A − F A + o (1) . (4.2)In the following we study I A ,A when ∂A is made either by two circles (§4.1.2) or by two superellipses or16y the boundaries of two two dimensional spherocylinders. Once A , A and their relative orientation havebeen fixed, we can only move their relative distance. A generic feature of the holographic mutual informationis that it diverges when A and A become tangent, while it vanishes when the distance between A and A is large enough. In this section we consider domains A whose boundary ∂A is made by two disjoint circles. The correspondingdisks can be either overlapping (in this case A is an annulus) [51–53] or disjoint [54, 55]. Let us consider the annular region A bounded by two concentric circles with radii R in < R out . The comple-mentary domain B is made by two disjoint regions and, since we are in the vacuum, S A = S B . The minimalsurfaces associated with this case have been already studied in [51, 53] as the gravitational counterpart ofthe correlators of spatial Wilson loops and in [52] from the holographic entanglement entropy perspective.In §D.2 we discuss the construction of the analytic solution in D dimensions for completeness, but herewe are interested in the D = 2 case. Because of the axial symmetry, it is convenient to introduce polarcoordinates ( ρ, φ ) at z = 0. Then, the profile of the minimal surface is completely specified by a curve in theplane ( ρ, z ).A configuration providing a local minimum of the area functional is made by the disjoint hemispheresanchored on the circles with radii R in and R out . In the plane ( ρ, z ), they are described by two arcs centeredin the origin with an opening angle of π/ ∂A that could give a local minimum of the area functional is the connected one having the same topologyof a half torus. This solution is fully specified by its profile curve in the plane ( ρ, z ), which connects thepoints ( R in ,
0) and ( R out , A , a transition occurs between these two types of surfaces, aswe explain below. This is the first case that we encounter of a competition between two saddle points of thearea functional.The existence of the connected solution depends on the ratio η ≡ R in /R out <
1. As discussed in §D.2,a minimal value η ∗ can be found such that for 0 < η < η ∗ only the disconnected configuration of twohemispheres exists, while for η ∗ < η <
1, besides the disconnected configuration, there are two connectedconfigurations which are local minima of the area functional (see Fig. 23). In the latter case, one has to findwhich of these two connected surfaces has the lowest area and then compare it with the area of the twodisconnected hemispheres. This comparison provides a critical value η c > η ∗ such that when η ∈ ( η c ,
1) theminimal surface is given by the connected configuration, while for η ∈ (0 , η c ) the minimal area configurationis the one made by the two disjoint hemispheres.Let us give explicit formulas about these surfaces by specifying to D = 2 the results found in §D.2 (inorder to simplify the notation adopted in §D.2, in the following we report some formulas from that appendixomitting the index D ). The profile of the radial section of the connected minimal surface in the plane ( ρ, z )is given by the following two branches ( ρ = R in e − f − ,K ( z/ρ ) ,ρ = R out e − f + ,K ( z/ρ ) , (4.3)where, by introducing ˜ z ≡ z/ρ , the functions f ± ,K (˜ z ) are defined as follows (from (D.11)) f ± ,K (˜ z ) ≡ ˆ ˜ z λ λ ± λ p K (1 + λ ) − λ ! dλ , (cid:54) ˜ z (cid:54) ˜ z m , ˜ z m = K + p K ( K + 4)2 . 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While the external radiusis kept fixed to R out = 1, for the internal one the values R in = 0 .
38 (red), 0 . . ε = 0 .
03 and, according to our regularization prescription, ∂A has been definedat z = ε in the numerical construction. Right panel: The sign of ∆ A establishes the minimal area surfacebetween the connected surface and the two disjoint hemispheres. The black curve is obtained from (4.12)by varying K > η = η ∗ , where the lower one corresponds tothe connected solution which is not the minimal one between the two connected ones. The data points havebeen found with Surface Evolver for various annular domains. Notice that in the left panel η < η c only forthe red curve.The integral occurring in f ± ,K can be computed in terms of the incomplete elliptic integrals of the first andthird kind (see §E), finding f ± ,K (˜ z ) = 12 log(1 + ˜ z ) ± κ r − κ κ − h F (cid:0) ω (˜ z ) | κ (cid:1) − Π (cid:0) − κ , ω (˜ z ) | κ (cid:1)i , (4.5)where we have introduced ω (˜ z ) ≡ arcsin ˜ z/ ˜ z m p κ (˜ z/ ˜ z m − ! , κ ≡ s z m z m . (4.6)The matching condition of the two branches (4.3) provides a relation between η (cid:62) η ∗ and the constant K ,namely (from (D.13))log( η ) = − ˆ ˜ z m λ (1 + λ ) p K (1 + λ ) − λ dλ = 2 κ r − κ κ − (cid:16) K (cid:0) κ (cid:1) − Π (cid:0) − κ , κ (cid:1)(cid:17) , (4.7)where K ( m ) and Π( n, m ) are the complete elliptic integrals of the first and third kind respectively.The relation (4.7) tells us η = η ( K ) and κ ∈ [1 / √ , η ∗ = 0 . η ∈ ( η ∗ , K fulfilling thematching condition (4.7). This means that, correspondingly, there are two connected surfaces anchored on18he same pair of concentric circles on the boundary which are both local minima of the area functional. Wehave to compute their area in order to establish which one has to be compared with the configuration ofdisjoint hemispheres to find the global minimum.Performing the following integral up to an additive constant (from (D.20) for D = 2) ˆ d ˜ z ˜ z p z − ˜ z /K = p (˜ z m − ˜ z )(˜ z m + ˜ z ˜ z m + ˜ z )˜ z ˜ z m + E (cid:0) arcsin(˜ z/ ˜ z m ) | κ (cid:1) + ( κ − F (cid:0) arcsin(˜ z/ ˜ z m ) | κ (cid:1) √ κ − , (4.8)one obtains the area of the connected surface [53, 61] A con = 2 π ˆ ˜ z m ε/R out d ˜ z ˜ z p z − ˜ z /K + ˆ ˜ z m ε/R in d ˜ z ˜ z p z − ˜ z /K ! (4.9)= 2 π ( R in + R out ) ε − π √ κ − (cid:16) E (cid:0) κ (cid:1) − (1 − κ ) K (cid:0) κ (cid:1)(cid:17) + O ( ε ) . (4.10)Plotting the O (1) term of this expression in terms of K , it is straightforward to realize that the minimalarea surface between the two connected configurations corresponds to the smallest value of K .As for the area of the configuration made by two disconnected hemispheres, from (D.23) one gets A dis = 2 π ˆ ∞ ε/R in d ˜ z ˜ z √ z + ˆ ∞ ε/R out d ˜ z ˜ z √ z ! = 2 π ( R in + R out ) ε − π + O ( ε ) . (4.11)We find it convenient to introduce ∆ A ≡ A dis −A con , which is finite when ε →
0. In particular, ∆
A → π ∆ R as ε →
0, where ∆ R is (D.27) evaluated at D = 2. From (4.10) and (4.11), we havelim ε → ∆ A = 4 π E (cid:0) κ (cid:1) − (1 − κ ) K (cid:0) κ (cid:1) √ κ − − ! . (4.12)Considering as the connected surface the one with minimal area, the sign of ∆ A determines the minimalsurface between the disconnected configuration and the connected one and therefore the global minimum ofthe area functional. The root η c of ∆ A can be found numerically and one gets η c = 0 .
419 [45, 51]. Thus, theconnected configuration is minimal for η ∈ ( η c , η ∈ (0 , η c ) the minimal area configuration is theone made by the disjoint hemispheres.By employing Surface Evolver, we can construct the surface anchored on the boundary of the annulus at z = 0 which is a local minimum, compute its area and compare it with the analytic results discussed above.This is another important benchmark of our numerical method.In the left panel of Fig. 11 we consider the profile of the connected configuration in the plane ( ρ, z ). Theblack dots correspond to the radial section of the surface obtained with Surface Evolver, while the solidline is obtained from the analytic expressions discussed above. Let us recall that the triangulated surface isnumerically constructed by requiring that it is anchored to the two concentric circles with radii R in < R out at z = ε and not at z = 0, as it should. Despite this regularization, the agreement between the analyticresults and the numerical ones is very good for our choices of the parameters. It is worth remarking that,when η (cid:62) η ∗ and therefore two connected solutions exist for a given η , Surface Evolver finds the minimalarea one between them. Nevertheless, it is not able to establish whether it is the global minimum. Indeed,for example, the red curve in the left panel of Fig. 11 has η ∗ < η < η < η ∗ , even if one beginswith a rough triangulation of a connected surface, Surface Evolver converges towards the configuration madeby the two disconnected hemispheres.In the right panel of Fig. 11 we compare the values of ∆ A obtained with Surface Evolver with the analyticcurve from (4.12), finding a very good agreement. Numerical points having η ∗ < η < η c are also found, forthe reason just explained. 19igure 12: The connected surface anchored on the boundary of an annulus at z = 0 (top left panel), whichis a local minimum of the area functional, can be mapped through (4.13) into one of the connected surfacesanchored on the configurations of circles at z = 0 shown in the remaining panels, depending on the value ofthe parameter of the transformation (4.13), as discussed in §4.1.2. The mapping preserves the color code.The green circle in the top left panel corresponds to the matching of the two branches given by (4.3) and(4.7) (see the point P m in Fig. 23) and it is mapped into the vertical circle in the bottom right panel. In this section we consider domains A made by two disjoint disks by employing the analytic results for theannulus reviewed in §4.1.1 and some isometries of H . This method has been used in [64] for the case of acircle, while the case of two disjoint circles has been recently studied in [54, 55]. The analytic results foundin this way provide another important benchmark for the numerical data obtained with Surface Evolver.Let us consider the following reparameterizations of H , which correspond to the special conformaltransformations on the boundary [64]˜ x = x + b x ( | v | + z )1 + 2 b · v + | b | ( | v | + z ) , ˜ y = y + b y ( | v | + z )1 + 2 b · v + | b | ( | v | + z ) , ˜ z = z b · v + | b | ( | v | + z ) , (4.13)being b ≡ ( b x , b y ) a vector in R and v ≡ ( x, y ).When z = 0 in (4.13), the maps ( x, y ) → (˜ x, ˜ y ) are the special conformal transformations of the Euclideanconformal group in two dimensions. These transformations in the z = 0 plane send a circle C with center c = ( c x , c y ) and radius R into another circle e C with center ˜ c = (˜ c x , ˜ c y ) and radius e R which are given by˜ c i = c i + b i ( | c | − R )1 + 2 b · c + | b | ( | c | − R ) i ∈ { x, y } , e R = R (cid:12)(cid:12) b · c + | b | ( | c | − R ) (cid:12)(cid:12) . (4.14)Notice that the center ˜ c is not the image of the center c under (4.13) with z = 0. Moreover, when c is suchthat the denominator in (4.14) vanishes, the circle is mapped into a straight line [64].20 Figure 13: Two examples of minimal surfaces (constructed with Surface Evolver) corresponding to A madeby two disjoint and equal disks ( ∂A is given by the red and blue circles). Only half of the surfaces is shownin order to highlight their section through a plane orthogonal to z = 0 and to the segment connecting thecenters. This section provides a circle whose radius and center are given in (4.20). In this figure ε = 0 . R = 1 and the distance between their centers is d = 2 .
16, while for the blue ones R = 0 .
75 and d = 1 . z = 0 with radii R in < R out , their images are two different circlesat z = 0 which do not intersect. In order to deal with simpler expressions for the mapping, let us placethe center of the concentric circles in the origin, i.e. c = (0 , η ≡ R in /R out < e R ≡ R in / | − | b | R in | and e R ≡ R out / | − | b | R out | the radii of the circles after the mapping, the distance between the two centers reads d = (1 − η ) β | (1 − β )( β − η ) | R in = (1 − η ) β | β − η | e R , (4.15)where β ≡ | b | R in . Thus, η and β fix the value of the ratio ˜ δ ≡ d/ e R . The final disks are either disjoint orfully overlapping, depending on the sign of the expression within the absolute value in the denominator of(4.15). In particular, when β ∈ ( η ,
1) the two disks are disjoint, while when β ∈ (0 , η ) ∪ (1 , + ∞ ) they21 (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1) ! ! ! 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The coloured solid lines are the numerical results found withSurface Evolver for the shapes indicated in the common legend in the right panel. Here R = 1 and ε = 0 . z = 0) correspond to the minimal surface for two disjoint circles andthey have been found by mapping the connected minimal surface for the annulus through the transformations(4.13) (see §4.1.2 and Fig. 12). The dashed curve corresponds to two infinite strips. Right: Zoom of the partof the left panel enclosed by the black rectangle.overlap. As for their ratio ˜ η ≡ e R / e R , we find˜ η = β − η η ( β − β ∈ (0 , η ) ∪ (1 , ∞ ) overlapping disks , β − η η (1 − β ) β ∈ ( η ,
1) disjoint disks . (4.16)Notice that ˜ η → /η > β → ∞ . Thus, given η and β , the equations (4.15) and (4.16) provide ˜ δ and˜ η . By inverting them, one can write η and β in terms of ˜ δ and ˜ η . The system is made by two quadraticequations and some care is required to distinguish the various regimes.When the disks after the mapping are disjoint, i.e. η < β <
1, an interesting special case to discussis e R = e R , namely when the disjoint disks have the same radius e R = R in / (1 − η ) = R out / ( η − − R in < R out the radii of the two concentric circles at z = 0 centered in the origin. Setting ˜ η = 1 in (4.16), onefinds that it happens for β = η , i.e. | b | = 1 / ( R in R out ). The distance corresponding to this value of β canbe found from (4.15) and it is given by d/R in = (1 + η ) / (cid:2) √ η (1 − η ) (cid:3) or, equivalently, by ˜ δ = (1 + η ) / √ η . Byinverting this relation, one finds η (˜ δ ) = (cid:8) ˜ δ − − (cid:2) (˜ δ − − (cid:3) / (cid:9) /
2, where the root η (˜ δ ) < δ > b = ( b x , b y ) = | b | (cos φ b , sin φ b ) is chosen by fixing the initial and final configurationsof circles at z = 0, the transformations (4.13) for the points in the bulk are fixed as well and they can beused to map the points belonging to the minimal surfaces spanning the initial configuration of circles. Inparticular, let us consider a circle given by ( R ? cos φ, R ? sin φ, z ? ) for φ ∈ [0 , π ), lying in a plane at z = z ? parallel to the boundary. This circle is mapped through (4.13) into another circle b C whose radius is given by b R = R ? p | b | ( z ? − R ? ) + | b | ( z ? + R ? ) , (4.17)22nd whose center ˆ c ≡ (ˆ c x , ˆ c y , ˆ c z ) has coordinatesˆ c i = | b | ( R ? + z ? ) + z ? − R ? | b | ( z ? − R ? ) + | b | ( z ? + R ? ) b i i ∈ { x, y } , ˆ c z = [1 + | b | ( R ? + z ? )] z ? | b | ( z ? − R ? ) + | b | ( z ? + R ? ) . (4.18)Setting z ? = 0, R ? = R and b R = e R in (4.17) and (4.18), the expressions in (4.14) with c = (0 ,
0) arerecovered. The circle b C lies in a plane orthogonal to the following unit vector v ⊥ = ( − cos φ b sin θ ⊥ , − sin φ b sin θ ⊥ , cos θ ⊥ ) , θ ⊥ ≡ arcsin(2 z ? | b | b R/R ? ) , (4.19)where 2 z ? | b | b R/R ? <
1, as can be easily observed from (4.17).In the top left panel of Fig. 12 we consider as initial configuration the annulus at z = 0 for some givenvalue of η and the corresponding connected minimal surface in the bulk anchored on its boundary, whichhas been discussed in §4.1.1. The transformation (4.13) with β = √ η maps this surface into the connectedsurface anchored on two equal and disjoint circles (bottom right panel in Fig. 12). It is interesting to followthe evolution of the former surface into the latter one as β ∈ [0 , √ η ] increases: in Fig. 12 we show twointermediate steps where the surfaces are qualitatively different and they correspond to different regimes of β separated by β = η . For 0 < β < η the disks at z = 0 are still overlapping but they are not concentric(top right panel of Fig. 12). Within this range of β , the radius of the largest disk, which is R out / | − β /η | ,increases with β and it diverges when as β → η . When η < β (cid:54) √ η , instead, the disks at z = 0 are disjointand the images of the initial surface through (4.13) are shown in the bottom panels of Fig. 12, where thesurface on the left has η < β < √ η , while the one on the right corresponds to the final stage of disjointequal disks ( β = √ η ). In Fig. 12 the mapping preserves the color code and we have highlighted the greencircle because in the top left panel it corresponds to the circle at z = z m along which the two branches givenby (4.3) match, as imposed by the condition (4.7). When β = √ η , this matching circle is mapped into thevertical one shown in the bottom right panel, whose radius e R v and whose coordinate z v > e R v of its centeralong the holographic direction are given respectively by e R v = 1 − η z m √ η e R , z v = (1 − η ) p z m z m √ η e R , (4.20)where e R is the radius of the two equal disjoint disks written above and ˜ z m is a function of η (see (4.4) and(4.7)). In Fig. 13 we show two examples of minimal surfaces constructed with Surface Evolver which providethe holographic mutual information of two equal disjoint disks. Considering the section of these surfacesthrough a vertical plane which is orthogonal to the boundary and to the line passing through the centers ofthe disks, we find a good agreement with (4.20).As for the finite part of the area, once η and β have been written in terms of ˜ η and ˜ δ by inverting(4.15) and (4.16), the limit ε → A or I A ,A (depending on whether the final disks are eitheroverlapping or disjoint respectively) is given by the r.h.s. of (4.12), where κ = κ ( η ) is obtained through thenumerical inversion of (4.7), being η = η (˜ δ, ˜ η ) found above.The special case of two equal disjoint disks corresponds to ˜ η = 1 and ˜ δ = (1 + η ) / √ η , and therefore thelimit ε → I A ,A depends only on the parameter ˜ δ , as expected. The relation ˜ δ = (1 + η ) / √ η can beused to find the critical distance d c between the centers beyond which the holographic mutual informationvanishes and also the distance d ∗ > d c beyond which the connected surface does not exist anymore. Theycorrespond to η c and η ∗ respectively and, in particular, one gets ˜ δ c = 2 .
192 and ˜ δ ∗ = 2 . z = 0), while thered curve is the section of the corresponding surface constructed by Surface Evolver (see also the red curvesin Fig. 10 for a similar construction with different A ). In Fig. 15 we have performed another comparisonbetween the analytic expressions and the numerical data of Surface Evolver by computing the holographic23 (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2)(cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3)(cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4)(cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5)(cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2)(cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:6)(cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7)(cid:8) (cid:8) (cid:8) (cid:8) (cid:8) (cid:8) (cid:8) (cid:8) (cid:8) (cid:8) (cid:8) (cid:8) (cid:8) (cid:8) (cid:8) (cid:8)(cid:9) (cid:9) (cid:9) (cid:9) (cid:9) (cid:9) (cid:9) (cid:9) (cid:9) (cid:9) (cid:9) (cid:9) (cid:9) (cid:9) (cid:9) (cid:9) (cid:9) (cid:9)(cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:6) (cid:2)(cid:3)(cid:4)(cid:5)(cid:2)(cid:6)(cid:7)(cid:8)(cid:9)(cid:6) !" squircles n = 3squircles n = 4squircles n = 6squircles n = 8circles d/R I A ,A R = 2 R = 1 Figure 15: Holographic mutual information of two disjoint and equal domains delimited by squircles forvarious n . The coloured points are the numerical data obtained with Surface Evolver, while the blacktriangles correspond to the solid black curve of Fig. 11 (right panel) mapped through the transformation(4.16) with β = η . The transition between the connected surface and the configuration of disconnectedsurfaces occurs at the zero of each curve. A point having I A ,A < A made by two equal disjoint disks. The black triangles have been found bymapping the black curve for the annulus in the right panel of Fig. 11 (which is given by the r.h.s. of (4.12))through η = η (˜ δ ) found above. The agreement with the corresponding data obtained with Surface Evolver(red curve) is very good. Notice that, as already observed for the annulus in §4.1.1, also in this case SurfaceEvolver finds a surface which is a local minimum of the area functional, even if it is not the global minimum.Let us conclude by emphasizing that, while this numerical method is very efficient in finding surfaces whichare local minima for the area functional when they exist, it is not suitable for studying the existence of asurface with a given topology. In §4.1.2 we have considered the holographic mutual information of two disjoint circular domains, for whichanalytic results are available. When A = A ∪ A is not made by two disjoint disks, analytic results for thecorresponding holographic mutual information are not known and therefore a numerical approach could bevery useful. Here we employ Surface Evolver to study I A ,A (defined in (4.1)) of disjoint regions delimitedby some of the smooth curves introduced in §3.1.The holographic mutual information of non circular domains depends on the geometries of their bound-aries, on their distance and also on their relative orientation. Independently of the shapes of ∂A and ∂A ,once the domains and their relative orientation have been fixed, the holographic mutual information vanishes24 /R d/R d/R d/R I A ,A I A ,A R R R R I A ,A !"! Figure 16: Holographic mutual information of two equal and disjoint domains delimited by ellipses (toppanels) or superellipses with n = 4 (bottom panels), which are defined by R and R (see the bottom panelof Fig. 1 and (3.9)), while d is the distance between their centers. The relative orientation is like in Fig. 10.Left panels: Density plots for I A ,A whose zero provides the corresponding transition curve (solid blackline) in the plane ( d/R , R /R ). The straight vertical line indicates the transition when A is made by twoequal and disjoint infinite strips whose width is 2 R and the distance between their central lines is d . Rightpanels: I A ,A in terms of d/R for various fixed values of R /R indicated by the horizontal dashed linesin the corresponding left panel, with the same color code. The lower curves (orange) in the right panelscorrespond to the squircles ( R = R ) with n = 2 (top) and n = 4 (bottom) and therefore they reproducethe red and orange curves in Fig. 15 respectively. The data reported here have been found with R = 1 andsome checks have been done also with R = 2. 25 A ,A R R d/R d/R Figure 17: Holographic mutual information of two equal and disjoint two dimensional spherocylindersoriented like the two ellipses in Fig. 10. The parameters R and R specify the domains (see the bottompanel of Fig. 1 and (3.10)) and d is the distance between their centers. The same notation and color codingof Fig. 16 has been adopted.when the distance between A and A is large enough. The critical distance d c beyond which I A ,A = 0depends on the configuration of the domains. This transition occurs because, for a generic distance d betweenthe centers of A and A , the global minimal area surface comes from a competition between a connected sur-face anchored on ∂A and a configuration made by two disconnected surfaces spanning ∂A and ∂A , whichare both local minima. Beyond the critical distance between the centers, the disconnected configurationbecomes the global minimum and therefore I A ,A vanishes.In Fig. 10 we show an example of a connected surface constructed with Surface Evolver where ∂A is madeby two equal and disjoint ellipses at z = 0. Let us recall that in our numerical analysis we have regularizedthe area by defining ∂A at z = ε , as discussed in §B. In the figure, we have highlighted two sections of thesurface suggested by the symmetry of this configuration of domains, which are given by the red curves andby the green one.We have constructed minimal area connected surfaces also for configurations of equal disjoint domainswith other shapes and in Fig. 14 we have reported the corresponding curves obtained from the section givingthe red curves in Fig. 10. The red curves in Fig. 14 are associated with circular domains and they canbe recovered analytically (black dots), as explained in §4.1.2. Instead, for the remaining curves analyticexpressions are not available and therefore they provide a useful benchmark for analytic results that couldbe found in the future.Besides the profiles for various sections, Surface Evolver computes also the area of the surfaces that itconstructs. Considering a configuration of disjoint domains with given shapes and relative orientation, wecan compute I A ,A while the distance d between their centers changes. In Fig. 15 we show the resultsof this analysis when ∂A and ∂A are squircles (i.e. (3.9) with R = R ≡ R ). As for their relativeorientation, drawing the squares that circumscribe ∂A and ∂A , their edges are parallel. Since I A ,A (cid:62) d c corresponds to the zero of the various curves and I A ,A vanishes for d (cid:62) d c . Thus, I A ,A is continuos with a discontinuous first derivative at d = d c . The points found numerically which have I A ,A < z Figure 18: Minimal surfaces obtained with Surface Evolver for a domain A = A ∪ A made by the interiorof two disjoint and equal squares. All the squares have the same size but the relative orientation of A and A is different in the two panels.Once the relative orientation has been chosen, a configuration of two equal and disjoint squircles iscompletely determined by two parameters: the distance d between the centers and the size R of the squircles.Instead, when A and A are two equal two dimensional spherocylinders or equal domains delimited by twodisjoint superellipses and the relative orientation has been chosen, we have three parameters to play with:the distance d between the centers and the parameters R and R which specify the two equal domains (seethe bottom panel of Fig. 1). In Fig. 16 we show I A ,A for two disjoint domains delimited by ellipses andsuperellipses with n = 4, whose relative orientation is like in Fig. 10. In the left panels, the black thick curveis the transition curve along which the holographic mutual information vanishes, while the continuos straightline identifies the transition value corresponding to two disjoint infinite strips [42]. Comparing the transitioncurve in the top left panel with the one in the bottom left panel, it is evident that the one associated with thesuperellipses having n = 4 is closer to the value corresponding to the infinite strips than the one associatedwith the ellipses. In Fig. 17 we study I A ,A for a domain A made by two equal and disjoint two dimensionalspherocylinders. In this case the transition curve is closer to the line corresponding to the transition for twoinfinite strips with respect to the transition curves of Fig. 16. Nevertheless, from our data we cannot concludethat the transition curve for the two dimensional spherocylinders approaches the value corresponding to theinfinite strips as R /R → ∞ . It would be interesting to have further data and some analytic argument tounderstand whether some bounds prevent the transition curves to approach the value associated with theinfinite strips for R /R → ∞ . Let us remark that the lowest curves (orange) in the right panels of Figs. 16and 17 correspond to disjoint squircles with n = 2 (i.e. circles) or n = 4 and therefore they reproduce thered and the orange curves of Fig. 15. Configurations of domains having smaller values of d than the onesshown in the plots provide unstable numerical results.By employing Surface Evolver, we could also study the holographic mutual information of disjoint domainswhose boundaries contain corners. In particular, one could take both A and A bounded by polygons,but also A bounded by a smooth curve and A by a polygon. In Fig. 18 we show the minimal areasurfaces corresponding to ∂A made by two equal and disjoint squares having different relative orientation.As discussed in §3.2, when ∂A has vertices a further logarithmic divergence occurs after the area law termin the ε → B A ∪ A = B A + B A for two disjoint regions, then the holographic mutual information is finite. Anexpression like (3.12) with the sum extended over the vertices of both the components of ∂A is additive,leading to a finite I A ,A . Also for these cases we could find plots similar to Figs. 16 and 17 but the curveswould not be suitable for a comparison with an analytic formula because of the regularization procedurethat we have adopted. Indeed, in our numerical computations ∂A is defined at z = ε and this regularizationaffects the O (1) term in (2.5) [61], as already mentioned in the closing part of §3.2.27igure 19: Minimal surface corresponding to three disjoint and equal red circles in the plane z = 0 (the z axis points downward). This surface has 13147 vertices and 26624 faces, while the number of edges is givenby Euler formula with vanishing genus and 3 boundaries. This kind of surfaces occurs in the computationof the holographic tripartite information for the union of three disjoint disks. In this paper we have studied the area of the minimal surfaces in AdS occurring in the computation of theholographic entanglement entropy and of the holographic mutual information, focussing on their dependenceon the shape of the entangling curve ∂A in the boundary of AdS .Our approach is numerical and the main tool we have employed is the program Surface Evolver, whichallows to construct triangulated surfaces approximating a surface anchored on a given curve ∂A which is alocal minimum of the area functional. We have computed the holographic entanglement entropy and theholographic mutual information for entangling curves given by (or made by the union of) ellipses, superellipsesor the boundaries of two dimensional spherocylinders, for which analytic expressions are not known. We havealso obtained the transition curves for the holographic mutual information of disjoint domains delimited bysome of these smooth curves (see Figs. 15, 16 and 17), providing a solid numerical benchmark for analyticexpressions that could be found in future studies. We focused on these simple examples, but the methodcan be employed to address more complicated domains.Besides the fact that the surfaces constructed by Surface Evolver are triangulated, a source of approx-imation in our numerical analysis is the way employed to define the curve spanning the minimal surface.Indeed, once the cutoff ε > ∂A at z = ε . It would be interesting to understandbetter this regularization with respect to some other ones and also to decrease ε in a stable and automaticallycontrolled way in order to get numerical data which provide better approximations of the analytic results.There are many possibilities to extend our work. The most important ones concern black hole geometriesand higher dimensional generalizations. An interesting extension involves domains A made by three or moreregions (see [65] for some results in two dimensional conformal field theories and [66–68] for a holographicviewpoint). In Fig. 19 we show a minimal surface anchored to an entangling curve made by three disjointcircles. The area of this surface provides the holographic entanglement entropy between the union of the threedisjoint disks and the rest of the plane, which is the most difficult term to evaluate in the computation of theholographic tripartite information [66]. Another important application of the numerical method employed28ere involves time dependent backgrounds modelling the holographic thermalization [11, 69–74].Surface Evolver is a useful tool to get numerical results for the holographic entanglement entropy, whichcan be used to test analytic formulas that could be found in the future. Acknowledgments
It is our pleasure to thank Ioannis Papadimitriou for collaboration in the initial part of this project and formany useful discussions during its development. We wish to thank Hong Liu, Rob Myers, Mukund Ranga-mani, Domenico Seminara, Tadashi Takayanagi and in particular Mariarita de Luca, Veronika Hubeny,Alessandro Lucantonio for useful discussions. We acknowledge Veronika Hubeny, Hong Liu, Mukund Ranga-mani, Tadashi Takayanagi and Larus Thorlacius for their comments on the draft. E.T. is grateful to PerimeterInstitute and to the Center for Theoretical Physics at MIT for the warm hospitality during parts of thiswork. L.G. has been supported by The Netherlands Organization for Scientific Research (NWO/OCW). A.S.has been supported by the Spanish Ministry of Economy and Competitiveness under grant FPA2012-32828,Consolider-CPAN (CSD2007-00042), the grant SEV-2012-0249 of the “Centro de Excelencia Severo Ochoa”Programme and the grant HEPHACOS-S2009/ESP1473 from the C.A. de Madrid. E.T. has been supportedby the ERC under Starting Grant 279391 EDEQS.
A Further details on minimal surfaces in H . Let us consider the area of a two dimensional surface γ A embedded in spatial slice t = const A [ γ A ] = ˆ γ A d A = ˆ U A √ h du du z , (A.1)where U A is a coordinate patch. As mentioned in §2, A can be interpreted as the energy of a two dimensionalinterface immersed in R endowed with a potential energy of density 1 /z . To find the surface ˜ γ A minimizing A we consider a small displacement along the normal direction N , parametrized as: R → R + w N , where R represents the position of a point on the surface and w is a small normal displacement. The linear areavariation can be straightforwardly calculated using classic differential geometry [75] δ A [ γ A ] = ˆ U A δ (cid:0) √ h du du (cid:1) z + ˆ U A δ (cid:18) z (cid:19) √ h du du = − ˆ U A z (cid:18) H + ˆ z · N z (cid:19) w du du , (A.2)where H is the surface mean curvature. Setting δ A [ γ A ] to zero yields (2.6).In a Monge patch ( u , u ) = ( x, y ) and the surface can be represented as the graph of the function z = z ( x, y ) representing the height of the surface above the ( x, y ) plane. In this case the mean curvaturereads H = z ,xx (1 + z ,y ) + z ,yy (1 + z ,x ) − z ,xy z ,x z ,y z ,x + z ,y ) / , (A.3)while the outward directed normal vector is given by N = − z ,x ˆ x + z ,y ˆ y − ˆ z q z ,x + z ,y . (A.4)Using Eqs. (A.3) and (A.4) in (2.6) yields the Cartesian equation (3.3).In §2 we argued that a surface described by (2.6) must be orthogonal to the z = 0 plane. This orthogo-nality implies that the boundary curve ∂ ˜ γ A is a geodesic of ˜ γ A . To see this we can recall that the curvature κ of a curve that lies on a surface can be decomposed as κ n = κ n N + κ g ( N × t ) , (A.5)29igure 20: Example of a typical evolution obtained by Surface Evolver in the case of a circular boundary.The initial configuration consists of an octagonal prism composed of 40 triangles (left). The shape is thenoptimized and refined as described in §B, finding the final configuration given by the rightmost surface,which consists of 10240 triangles and yields e F A = 1 . π whereas F A = 2 π is the exact value from theanalytic result (3.1). In this example the radius of the circle is R = 1 and ε = 0 . t the tangent vector of ∂ ˜ γ A , κ n = t ,s (with s the arc lenght) and κ n and κ g the normal and geodesiccurvature respectively. Since ∂ ˜ γ A lies on the z = 0 plane and ˆ z · N = 0 at z = 0, then N = ± n where thechoice of the sign is conventional. By virtue of (A.5) this implies that κ g = 0. Thus ∂ ˜ γ A is a geodesic over˜ γ A .An interesting consequence of the previous statement is that the total Gaussian curvature of the surfaceis constant, regardless the shape of the boundary in the z = 0 plane. The Gauss-Bonnet theorem tells usthat ˆ ˜ γ A K G √ h du du + ˛ ∂ ˜ γ A κ g ds = 2 πχ , (A.6)where K G is the Gaussian curvature and χ is the Euler characteristic. Since κ g = 0 in our case, we have ˆ ˜ γ A K G √ h du du = 2 πχ . (A.7)Let us recall that the Euler characteristic is χ = 2 − g − b , where g is the genus of the surface and b is thenumber of its boundaries. B Numerical Method
The numerical results presented in §3 and §4 have been obtained with Surface Evolver [56, 57]. This is amultipurpose shape optimization program created by Brakke [56] in the context of minimal surfaces andcapillarity and then expanded to address generic problems on energy minimizing surfaces. A surface isimplemented as a simplicial complex, i.e. a union of triangles. Given an initial configuration of the surface,the program evolves the surface toward a local energy minimum by a gradient descent method. The energyused in our calculations is the H area function given in (2.3).The initial configuration is preferably very simple and contains only the least number of triangles necessaryto achieve a given surface topology (Fig. 20). A typical evolution consists in a sequence of optimization andmesh-adjustment steps. During an optimization step, the coordinates of the vertices are updated by a localminimization algorithm (conjugate gradient in our case), resulting in a configuration of lower energy. Thetopology of the mesh (i.e. the number of vertices, faces and edges) is not altered during minimization.A mesh-adjustment step, on the other hand, consists of a set of operations whose purpose is to renderthe discretized surface smooth and uniform. These operations can be broadly divided in two class: mesh-refinements and mesh-repairs. In a mesh-refinement operation a finer grid is overlaid on the coarse one.This is obtained, for instance, by splitting a triangle in four smaller triangle obtained by joining the midpoints of the original edges. In a mesh-repair operation, the triangles that are too distorted compared to theaverage are eliminated. This operation can change the topology of the mesh and possibly also the topologyof the surface which can then breakup into two or more connected parts. This happens, for instance, in the30 (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1)(cid:1) (cid:1)(cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) !" ! +&,--%, . / /&/+ /&/( /&/% /&/* /&/, /&/0 /&/' /&/) /&/-%&0/%&0,%&'/ (cid:1) ! 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" " (cid:2) ! " " " (cid:3) ! " " $ (cid:4) ! " " R = 1 R = 2 R = 5 R = 10 ε/R F A R Figure 21: The quantity e F A (see (3.8)) computed with Surface Evolver for ellipses having R = 2 R (see thebottom panel of Fig. 1), for various R and ε . When ε/R is too small, our numerical data are not stable.The fitted value on the vertical axis is 3 . d A = √ h/z du du at z = 0, the boundary curves used in thenumerical work have been defined on the plane z = ε . In order to maximize the accuracy of the numericalsolution, it is preferable to choose value of ε that is much smaller than any other length scale in the problemand yet large enough to allow the convergence of the optimization steps. With this goal in mind, we haveadopted an empirical selection criterion based on the following procedure. Let ∂ ˜ γ A be an ellipse and let R and R = R / e F A for various choices of ε and R . In the limit of ε → e F A /R is expected toapproach a finite value, but from the data shown in Fig. 21 we see that for ε/R < .
02, the accuracy of thenumerical calculation starts to drop. Based on this numerical evidence we have set in most of our numericalcalculations ε/R = 0 .
03, where R is the typical length scale of the boundary. It is worth remarking that inour numerical computations it is easier (namely the evolution is more stable) to deal with smaller values of ε/R by increasing R than by decreasing ε . Smaller values of ε/R obtained by decreasing ε keeping R fixedcan be achieved by setting up ad hoc evolutions, tailored for a specific type of boundary shape. This hasbeen done only for the triangles in Fig. 4, while in the remaining figures we have increased R keeping ε = 0 . ε fixed, numerical instabilities are encountered when R is too large as well. Thevalues of ε/R adopted in our numerical calculations have been chosen to guarantee both stable evolutionsand a satisfactory precision to compare the data with the analytic results, when they are available.Other alternative methods are available to construct minimal surfaces. A popular one by Chopp [76]consists of evolving the surface level sets under the surface mean curvature flow. A variant of this methodhas been employed in [50] to study minimal surfaces in the Schwarzschild-AdS D +2 background.31 Superellipse: a lower bound for F A In this appendix we provide a lower bound for the quantity F A (see (2.4)) associated with the entanglingcurves ∂A given by the superellipses (3.9), that we have discussed in §3.1.If A is a simply connected domain without corners in its boundary, let us consider a surface γ ∗ A anchoredon ∂A , but different from ˜ γ A , and such that A [ γ ∗ A ] = P A /ε − F ∗ A + o (1) as ε →
0. Being ˜ γ A the minimal areasurface anchored on ∂A , it is immediate to realize that F ∗ A < F A . Here we consider the superellipses (3.9),whose perimeter is given by P A = 4 R ˆ q R /R ) h n (˜ x ) d ˜ x , h n (˜ x ) ≡ ˜ x n − (1 − ˜ x n ) − /n , (C.1)where the integration variable ˜ x = x/R as been employed. Let us adapt to this case the choice of the trialsurface suggested in [28] for the ellipse, namely we consider γ ∗ A such that any section along the x directionprovides the profile of the infinite strip whose width is given by y ( x ) obtained from (3.9), i.e. y (˜ x ) = R (1 − ˜ x n ) /n . (C.2)Given the symmetries of the superellipse, we are allowed to restrict ourselves to x > y >
0. From(D.2) for D = 2, we construct the trial surface γ ∗ A by requiring that we have that any section at x = constis given by y ( z, ˜ x ) = z ∗ (˜ x ) ˆ z/z ∗ (˜ x ) Z √ − Z dZ , z ∗ (˜ x ) ≡ y (˜ x ) √ s ∞ , (C.3)where the integration variable Z ≡ z/z ∗ has been employed and z ∗ (˜ x ) has been introduced by taking z ∗ in(3.5) with s ∞ defined in (3.6) and replacing R with y (˜ x ) defined in (C.2). From (C.3), it is straightforwardto show that y (0 , ˜ x ) = y (˜ x ) and this guarantees that the trial surface is anchored on the superellipse (C.2).The occurrence of the cutoff ε in the holographic direction influences the integration domain along the x direction. In particular, by employing (C.2) and (C.3), the requirement z ∗ (˜ x ) (cid:62) ε becomes ˜ x (cid:54) ˜ x ε , where˜ x ε ≡ (cid:20) − (cid:18) √ s ∞ R ε (cid:19) n (cid:21) /n . (C.4)Plugging (C.3) inside the area functional, being y written in terms of x and z , we get A [ γ ∗ A ] = 4 ˆ ˜ x ε d ˜ x ˆ z ∗ (˜ x ) ε dz p ∂ z y ) + ( ∂ x y ) z = 2 R √ s ∞ R ˆ ˜ x ε M ε (˜ x )(1 − ˜ x n ) /n d ˜ x , (C.5)where M ε (˜ x ) ≡ ˆ ε/z ∗ (˜ x ) q R /R ) h n (˜ x ) C ( Z ) Z √ − Z dZ , C ( Z ) ≡ √ s ∞ ˆ Z r − Z − u u du − Z ! . (C.6)Computing (C.5) analytically is too hard, but one can check that the area law is satisfied. When ε → x ε = 1 + O ( ε n ). In this limit, the most divergent term of M ε (˜ x ) comes from thelimit of integration ε/z ∗ (˜ x ) and it can be found by considering an integration on the interval [ ε/z ∗ (˜ x ) , a ],where Z is infinitesimal if a (cid:28)
1. The remaining integral provides O (1) terms. For Z → C (0) = 1 and therefore the leading term in (C.5) is given by A [ γ ∗ A ] = 2 R √ s ∞ R ˆ d ˜ x q R /R ) h n (˜ x ) (1 − ˜ x n ) /n ˆ aε/z ∗ (˜ x ) dZZ + O (1) = P A ε + O (1) , (C.7)32here P A given in (C.1) can be recognized after (C.3) and (C.2) have been employed. We are not able tofind F ∗ A analytically but it can be obtained numerically as F ∗ A = lim ε → ( P A /ε − A [ γ ∗ A ]), with A [ γ ∗ A ] givenby (C.5), getting a lower bound for F A associated with the superellipse.It is interesting to consider F ∗ A in the limit of a very elongated superellipses, namely when R /R → ∞ .This means that (C.5) must be studied in the double expansion ε → R /R →
0. Assuming thatthe order of this two limits does not matter, let us set R /R = 0 in the expressions of M ε (˜ x ) in (C.6) andexpand it for small ε , finding M ε (˜ x ) (cid:12)(cid:12) R /R =0 = z ∗ (˜ x ) ε − √ s ∞ O ( ε ) , (C.8)where z ∗ (˜ x ) is given in (C.3). By plugging (C.8) into (C.5) and expanding the resulting expression for ε → A [ γ ∗ A ] = 4 R ε − s ∞ R R ˆ ˜ x ε d ˜ x (1 − ˜ x n ) /n + o ( ε ) = 4 R ε − πs ∞ n sin( π/n ) R R + o ( ε ) . (C.9)Notice that, from (C.1), one can observe that P A = 4 R (cid:2) o (1) (cid:3) when R /R → ∞ . We conclude that theleading term of F ∗ A as R /R → ∞ reads F ∗ A = πs ∞ n sin( π/n ) R R + . . . . (C.10)When n = 2, the result of [28] is recovered, as expected. Moreover, the expression (C.10) in the special casesof n = 2 and n = 3 has been checked in Fig. 2 against the data obtained with Surface Evolver (see respectivelythe red and the blue dotted horizontal lines), finding a good agreement. Notice that the expression in ther.h.s. of (C.10) is strictly larger than the value of F A corresponding to the infinite strip (see (3.6)), which isapproached as n → ∞ . D Some generalizations to AdS D +2 D.1 Sections of the infinite strip
In this section we discuss the computation of the area of the domain identified by an orthogonal section ofthe minimal surfaces associated with the infinite strip.The metric of AdS D +2 in the Poincaré coordinates reads ds = − dt + dz + dx + · · · + dx D z . (D.1)Considering an infinite D -dimensional strip on the spatial slice t = const extended along the x , . . . , x D directions whose width is given by 2 R , i.e. | x | (cid:54) R , the minimal area surface associated with this domainis characterized by the profile z = z ( x ). Because of the symmetry of the problem, z ( x ) is even and thereforewe can restrict to 0 (cid:54) x (cid:54) R . The profile is obtained by solving the following differential equation [9, 10] z = − p z D ∗ − z D z D . (D.2)where z ∗ is the maximum value of z , which is reached at x = 0.A way to get an orthogonal section of the infinite strip is defined by x = · · · = x D = const. Then, oneconsiders the two dimensional region enclosed by the profile z ( x ) and the cutoff z = ε in the plane ( x , z ).The domain along the x axis is | x | (cid:54) R − a , where a is defined by z ( R − a ) = ε . Its area readsˆ A = 2 ˆ R − a dx ˆ z ( x ) ε dzz = 2( R − a ) ε − D (cid:20) π − arctan (cid:18) ε D p z D ∗ − ε D (cid:19)(cid:21) = 2 R ε − πD + o (1) , (D.3)33here (D.2) has been employed.Another section of the infinite strip to study is defined by x i = const for some 2 (cid:54) i (cid:54) D and | x j | (cid:54) R for j = i . In this case we are interested in the volume of the D dimensional region enclosed by the profile z ( x ) and z = ε , whose projection on the z = 0 hyperplane is included within the section of the infinite stripwe are dealing with. It is given byˆ A = 2(2 R ) D − ˆ R − a dx ˆ z ( x ) ε dzz D = (2 R ) D − ( D − (cid:20) R ε D − − √ π Γ(1 + 1 /D ) z D − ∗ Γ(1 / /D ) + o (1) (cid:21) . (D.4)Notice that for D = 2 the expressions (D.3) and (D.4) coincide, as expected, and the result is employed in§3 to study the auxiliary surface, which corresponds to the dashed curve in Fig. 2. D.2 Annular domains
In this appendix we consider the surfaces anchored on the boundaries of annular domains which are localminima of the area functional because some analytic expressions can be found for them.The metric of AdS D +2 in Poincaré coordinates (2.2) written by employing spherical coordinates for thespatial part R D of the boundary z = 0 is ds = dz − dt + dρ + ρ d Ω D − z , (D.5)where ρ ∈ [0 , ∞ ) and the AdS radius has been set to one.A spherically symmetric spatial region A in the AdS boundary is completely specified by an interval inthe radial direction. Because of the symmetry of A , the minimal surface anchored on ∂A is given by z = z ( ρ )and, for a generic profile z = z ( ρ ), the corresponding area of the two dimensional surface γ A reads A [ γ A ] = Vol( S D − ) R D , R D ≡ ˆ ρ D − z D p z ) dρ , (D.6)where Vol( S D − ) is the volume of the ( D − R D is the integral in the radialdirection. We remark that the integration domain in R D is not necessarily the interval defining A in theradial direction, as it will be clear from the case discussed in the following. In order to find the minimalsurface ˜ γ A , one extremizes the area functional (D.6), obtaining zz + (1 + z ) (cid:20) D + ( D − zz ρ (cid:21) = 0 . (D.7)When A is a sphere of radius R , we have that 0 (cid:54) ρ (cid:54) R and it is well known that the corresponding minimalsurface is a hemisphere [9, 10].Here we consider the region A delimited by two concentric spheres, whose radii are R in and R out , with0 < R in < R out . In this case R in (cid:54) ρ (cid:54) R out and A is not simply connected. For D = 2 and D = 3, thecorresponding minimal surface extending in the bulk and anchored on ∂A has been studied in [51–53]. Inorder to solve (D.7) for this configuration, we find it convenient to introduce [51, 52] z ( ρ ) ≡ ρ ˜ z ( ρ ) , u ≡ log ρ , ˜ z u ≡ ∂ u ˜ z . (D.8)Notice that ˜ z = tan θ is the angular coefficient of the line connecting the origin to a point belonging to thesurface. Given (D.8), the differential equation (D.7) becomes˜ z ˜ z u (cid:0) ∂ ˜ z ˜ z u (cid:1) + (cid:2) z + ˜ z u ) (cid:3)(cid:2) D + ( D − z (˜ z + ˜ z u ) (cid:3) = 0 . (D.9)Integrating this equation, we find two solutions, namely˜ z u, ± (˜ z ) = − z ˜ z " ± ˜ z D − p K (1 + ˜ z ) − ˜ z D − , K > , (D.10)34 " D η ∗ D = 2 D = 3 D = 4 D = 5 D = 6 D = 7 η K / Figure 22: Curves for η as function of K obtained from the matching condition (D.13) for various dimensions2 (cid:54) D (cid:54)
7. For any D , a minimal value η ∗ > η ∈ ( η ∗ , K correspond to it, providing two different radial profiles (see an example for D = 2 in Fig. 23).which correspond to two different parts of the profile. As for the integration constant K , it must be strictlypositive because ˜ z = 0 corresponds to the boundary z = 0, which is included in the range of z . The domainfor ˜ z is 0 (cid:54) ˜ z (cid:54) ˜ z m , where ˜ z m is the first positive zero of the polynomial under the square root in (D.10).For D = 2 we are lead to solve a biquadratic equation, which gives ˜ z m = (cid:0) K + p K ( K + 4) (cid:1) /
2. Notice that˜ z m → K → f ( D ) ± ,K (˜ z ) ≡ ˆ ˜ z λ λ " ± λ D − p K (1 + λ ) − λ D dλ . (D.11)Then, the profile of the radial section is given by the following two branches ( ρ = R in e − f ( D ) − ,K (˜ z ) ,ρ = R out e − f ( D )+ ,K (˜ z ) . (D.12)Imposing that these two branches match at the point P m , whose ( ρ, z ) coordinates are ( ρ m , z m ≡ z ( ρ m )),where z m has been found above, we get the following relation − log( η ) = f ( D )+ ,K (˜ z m ) − f ( D ) − ,K (˜ z m ) = ˆ ˜ z m λ D (1 + λ ) p K (1 + λ ) − λ D dλ , η ≡ R in R out . (D.13)Since ˜ z m depends on K , from (D.13) we get a relation between η and K , which is represented in Fig. 22 for2 (cid:54) D (cid:54)
7. The first feature to point out about (D.13) is the existence of a minimal value for η that will be35 !" z ρ P P P m P m P ∗ P ∗ R out R in Figure 23: Radial profiles in the ( ρ, z ) plane for the connected surfaces anchored on the boundary of thesame annulus A having R in < R out . They correspond to local minima of the area functional and they arecharacterized by the two different values of K associated with the same η . These connected surfaces areobtained through (D.12) and (D.13), where different colours are used for the various branches. The dashedcurves represent the two concentric hemispheres anchored on ∂A as well. The continuous grey curves arethe paths in the ( ρ, z ) plane of the points P , P m and P ∗ as K ∈ (0 , ∞ ). Here D = 2, R in = 0 . R out = 1and the values of K are K = 0 .
81 (global minimum) and K = 2 .
05 (local minimum). Comparing the areaof the two connected surfaces, we find that the one having minimal area has P ∗ closer to the boundary.denoted by η ∗ >
0. For instance, we find η ∗ = 0 . η ∗ = 0 .
542 and η ∗ = 0 .
643 for D = 2, D = 3 and D = 4respectively (see the inset in Fig. 22 for other D ’s). Then, for any η ∗ < η <
1, there are two values of K giving the same η , while for 0 < η < η ∗ connected solutions do not exist. The two different K ’s associatedwith the same η ∗ < η < ∂A . In order to find the global minimum of the area functional, we have to evaluate their area.Through a numerical analysis, one observes that z m is an increasing function of K .Beside P m , another interesting point of the profile is P = ( ρ , z ≡ z ( ρ )), where | z ( ρ ) | diverges. From(D.10), this divergence occurs when q K (1 + ˜ z ) − ˜ z D ± ˜ z D − = 0 = ⇒ K = ˜ z D − ≡ (tan θ ) D − . (D.14)This tells us that K has a geometric meaning because it provides ˜ z .36et us also introduce the point P ∗ , with coordinates ( ρ ∗ , z ∗ ≡ z ( ρ ∗ )) as the point having the maximumvalue of z , which corresponds to the maximal penetration of the minimal surface into the bulk. The coordinate z ∗ can be found by considering the branch z ( ρ ) characterized by f ( D )+ ,K in (D.12) and then computing itsderivative w.r.t. ρ , which is given by dzdρ = d (˜ zρ ) d ˜ z (cid:18) dρd ˜ z (cid:19) − = ˜ z − df ( D )+ ,K (˜ z ) d ˜ z ! − , (D.15)where in the last step (D.12) has been used. When D = 2 the root of (D.15) can be found and it reads˜ z ∗ = K / . (D.16)An explicit example in D = 2 is given in Fig. 23, where we have shown the two connected radial profileshaving the same η > η ∗ but different values of K . The two different branches in (D.12) at fixed K , supportedby the matching condition (D.13), have been denoted with different colours: the red and cyan curves areobtained through f (2)+ ,K while the blue and the green ones through f (2) − ,K . In Fig. 23 the grey curves denotethe paths described by the three points P m , P and P ∗ introduced above as K assumes all the positive realvalues.We find it instructive to consider the limit K → + ∞ . From (D.11), in this limit one finds f ( D )+ , ∞ = f ( D ) − , ∞ for any D , which reads lim K →∞ f ( D ) ± ,K (˜ z ) = ˆ ˜ z λ λ dλ = 12 log(1 + ˜ z ) , (D.17)and therefore η → R in = R out ≡ R (see also Fig. 22). From (D.17), both the branches in(D.12) become ρ = R √ z , (D.18)which is the well known spherical solution z = R − ρ . As for the points P m , P and P ∗ , they tend to thesame point when η →
1, as can be seen from Fig. 23, where the gray lines show the paths of these points inthe ( ρ, z ) plane as K varies in (0 , ∞ ).Given the radial profile (D.12), we can compute the area of the corresponding surface obtained byexploiting the rotational symmetry. From (D.8), the radial integral in (D.6) can be written as R con D = ˆ ˜ ε + ˜ z m p z + ˜ z u, + ) ˜ z D ˜ z u, + d ˜ z + ˆ ˜ ε − ˜ z m p z + ˜ z u, − ) ˜ z D ˜ z u, − d ˜ z , ˜ ε + ≡ εR out , ˜ ε − ≡ εR in , (D.19)where ˜ z u, ± have been defined in (D.10) and 0 < ε (cid:28) R con D = ˆ ˜ z m ε/R out √ K d ˜ z ˜ z D p K (1 + ˜ z ) − ˜ z D + ˆ ˜ z m ε/R in √ K d ˜ z ˜ z D p K (1 + ˜ z ) − ˜ z D (D.20)= 2 ˆ ˜ z m ε/R in √ K d ˜ z ˜ z D p K (1 + ˜ z ) − ˜ z D + ˆ ε/R in ε/R out √ K d ˜ z ˜ z D p K (1 + ˜ z ) − ˜ z D . (D.21)In the second integral of (D.21), we can employ the expansion of the integrand for ˜ z ∼
0, which reads1˜ z D p z − ˜ z D /K = 1˜ z D + γ D,D − ˜ z D − + γ D,D − ˜ z D − + · · · + γ D, log ˜ z + O (˜ z ) odd D , γ D, − + O (˜ z ) even D , (D.22)finding that it provides a non trivial contribution γ D, log log( R out /R in ) to the finite term for odd D .37iven R in and R out , besides the two connected surfaces having the same η but different K , we have alsoanother surface γ A which is a local minimum for the area functional (D.6) such that ∂γ A = ∂A : it is made bytwo disjoint concentric hemispheres in the bulk with radii R in and R out which are anchored on the boundariesof the concentric spheres in the boundary (see the dashed curves in Fig. 23). The area of a hemisphere ofradius R in the bulk anchored on the boundary of a sphere with the same radius at z = 0 can be found byintegrating (D.6) for 0 (cid:54) ρ (cid:54) R − a , where z ( ε ) ≡ a , finding R sph D ( R ) = ˆ ε/R ∞ p z + ˜ z u ) ˜ z D ˜ z u d ˜ z = ˆ ∞ ε/R d ˜ z ˜ z D √ z , ε = p R − ( R − a ) (cid:28) , (D.23)where ˜ z u is (D.10) in the limit K → + ∞ , namely ˜ z u = − (1 + ˜ z ) / ˜ z .Thus, the factor coming from the radial integration in (D.6) for this configuration of two disjoint hemi-spheres is R dis D = R sph D ( R out ) + R sph D ( R in ).Having found three surfaces anchored on ∂A for any given R in < R out such that η ∗ < η < R D ≡ lim ε → ( R dis D − R con D ) . (D.24)From (D.20) and (D.23), it can be written as∆ R D = J ( in ) D + J ( out ) D , (D.25)where we have introduced J ( j ) D = lim ε → ˆ ∞ ε/R j d ˜ z ˜ z D √ z − ˆ ˜ z m ε/R j d ˜ z ˜ z D p z − ˜ z D /K ! . (D.26)Splitting the second integral, we can take the limit, finding that J ( in ) D = J ( out ) D and then∆ R D = 2 " ˆ ∞ ˜ z m d ˜ z ˜ z D √ z − ˆ ˜ z m z D √ z p − ˜ z D / [ K (1 + ˜ z )] − ! d ˜ z . (D.27)Since z m = z m ( K ) and K depends on the ratio η only, also ∆ R D is a function of η . Nevertheless, as discussedabove, there are two values of K associated with the same η and, by computing ∆ R D for both of them, wecan easily find which surface has the minimal area between the two connected ones. It turns out that it isthe one associated with the lowest value of K . Since z m is an increasing function of K , the minimal areasurface between the two connected ones has the lowest z m . In the example in Fig. 23 for D = 2, both theradial profiles of the two connected surfaces which are local minima of the area functional and which havethe same η are shown. The one described by the red and the blue curves characterizes the minimal areasurface between the two connected ones.Once the connected surface having minimal area has been found, the sign of the corresponding ∆ R D de-termines the configuration with minimal area, providing therefore the global minimum of the area functional,and its root (which can be found numerically) gives the value of η = η c which characterizes the transition.For D = 2, D = 3 and D = 4 we get respectively η c = 0 .
419 [45, 51], η c = 0 .
562 [52] and η c = 0 . η ∈ ( η ∗ , η c > η ∗ and ∆ R D > η ∈ ( η c , η < η c theconfiguration occurring in the holographic entanglement entropy for the annular domains is the one madeby two disjoint hemispheres. 38 Elliptic integrals
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