You are an expert at summarizing long articles. Proceed to summarize the following text: 
the famous bekenstein  hawking area law @xcite @xmath0 for the entropy of black holes connects thermodynamics , gravity and relativistic quantum field theory . this relation remains valid not only in einstein s gravity in four dimensions but in higher dimensions too , as long as the gravitational constant @xmath1 is @xmath2-dimensional and the area @xmath3 is understood as the volume of the horizon surface .    in quantum field theory ( qft ) an entanglement entropy can be attributed to any surface formally dividing the system in two parts . the leading uv contribution of quantum fields to the entanglement entropy is proportional to the area of the dividing surface @xcite . this property is strikingly similar the bekenstein  hawking area law . the analogy with black hole entropy is not accidental . one may consider a black hole horizon as a surface separating the interior of the black hole from its exterior . to define a wave function of all quantum fields in the black hole spacetime @xcite one can use an analogue to the hartle - hawking no - boundary proposal . using this wavefunction one can construct the density matrix for the fields inside the black hole and derive the corresponding entanglement entropy , which is proportional to the area of the horizon , but the coefficient of proportionality formally diverges . however , one has to take into account that quantum fields on a curved background spacetime also contribute to the effective gravitational constant . it is amazing that quantum contributions to the entropy per unit area of a horizon are described by the same functions as quantum corrections to the gravitational coupling @xcite . the interpretation of the bekenstein  hawking formula as the entanglement entropy becomes even more striking in the framework of induced gravity models @xcite where the gravitational coupling is completely defined by quantum field contributions . in these models the leading uv contribution to the entanglement entropy of the horizon @xcite is given by the formula @xmath4 and is finite as soon as the induced gravitational constant @xmath5 is finite . it was also proposed @xcite that in generic static spacetimes with horizons , the minimal area surface inside the slice of a constant time may play an important role in the definition of the entanglement entropy of a black hole . recently , holographic computations of the entanglement entropy in conformal field theory ( cft ) at infinity of the anti - de  sitter ( ads ) spacetime have seen a lot of attention and development . the original conjecture for entanglement entropy by ryu and takayanagi @xcite is that in a static configuration the entanglement entropy of a subsystem localized in a domain @xmath6 is given by the formula system of units . ] @xmath7 given a static time slice ( the bulk space ) , the domain @xmath8 belongs to an infinite boundary @xmath9 of the bulk and the area @xmath10 in eq .   is to be understood as the area of a @xmath11-dimensional minimal surface @xmath12 in the bulk spanned on the boundary @xmath13 of the subsystem ( i.e. , @xmath14 ) .    the holographic derivation of the ryu takayanagi formula for the entanglement entropy was proposed in @xcite using the replica trick . in the replica method there naturally appears a more general notion of the renyi entanglement entropy . but the qft derivation of the ryu takayanagi relation based on the calculation of renyi entropies requires a different approach @xcite . in qft with gravity duals , formula was proven for @xmath15 @xcite . in a more general case of euclidean gravity solutions without killing vectors , arguments supporting the validity of the ryu - takayanagi formula were given in @xcite . in the last few years the conjecture by ryu and takayanagi has been generalized to gravity theories with higher curvature interactions @xcite or some other deformations of the gravity theory @xcite . an excellent up - to - date review of the entanglement entropy and black holes can be found in @xcite . calculation of entanglement entropy for a subsystem localized in two disjoint regions is particularly interesting , since it can be used as a probe of confinement @xcite . it was demonstrated @xcite that in confining backgrounds there are generally more solutions for minimal surfaces in the bulk spanned on the boundaries of these disjoint regions . however , there is a maximum distance between the regions beyond which the tube - like minimal surface connecting both components ceases to exist . there is also a critical scale beyond which a solution with disconnected minimal surfaces dominates over the connected one . in the qft language this critical behavior is analogous to a deconfinement transition at a finite temperature .    in the case of a few disjoint regions the minimal surfaces connecting their boundaries are generally not unique and their areas differ . the conventional wisdom is that the entanglement entropy is related to the surfaces of minimum area . this choice guarantees the required strong subadditivity property @xcite of the entanglement entropy . there were some proposals @xcite how to modify this `` least area '' rule while still satisfying the strong subadditivity property . in this paper we study minimal surfaces in the pure ads spacetime . we show that many properties of the entanglement entropy , such as the critical behavior @xcite demonstrated for the asymptotically ads spacetimes with black holes in the bulk , exist already in the pure ads . the main result is that we are able to find exact solutions for all minimal surfaces spanned on one or two spherical boundaries positioned arbitrarily at conformal infinity @xmath16 . in this short paper we give analytical formulas for the regularized and renormalized area of these minimal surfaces . the explicit form of the surfaces and its derivation is presented in a more detailed paper @xcite . we also shortly discuss the spacetime character of a minimal surface spanned on two accelerated spherical domains . we start with geometrical preliminaries concerning the bulk space and with the characterization of the spherical domains at infinity . we will discuss only a ads spacetime although most of the discussion can be extended to higher dimensions . the ads spacetime has many killing symmetries and can be viewed as a static spacetime in various ways . however , in all cases the spatial section  the bulk space  has the hyperbolic geometry of lobachevsky space . to describe it , we use cylindrical coordinates @xmath17 and poincar coordinates @xmath18 in which the metric reads @xmath19 here , @xmath20 is the characteristic scale describing the radius of curvature of ads , as well as of its spatial section . the coordinates are related by @xmath21 , @xmath22 , @xmath23 , with @xmath24 . the conformal infinity @xmath16 of the spatial section is the conformal sphere . in cylindrical coordinates it is given by @xmath25 , @xmath26 . in poincar coordinates it is represented by the plane @xmath27 plus one improper point @xmath28 .    by the _ circular _ ( in higher dimension , _ spherical _ ) boundary @xmath29 of a ball - like domain @xmath8 at infinity @xmath16 we mean a surface given by infinite points of a hyperplane in the bulk ( hyperplane in the sense of hyperbolic geometry , a hypersurface with zero extrinsic curvature ) . the circular boundaries at infinity are thus in one - to - one correspondence with hyperplanes in the bulk . visualization of such a hyperplane and the corresponding circular boundary can be found in fig . [ fig : hyperplane ] . the boundary @xmath29 can be understood also as the boundary of the complementary domain @xmath30 .    * hyperplane corresponding to the circular boundary at infinity . * the hyperplane ( the hypersurface of zero extrinsic curvature ) reaches infinity at the circular boundary which divides infinity into two domains . the poincar spherical ( left ) and spherical half - space ( right ) representation of hyperbolic space is shown.,title="fig:",width=160]*hyperplane corresponding to the circular boundary at infinity . * the hyperplane ( the hypersurface of zero extrinsic curvature ) reaches infinity at the circular boundary which divides infinity into two domains . the poincar spherical ( left ) and spherical half - space ( right ) representation of hyperbolic space is shown.,title="fig:",width=160 ] +    from a point of view of the conformally spherical geometry on @xmath16 , all such boundaries are equivalent . it is a reflection of the trivial fact that all hyperplanes in the bulk are isometric . therefore we do not have any quantity measuring a ` size ' of spherical boundaries at infinity . however , in many calculations , both in the bulk or at infinity , we need to regularize various quantities . instead of working at @xmath16 we restrict on some cut - off surface at large finite size . then we can measure the sizea of the circular boundaries using the geometry on the cut - off surface . but , since the choice of the cut - off can be rather arbitrary , the regularized size of the spherical boundary can be only an intermediate quantity , and physically measurable quantities should be cut - off independent . two circular boundaries can be in three qualitatively distinct positions : ( i ) disjoint boundaries ( corresponding hyperplanes are ultraparallel ) , ( ii ) boundaries crossing each other ( the hyperplanes intersect in a line ) , and ( iii ) boundaries touching in one point ( the corresponding hyperplanes are asymptotic ) .    in the first case we can define the distance of the boundaries as a distance of the corresponding hyperplanes . to the crossing circular boundaries we can assign an angle of the corresponding hyperplanes . finally , all pairs of touching boundaries are equivalent . indeed , all pairs of asymptotic hyperplanes in an arbitrary position are isometric to each other . in global hyperbolic space there is no measure which could distinguish them . now we review known results for a minimal surface spanned on one circular boundary . such a minimal surface is trivial : it is the hyperplane which defines the boundary . if we choose the axis of the cylindrical coordinates perpendicular to the hyperplane , the hyperplane is given by @xmath31 . if we choose the axis inside the hyperplane , the hyperplane is given by @xmath32 . in the poincar coordinates the hyperplane is represented as a plane orthogonal to the infinity surface @xmath33 or as a hemisphere with the center at @xmath33 ( here we used a language of the conformally related euclidian geometry with cartesian coordinates @xmath18 ) .    to demonstrate different regularizations used later , we can write down the area of a hyperplane measured up to a cut - off . for the hyperplane othogonal to the axis we have @xmath34\;,\ ] ] where @xmath35 , with @xmath36 , is the circumference of the circular boundary on the cut - off surface @xmath37 .    for the hyperplane which contains the axis we have @xmath38\;,\ ] ] where @xmath39 , with @xmath40 , is the length of the boundary at the cut - off surface . in this case , the circular boundary is represented by two lines @xmath41 , @xmath42 , @xmath32 . we thus have to introduce two cut - offs : an ultraviolet one , @xmath43 , in the direction away from the axis , and an infrared one , @xmath44 , along the axis . finally , the area of the hyperplane represented by a half - plane in the poincar coordinates , say @xmath45 , is @xmath46 @xmath47 is again the length of the circular boundary at the cut - off surface @xmath48 . it is also infrared divergent : one has to cut - off the @xmath49 direction at @xmath50 .    in all three cases we recognize the well - known property that the leading diverging term of the minimal surface is ( up to a constant scale ) given by the regularized size of the boundary at infinity . clearly , the exact expression for the divergent term depends on the regularization scheme , however , in all cases it can be interpreted as the regularized size of the boundary at infinity @xcite .    the area of the trivial minimal surface spanned on one circular boundary can be used to eliminate the infinite contributions to the area for more complicated surfaces . we define the renormalized area of a surface by subtracting the area of hyperplanes spanned on the same boundaries at infinity . in this sense , the trivial minimal surface has vanishing renormalized area . +   + ( a)(b)(c ) +    * * disjoint boundaries.**given two circular boundaries at infinity , we can always find the unique line perpendicular to the corresponding hyperplanes in the bulk . if we adjust the cylindrical coordinates to this axis , the circular boundaries are represented by two circles @xmath41 , @xmath51 . it is possible to find @xcite a tube - like minimal surface joining these two boundaries , see fig .  [ fig : ms]a . it is described by the function @xmath52 with @xmath53 , and it is parametrized by @xmath54 , where @xmath55 is the closest approach of the surface to the axis : @xmath56\;. \end{split}\raisetag{4ex}\ ] ] setting @xmath57 we can read out the coordinates @xmath58 of the circular boundaries : @xmath59\;. \end{split}\raisetag{8ex}\ ] ] the distance between both boundaries , @xmath60 , as a function of the parameter @xmath61 is depicted in fig . [ fig : area]a . it reveals that the tube exists only for distances smaller than the maximal distance @xmath62 , and for these small distances there actually exist two tube - like minimal surfaces , one shallow one , remaining at large distances from the axis , and a deep one , approaching the axis . if the distance of circular boundaries is enlarged , the tube tears off and the minimal surface discontinuously splits into two trivial hyperplanes spanned on both boundaries .    to estimate which surface is the smallest one , we have to write down the regularized area : @xmath63 the divergent term @xmath64 is given by , the finite part @xmath65 reads @xmath66     + ( a)(b)(c ) +    the renormalized area as a function of @xmath61 or of the distance @xmath67 is shown in fig .  [ fig : area ] . we see that the shallow tube has always smaller area than the deeper one . however , for @xmath68 , the renormalized area of the tube is positive , i.e. , the tube has larger area than the trivial solutions of two hyperplanes . the tube is thus the smallest minimal surface only for @xmath69 . all these exact results confirm the previously conjectured properties based on numerical and approximate analysis @xcite .    * * crossing circular boundaries.**in the case of two crossing circular boundaries at infinity we naturally adjust the cylindrical coordinates to the axis going through the intersection points . thus , the semicircles between these intersection points are represented by lines @xmath41 , @xmath70 . in poincar coordinates they are half - lines in the plane @xmath27 starting at @xmath71 . the minimal surface spanned on two such semicircles is depicted in fig . [ fig : ms]b . its explicit form can be found in @xcite . it exists for any angle @xmath72 between both semicircles and can be parametrized by @xmath73 with @xmath55 corresponding to the closest approach of the surface to the axis . the relation between of @xmath72 and @xmath74 is one - to - one @xcite . the regularized area takes form : @xmath75\;. \end{split}\ ] ] the leading term @xmath64 is given by . it is divergent because of both uv and ir divergences . the next term is proportional to the ir cut - off @xmath76 . the reason is that the minimal surface is invariant under the translation along the axis . however , we can write down the finite renormalized area density @xmath77 : @xmath78\,.\ ] ] it is always negative . naturally , the surface has always smaller area then two half - hyperplanes starting at the axis reaching the semi - circles at infinity .    * * touching circular boundaries**tangent circular boundaries are trivially represented in poincar coordinates . if oriented in the @xmath49 direction , they are given by @xmath27 , @xmath79 . the minimal surface spanned on such a ` strip ' is in fig . [ fig : ms]c , @xcite . it reaches the maximal value of the coordinate @xmath80 for @xmath81 , which we call the ` top - line ' of the surface . the constant @xmath82 is given by @xmath83 . the area regularized at @xmath84 is @xmath85\\      & = a_{{\scriptstyle\mathrm{hp}}}+   l_0 \bigl[-2x_0\ell + o({{\bar{z}}}^3)\bigr]\ , . \end{split}\raisetag{3ex}\ ] ] the leading divergent term @xmath64 is given by . the next term is ir divergent since the surface has the horocyclic symmetry @xmath86 . it is thus proportional to the length @xmath87 measured on the ` top - line ' of the surface . the renormalized area density @xmath88 is , as expected , a constant independent of the position of the touching circular boundaries . returning to the conjecture , we can now associate entanglement entropy with any two generally positioned spherical domains at infinity . the most interesting case occurs for two disjoint domains . for boundaries closer than @xmath89 there are three possible minimal surfaces , which corresponds to three possibilities ( phases ) for the holographic entanglement entropy in cft . the physical choice would correspond to the surface of the smallest area . inspecting fig . [ fig : area]b , one can see that the transition between these phases occurs at the distance @xmath90 , when the area of the tube - like surface starts to exceed the area of the trivial solution with two hyperplanes . if we accept that the entanglement entropy for disjoint subsystems is given by the absolute minimal surface according to , then the renormalized area is directly related to the mutual information @xmath91 which quantifies correlations between the disjoint subsystems . indeed , since the entanglement entropy @xmath92 of a single spherical domain @xmath8 is given by the area @xmath64 of the trivial hyperplaneboundary @xmath29 , , the renormalized area @xmath65 of the tube joining the boundaries of two such domains gives directly the mutual information @xmath93 , provided that the tube does give the minimal area , i.e. , for @xmath94 . although the entanglement entropy changes continuously with the distance between the boundaries at @xmath90 , the corresponding minimal surface changes discontinuously . to move from the trivial phase to the tube - like phase continuously , one would have to start with two very close hyperplanes . at a point , where they almost touch , a very deep tube - like surface can appear . by enlarging the distance of the boundaries , the tube starts to grow wider . it follows the upper branch of the curve in fig .  [ fig : area]c ( i.e. the non - physical phase ) up to the maximal possible distance @xmath89 of the boundaries . here , one has to start decreasing the distance of the boundaries in such a way that the tube grows even wider ( following the lower branch in fig . [ fig : area]c ) . after decreasing the distance under @xmath95 one obtains , in a continuous way , the physical tube - like phase .    * world - sheets of minimal surfaces . * the vertical cylinder represents a 3 + 1 dimensional ads spacetime with the angular direction @xmath96 suppressed . the world - sheet of one circular boundary is thus reduced only to two curves at infinity . one circular boundary is localized in the left static region , the other in the right one . ( a ) two uniformly accelerated hyperplanes spanned on these circular boundaries . killing horizons are indicated . ( b ) the world - sheet of the tube - like minimal surface joining the same circular boundaries.,title="fig:",width=148]*world - sheets of minimal surfaces . * the vertical cylinder represents a 3 + 1 dimensional ads spacetime with the angular direction @xmath96 suppressed . the world - sheet of one circular boundary is thus reduced only to two curves at infinity . one circular boundary is localized in the left static region , the other in the right one . ( a ) two uniformly accelerated hyperplanes spanned on these circular boundaries . killing horizons are indicated . ( b ) the world - sheet of the tube - like minimal surface joining the same circular boundaries.,title="fig:",width=148 ] + ( a)(b ) +    the fact that the tube - like minimal surface does not exist for too distant boundaries can be explored also in a dynamical way . although we consider only static situations calculation of the minimal surface area given in a spatial section of a static region of the ads spacetime  we can take advantage of the rich structure of ads symmetries and investigate the situation which looks rather dynamical in the global picture . let us consider a static killing vector with orbits that have an acceleration larger than @xmath97 . this killing vector has a bifurcation character similar to the boost killing vector in the minkowski spacetime . its killing horizons divide the ads space into pairs of static regions positioned acausally with respect to each other , with non - static regions between , cf . [ fig : ads]a . the hyperbolic space in which we found the tube - like solution is a spatial section of both opposite static regions . we can position one circular boundary at infinity of one static region and the other one at infinity of the opposite static region . the world - sheets of the corresponding hyperplanes describe uniformly accelerated motion along the killing vector , see fig . [ fig : ads]a . the tube - like minimal surface can be also evolved into both static regions . however , it does reach the killing horizons and there it must be extended into non - static regions . the resulting surface is depicted in fig . [ fig : ads]b . we see that the surface is non - smooth along two spatial edges , one describing the formation of the surface in the past , the other its termination in the future . if the surface is viewed from the perspective of a globally static observer ( the vertical direction in the figure ) , the future edge can be interpreted as a tear - off line for the boundaries positioned too far from each other and the subsequent motion of the separate pieces . for more details , see @xcite .    beside the case of two spherical domains we can investigate even more complicated situations : let us consider spherical domains @xmath98 , each of them a subdomain of all the subsequent ones : @xmath99 for @xmath100 . they do not have to be all simultaneously concentric . the circular boundaries of these domains correspond to ultraparallel hyperplanes in the bulk . for such a configuration we know the minimal surfaces for any pair of the boundaries . employing we find that the renormalized entropy depends only on the distance of the boundaries , cf .  , . we can thus test properties of the entropy for domains obtained by a combination of several subdomains . namely , one can check the strong subadditivity inequalities to find that they are satisfied , as expected from general considerations @xcite . similarly one can study systems of strips between several semicircles joined at the same poles .    summarizing , we have found exact analytical solutions for minimal surfaces in ads for two disjoint domains at infinity . these classical geometrical solutions reveal the existence of different phases that reflect a phase transition in the corresponding quantum cft , similar to the confinement / deconfinement phase transition at a finite temperature @xcite . the holographic entanglement entropy becomes an effective tool for testing phase transitions in cft . note that calculations in purely classical gravity provide an insight to non - trivial quantum properties of the corresponding field theories . p.  k. was supported by grant gar p203/12/0118 and appreciates the hospitality of the tpi of the university of alberta . a.  z. thanks the natural sciences and engineering research council of canada and the killam trust for financial support and appreciates the hospitality of the itp fmp of charles university in prague . the authors thank don page and martin ofka for usefull comments on the paper . 35ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ]  + 12$12&12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop * * ,   ( ) link:\doibase 10.1007/bf01877517 [ * * ,   ( ) ] link:\doibase    10.1103/physrevd.34.373 [ * * ,   ( ) ] link:\doibase    10.1103/physrevd.48.4545 [ * * ,   ( ) ] ,   link:\doibase 10.1103/physrevlett.71.666 [ * * ,   ( ) ] ,   link:\doibase 10.1103/physrevd.51.1741 [ * * ,   ( ) ] ,   link:\doibase 10.1103/physrevd.50.2700 [ * * , ( ) ] ,   @noop * * ,   ( ) @noop   ( ) ,  ,   link:\doibase 10.1016/s0550 - 3213(96)00678 - 5 [ * * ,   ( ) ] ,   link:\doibase 10.1016/s0920 - 5632(97)00373 - 3 [ * * , ( ) ] @noop * * ,   ( ) ,   link:\doibase 10.1103/physrevlett.96.181602 [ * * ,   ( ) ] ,   link:\doibase 10.1088/1126 - 6708/2006/08/045 [ * * , ( ) ] ,   link:\doibase 10.1088/1751 - 8113/42/50/504008 [ * * , ( ) ] ,   link:\doibase 10.1088/1126 - 6708/2006/09/018 [ * * , ( ) ] ,   @noop   ( ) ,  ,   @noop   ( ) ,  ,   link:\doibase 10.1007/jhep08(2013)090 [ * * , ( ) ] ,   @noop   ( ) ,  , link:\doibase 10.1007/jhep04(2011)025 [ * * ,   ( ) ] ,   link:\doibase 10.1007/jhep05(2011)036 [ * * ,   ( ) ] ,   link:\doibase 10.1007/jhep06(2013)013 [ * * ,   ( ) ] ,   link:\doibase 10.1007/jhep08(2013)012 [ * * ,   ( ) ] ,   link:\doibase 10.1007/jhep08(2011)039 [ * * ,   ( ) ] ,   @noop * * ,   ( ) ,   link:\doibase 10.1016/j.nuclphysb.2007.12.017 [ * * ,   ( ) ] ,   link:\doibase 10.1007/jhep05(2012)032 [ * * , ( ) ] ,   link:\doibase 10.1088/1126 - 6708/2007/02/042 [ * * , ( ) ] ,   link:\doibase 10.1103/physrevd.76.106013 [ * * ,   ( ) ] ,   link:\doibase 10.1088/1126 - 6708/2008/03/006 [ * * , ( ) ] ,   @noop `` , ''  ,   link:\doibase 10.1007/jhep01(2011)125 [ * * ,   ( ) ] ,   link:\doibase 10.1088/0264 - 9381/29/15/153001 [ * * ,   ( ) ] , @noop _ _  ( ,  ,  ) link:\doibase 10.1007/jhep05(2011)004 [ * * ,   ( ) ] ,
it was proposed by ryu and takayanagi that the entanglement entropy in conformal field theory ( cft ) is related through the ads / cft correspondence to the area of a minimal surface in the bulk . 
 we apply this holographic geometrical method of calculating the entanglement entropy to study the vacuum case of a cft which is holographically dual to empty anti - de  sitter ( ads ) spacetime . 
 we present all possible minimal surfaces spanned on one or two spherical boundaries at ads infinity . 
 we give exact analytical expressions for the regularized areas of these surfaces and identify finite renormalized quantities . in the case of two disjoint boundaries 
 the existence of two different phases of the entanglement entropy is confirmed @xcite . 
 a trivial phase corresponds to two disconnected minimal surfaces , while the other one corresponds to a tube connecting the spherical boundaries . 
 a transition between these phases is reminiscent of the finite temperature deconfinement transition in the cft on the boundary . 
 the exact analytical results are thus consistent with previous numerical and approximate computations 
 . we also briefly discuss the character of a spacetime extension of the minimal surface spanned on two uniformly accelerated boundaries .