A note on the extensivity of the holographic entanglement entropy
aa r X i v : . [ h e p - t h ] M a y IFTE UAM-CSIC/2007-66
A note on the extensivity of the holographic entanglemententropy
Jos´e L.F. Barb´on and Carlos A. Fuertes
Instituto de F´ısica Te´orica IFTE UAM/CSICFacultad de Ciencias C-XVIC.U. Cantoblanco, E-28049 Madrid, Spain [email protected] , [email protected] Abstract
We consider situations where the renormalized geometric entropy, as defined by theAdS/CFT ansatz of Ryu and Takayanagi, shows extensive behavior in the volume of theentangled region. In general, any holographic geometry that is ‘capped’ in the infraredregion is a candidate for extensivity provided the growth of minimal surfaces saturates atthe capping region, and the induced metric at the ‘cap’ is non-degenerate. Extensivity iswell-known to occur for highly thermalized states. In this note, we show that the holo-graphic ansatz predicts the persistence of the extensivity down to vanishing temperature,for the particular case of conformal field theories in 2 + 1 dimensions with a magnetic fieldand/or electric charge condensates.October 31, 2018
Introduction
A very interesting proposal for a holographic [1] computation of geometric (entangle-ment) entropy [2] in conformal field theories was put forward in [3] (see [4] for subsequentwork). The proposed procedure bears a close similarity with the holographic calculationof Wilson and ’t Hooft loops in gauge theories and applies strictly to a large- N , strongcoupling limit of the quantum field theories (QFT) in question.In recent years, entanglement entropy has gradually emerged as a useful non-local orderparameter to characterize new phases in strongly coupled systems at zero temperature(cf. for example [5]). It is therefore of great interest to parametrize the different typesof qualitative behavior that can be identified from the AdS/CFT side. In this direction,the authors of [6] have made the interesting observation that the scaling of geometricentanglement entropy is a good order parameter of confinement in gauge theories thatadmit standard AdS-like duals (see also the earlier work of Ref. [7]). In particular theyfound that for regions of size bigger than the confining scale the renormalized entanglemententropy suddenly becomes strictly proportional to the area of the region with a constantof proportionality independent of the size, unlike conformal field theories. In this paper,we follow this program, focusing on the property of extensivity , an admittedly non generic feature of geometric entropy in standard weak-coupling calculations at zero temperature.In this note, we begin with a review of the holographic proposal of [3], emphasizing thenatural emergence of the non-extensive ‘area law’ for the geometric entropy in models withconformal ultraviolet fixed points. It is then argued that extensivity of the renormalizedfinite entropy is quite a generic feature of the holographic models with a gapped spectrum.We discuss in some detail the conditions that ensure this behavior, and comment on therelation to the confinement property, making contact with the results of [6].Finally, we focus on a particular model of especial interest, namely the case of confor-mal field theories (CFT) in 2 + 1 dimensions with a magnetic field and/or electric chargecondensates, holographically represented by zero-temperature extremal black holes withfinite horizon area in Anti-de Sitter space (AdS). The relevance of this system stems fromthe fact that the extensive law, characteristic of high-temperature states, does persist allthe way down to zero temperature. The starting point of the holographic proposal of [3] is the replica trick calculationof the entanglement entropy [9]. Consider a quantum field theory defined on a Hamilto-nian spacetime of the form R × X d , with X d a d -dimensional spatial manifold. Given ageneral state ρ defined in terms of the Hamiltonian quantization on X d , let A denote a d -dimensional region of X d , with smooth boundary ∂ A, and consider the inclusive density We understand by renormalized entanglement entropy the quantity obtained by subtracting theultraviolet divergent term to the entanglement entropy. A more refined quantity is the entropic C-function [8] to which all our comments equally apply. ρ A = Tr A ρ . The associatedentanglement entropy can be computed as [9] S A = − Tr ρ A log ρ A = − ddn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n =1 Tr ( ρ A ) n . (1)The path-integral representation of Tr ( ρ A ) n can be manipulated into a frustrated (Eu-clidean) partition function on a theory with n copies of the original QFT,Tr ( ρ A ) n = D T ( n ) ∂ A E QFT ⊗ n , (2)where the twist operator T ∂ A is defined by introducing the following boundary conditionsin the path integral, Φ (+) i = n X j =1 Γ ij Φ ( − ) j , (3)with Γ ij = δ i,j +1 + δ n, identifying fields of consecutive copies across the set A. Thesuperscripts ( ± ) in (3) denote the ‘two sides’ of A, when embedded in the completeEuclidean manifold R × X d . Alternatively, for any point P ∈ A, Φ (+) ( P ) is obtained fromΦ ( − ) ( P ) by transporting the field from the ‘lower side’ to the ‘upper side’ of A along aclosed path of linking number one with the boundary ∂ A. In particular, the constructionshows that the twist operator in (2) is ‘instanton-like’, i.e. it is always sharply localizedin the time direction.The twist operator T is locally supported on ∂ A. The simplest particular case occursin 1 + 1 dimensions, where the non-local twist operator is given by a bilocal product oftwo standard (local) twist operators, i.e. (2) is a two-point function. In 2 + 1 dimensions, T is supported on a one-dimensional curve, just like standard Wilson and ’t Hooft loopoperators. Taking inspiration from the case of Wilson loops, one’s natural guess for theultraviolet contribution to (2) is D T ( n ) ∂ A E UV ∼ exp (cid:0) − α n ε − d | ∂ A | (cid:1) , (4)with ε a short-distance cutoff and | ∂ A | the volume of ∂ A in the metric of the spatialmanifold X d . Although all previous expressions hold in an arbitrary QFT in a formalsense, Eq. (4) assumes that ε − is taken beyond any mass scale in the theory, so that weare close to some ultraviolet (UV) fixed point. Since (2) is almost an n -fold product ofpartition functions, we expect α n /n to be finite in the n → ∞ limit. Furthermore, weknow that α = 0, since the partition function is not ‘frustrated’ for n = 1. The resultingentanglement entropy has the form S A = dα n dn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n =1 | ∂ A | ε d − + . . . , (5)where the dots stand for less divergent terms, which will depend in general on the size andshape of A and any intrinsic mass scales of the QFT. In Eq. (5) we see the expected UVscaling of the geometric entropy, i.e. non-extensive behavior and ‘area law’ with respectto the entangled region [2]. 2 .1 AdS and area law By direct analogy with the treatment of Wilson and ’t Hooft loops in gauge theories,the authors of [3] propose to compute (2) in an AdS/CFT prescription involving a minimalhypersurface with boundary data given by ∂ A. This procedure can be strictly justified forthe particular case of two-dimensional conformal models, for then (2) becomes a two-pointfunction of local conformal operators. More generally, we have the ansatz D T ( n ) ∂ A E QFT ⊗ n ≈ exp (cid:0) − c n Vol ( ¯A) (cid:1) , (6)where A is the minimal d -dimensional hypersurface dropped inside the bulk of the AdSspace, with boundary conditions ∂ A = ∂ A at the UV boundary of AdS. Vol (cid:0) ¯A (cid:1) standsfor the volume of the hypersurface as induced from the ambient AdS metric. Computingthe entropy via the replica-trick formula we find S A ≈ dc n dn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n =1 Vol (cid:0) ¯A (cid:1) = Vol (cid:0) ¯A (cid:1) G d +2 , (7)where the precise coefficient is fixed in [3] by comparison with known standard results intwo dimensions. Considering bulk spaces with AdS asymptotics near the boundary: ds /R −→ u ( dτ + d~x ) + du /u , (8)any minimal hypersurface A with a boundary component at infinity is asymptoticallyperpendicular to the boundary, with a volume divergence at large u of the form R d u d − ε | ∂ A | G d +2 ∝ N eff | ∂ A | ε d − , (9)where u ε = ε − and N eff = R d /G d +2 is the effective number of degrees of freedom ofthe conformal UV fixed point. Hence, the holographic ansatz obtains the expected UVstructure (4).In addition to the UV asymptotics, one can determine a UV-finite part that is alwayspresent in the conformal approximation of (8), the renormalized entanglement entropy.To find this term, let us focus on a simple situation where X d = R d and let A be the d -dimensional strip , i.e. the product of a finite interval of length ℓ times an infinite hyper-plane of codimension one in space: A = [ − ℓ/ , ℓ/ × R d − . By translational invarianceon R d − we know that the entropy will be extensive along these d − ~x , induces a rescaling of ℓ which canbe absorbed in a rescaling of u in (8) leaving the metric invariant. Hence, the induced For example, for models governed by a large- N gauge theory we have N eff ∼ N . Other models,such as the theory on a stack of N M2-branes, have N eff ∼ N / , whereas N eff ∼ N for a stack of N M5-branes. C A ( ℓ ) = ℓ ddℓ S A ∝ N eff | R d − | ℓ d − , (10)where the factor of N eff is ensured by dimensional analysis and the overall factor of G − d +2 .The quantity defined in (10), designed to remove the leading UV term, is also called the‘entropic C-function’ [8]. Therefore, a form of ‘area law’ also holds at conformal fixedpoints, at the level of the renormalized entropy.If the UV fixed point is associated to a holographic model of the form AdS d +2 × K , with K some compact Einstein manifold of dimension d K = D − d −
2, this extra structure onlyenters (7) through the Kaluza–Klein reduction of Newton’s constant G − d +2 = G − D Vol ( K ).Therefore, we can generalize the prescription to the higher-dimensional description via aminimal ( D − u . If R and G d +2 denote the asymptotic values of curvatureradius and Newton’s constant, the ansatz (7) remains valid, ensuring UV asymptoticscontrolled by N eff ∼ R d /G d +2 as before, but with modified behavior at the level of therenormalized entropy. Models arising from ten-dimensional string backgrounds are treatedin the same way, provided we remember to use the Einstein-frame metric in the ten-dimensional set up before the Kaluza–Klein reduction. In order to fix the notation, let us consider ( d + 2)-dimensional spaces with Einstein-frame metric ds = R α ( u ) (cid:18) β ( u ) h ( u ) du + d~x + h ( u ) dτ (cid:19) , (11)which are assumed to asymptote AdS d +2 at u → ∞ with radius R , i.e. the backgroundprofile functions, α ( u ) /u , β ( u ) u and h ( u ) all approach unity at infinity. More generalsituations, where the model has no smooth UV fixed point, can be dealt with by defininga running number of degrees of freedom N eff ( u ). We include the Lorentz-violating profilefunction h ( u ) in order to extend the analysis to black-hole spacetimes.One can further reduce the expression (7) for the particular situation of the strip:A( ℓ ) = [ − ℓ/ , ℓ/ × R d − , leading to a bulk hypersurface A called the ‘straight belt’ in[3]. The ‘longitudinal’ entropy density along the strip, s A = S A / | R d − | , is given by s A = R d G d +2 Z ℓ − ℓ dx γ ( u ) s β ( u ) ∂ x u ) h ( u ) , (12)with γ ( u ) ≡ α ( u ) d . This functional defines a variational problem for the profile u ( x ),which is parametrized by the radial position u ∗ of the turning point ∂ x u = 0. Thefunctional relation between ℓ and u ∗ is given by ℓ ( u ∗ ) = 2 γ ( u ∗ ) Z ∞ u ∗ du β ( u ) p h ( u ) ( γ ( u ) − γ ( u ∗ ) ) . (13) The difference between both frames amounts to a factor of exp( − φ ) in the induced volume 8-form. ℓ ( u ∗ ) ∼ /u ∗ , characteristic of the conformal case, can be readily obtainedby a heuristic argument. Let us model the surface A as the union of two components.The first component is a cylinder extending from radial coordinate u m up to u ε , andsubtending a strip of length ℓ in the boundary metric. The second component is the ‘cap’at u = u m . The entropy functional is obtained by adding the volumes of each component.Factoring out the longitudinal volume of the strip we find s A ( u m ) ∼ N eff ( u d − ε − u d − m ) + N eff u dm ℓ , with the first term coming from the cylinder and the second one from the cap at u = u m .Minimizing with respect to u m we find an optimal turning point satisfying ( d − u d − ∗ − ℓ d u d − ∗ = 0, which yields the desired relation ℓu ∗ ∼ In a model that combines a UV fixed point and an intrinsic energy scale M , the holo-graphic dual geometry will remain approximately AdS until we reach radial coordinates oforder u ∼ M , and the form (10) is a good approximation as long as ℓM ≪
1. A typicalinstance of nontrivial infrared (IR) behavior is that the radial coordinate ‘terminates’ at u = M , either because of the existence of some sort of sharp wall, a ‘repulsive’ singu-larity, or perhaps a vanishing cycle along the rest of the dimensions. In this section westudy the different types of qualitative behavior to be expected when the turning point u ∗ approaches the wall position u . By the conformal UV/IR relation, ℓu ∗ ∼
1, this willoccur roughly around ℓ ∼ /M . The precise functional dependence ℓ ( u ∗ ) is specified in Eq. (13), which we may rewriteas ℓ = Z ∞ dx ζ ( x ) √ x − , (14)where we have made a change of variables x = γ ( u ) /γ ( u ∗ ) and we have defined thefunction ζ ( x ) by ζ ( x ) = 2 γ (1) β ( x ) γ ′ ( x ) p h ( x ) , (15)where γ ′ ≡ dγ/du . The turning point u = u ∗ lies at x = 1 in the new variables, so that γ (1) = γ ( u ∗ ), and the wall sits at some x = x <
1. In the following, we use the notation γ ( u ) ≡ γ and γ ′ ( u ) ≡ γ ′ . There are three basic types of qualitative behavior as theturning point of smooth surfaces approaches the wall, x →
1, depending on whether ℓ diverges, approaches a constant value, or vanishes in this limit. These three alternatives5 igure 1: Schematic plot of the typical configuration of a smooth minimal surface that gives usthe entanglement entropy in the presence of infrared walls. Close to the position of the infraredwall, the surface prefers to lean on it, giving an extensive contribution to the entropy. We alsoshow the ‘capped cylinder’ in dashed lines, which becomes a better and better approximationto the minimal surface as ℓ → ∞ . can be translated into corresponding qualitative properties of the background, throughthe structure of the function ζ ( x ) in the vicinity of the wall x ∼ x .In our analysis, we shall adopt some technical assumptions derived from experiencewith concrete examples of holographic duals. First, we assume that β ( u ) is smooth andpositive for all u ≥ u . The Schwarzschild-like factor h ( u ) will be positive as well, exceptfor a possible single zero at u = u . Finally, the warp factor γ ( u ) is taken to be positiveand monotonically increasing for all u > u , but we allow for the possibility that it mayvanish right at the wall position. Within these technical assumptions, the only possiblesingularity structure of ζ ( x ) is either a pole at x = x , coming from a zero of the warp-factor derivative, γ ′ ( u ) = 0, or a square root singularity ζ ( x ) ∼ ( x − x ) − / , comingfrom a regular black-hole horizon, h ( u ) = 0.Since we are assuming that the model has an UV fixed point, we know that thecontribution of large values of x to (14) is well approximated by the conformal law ℓ high ∼ /u ∼ /M . Then, if ζ ( x ) is smooth in the vicinity of x = 1 we have a finite ℓ ∝ /u as u ∗ → u , perhaps decorated with some numerical factors involving dimensionlessparameters of the background. A vanishing ℓ in the wall limit can only result from avanishing warp factor γ = 0.Finally, when ζ ( x ) diverges as ( x − x ) − / or faster, we have ℓ → ∞ as x → saturates at the wall. In this case we have thesituation depicted in Fig. 1, where a smooth surface asymptotically rests over the wall,forming a cap that is smoothly joined by an almost cylindrical surface of section ∂ A,reaching out to the boundary. In the limit of very large ℓ the resulting entropy is verywell approximated by that of the ‘capped cylinder’, obtained by adjoining a ‘cap’ u × Ato the cylinder [ u , ∞ ] × ∂ A (in the case of the strip, the cylinder is just given by the twostraight [ u , ∞ ] × R d − panels, suspended a distance ℓ apart).We are now ready to detail the casuistics of different types of walls regarding theirsaturation properties. These walls are characterized by a local minimum of the warp factor, γ ′ = 0, with6 igure 2: Qualitative behavior of the warp factor near a soft wall, with γ ′ = 0 and γ >
0. Onthe right, the corresponding UV/IR relation with saturation, ℓ → ∞ , at the wall. γ > h ( u ) = 1 for simplicity). Then ζ ( x ) ∼ ( x − x ) − and the smoothsurfaces saturate (cf. Fig 2), ℓ → ∞ as u ∗ → u . We call these walls ‘soft’ because thefunction γ ( x ) usually extends for x < x , albeit with negative derivative. Static massesfeel a repulsive gravitational force that prevents physical probes from reaching far intothe x < x region. Particular examples include the confinement models of [10] based onbackgrounds with naked singularities. Hence, these models are usually regarded as formalqualitative tools at best. These walls are defined by γ , γ ′ >
0, i.e. they behave formally as the ‘sharp cutoff’regularization of a simple AdS background and are commonly used in phenomenologicaldiscussions of AdS/QCD, as in [11]. In a formal sense, they can be obtained as the ‘stiff’limit of the soft walls in the previous subsection, that is to say the limit in which theminimum of γ ( u ) at u = u is made sharper and sharper.Regarding the entanglement entropy, the most striking property of these walls is aUV/IR relation ℓ ( u ∗ ) with a maximal value of ℓ (cf. Fig. 3). Indeed, for ℓ > ℓ max theminimizing surface is not smooth. By explicit inspection, we can see that the cylinder[ u , ∞ ] × ∂ A, with a cap u × A right at the wall is a minimal surface (any perturbationtakes an element of the surface to larger u , so that it increases the induced volume).Notice that this surface is not smooth because it has sharp edges at the juncture betweenthe cap and the cylinder, but it is clearly connected and has the same topology as thesmooth surfaces that extremize the problem for ℓ < ℓ max . These walls are characterized by a vanishing warp factor, γ = 0, at the location ofthe wall. As a result, the function ℓ ( u ∗ ) vanishes as u ∗ → u . As depicted in Fig. 4 thereis again a maximal ℓ max but now there are two smooth extremal surfaces for any ℓ < ℓ max .This is the case studied at length in the examples of Ref. [6]. It corresponds to most welldefined models of confinement, in which the vanishing induced metric γ = 0 arises from avanishing cycle in some extra dimension, in such a way that the higher-dimensional metric7 igure 3: Qualitative behavior of the warp factor near a hard wall, with γ ′ > γ > ℓ . is completely smooth at u = u . For this reason we refer to these walls as ‘resolved’ bythe higher-dimensional uplifting.In addition to the mentioned examples studied in [6], there is a different type ofmodel that belongs to this class, namely that of spherical distributions of D-branes inthe Coulomb branch of N = 4 super Yang–Mills theory (cf. [12]). The dual supergravityconfiguration consists of a standard AdS × S background for u > M , matched with aflat ten-dimensional metric inside the sphere at u = M , where M is the mass scale setby the Higgs mechanism in the Coulomb branch. The hypersurface entering the entropyansatz wraps the S spheres, which remain of constant volume for all u > M but shrinkto zero size as a standard angular sphere in polar coordinates for u < M . Since thefive-dimensional warp factor γ ( u ) is proportional to the volume of the internal manifoldat fixed u , we find ourselves in the qualitative situation of Fig. 4.Among the two solutions for each ℓ < ℓ max , the standard minimal embeddings cor-respond to the solution with the larger value of u , whereas the branch of solutions with ℓ → u → u corresponds to locally unstable surfaces. The absence of stable minimalsurfaces for ℓ > ℓ max makes this case similar to that of the hard wall, and the problem isresolved in the same fashion. Namely, the minimal surface at very large ℓ is not smooth,consisting of the capped cylinder lying right at the wall. A crucial difference withrespect to the hard wall case is that γ = 0 and now the induced volume of the cap isnot just minimal, but in fact it vanishes. Hence, the contribution of the minimal surfaceto the entropy is just given by the cylindrical portion, which was referred to in [6] as the‘disconnected surface’, in the particular case of the strip.It is important to stress, however, that the minimal surface with degenerate cap isjust a member of a continuous set of surfaces with standard topology and with inducedvolume arbitrarily close to the minimal one. For this reason we do not think that thepicture of a ‘disconnected surface’ is relevant or appropriate in general, despite the factthat it gives the same answer as the capped cylinder for the particular case of these walls(more on this in a subsequent section). As shown in [6] the capped cylinder remains the absolute minimum down to ℓ c , a critical length oforder ℓ max , albeit somewhat smaller. igure 4: Qualitative behavior of the warp factor near a resolved wall, with γ ′ > γ > ℓ and two smoothextremal surfaces for each ℓ < ℓ max . Finally, the remaining case with a clear physical interpretation is that of a black holehorizon, h ( u ) = 0, corresponding to a plasma phase in the dual theory on the boundary.A regular (non-extremal) horizon will have h ′ ( u ) = 0 and the effective function divergesas ζ ( x ) ∼ ( x − x ) − / in the vicinity of the wall. This translates into a logarithmicallydivergent ℓ in the limit u ∗ → u , as defined by (14). In other words, we have saturationby smooth surfaces and a behavior qualitatively similar to that of the soft walls.The full Euclidean metric (11) of the black hole backgrounds is smooth at the wall,since the compact thermal cycle with the inverse temperature identification τ ≡ τ + β ,attains vanishing size at u = u in a smooth way. In this sense, this metric is similar tothe higher-dimensional versions of the metric in the case of the resolved walls, discussed inthe previous subsection. It is very important, however, to distinguish the two situationsby recalling that the embedded surface A is always localized in the τ direction, so thatthe vanishing thermal cycle does not translate in a vanishing induced metric at the wall,i.e. we still have γ > γ there. We conclude that in all cases the minimal surfaces effectively saturate at the wall. Insome cases the saturating surface is smooth and well described by Eq. (14), whereas insome others the minimal surface is not smooth, given by the ‘capped cylinder’ at the wall.In either case, the entropy is well approximated by that of the capped cylinder, and inthe cases of non-smooth minimal surfaces, it is exactly given (in the classical limit) bythe volume of the capped cylinder. In fact, we may carry the complete discussion at aqualitative level in terms of a restricted family of surfaces, corresponding to cylinders ofbase ∂ A with a cap at some u = u m ≥ u . The resulting entropy takes the form S ( u m ) = 14 N eff " | A | α ( u m ) d + | ∂ A | Z u ε u m du p h ( u ) β ( u ) α ( u ) d , (16)9here the first term is the contribution of the cap at u = u m and the second termcorresponds to the cylinder extending from u = u m up to the cutoff scale u = u ε . Recallthat | A | and | ∂ A | denote the volumes of A and its boundary in the field-theory metric.For the case of the strip 2 | A | = ℓ | ∂ A | . Taking the first derivative we find dSdu m = N eff | ∂ A | " ℓ d γ ′ ( u m ) − β ( u m ) γ ( u m ) p h ( u m ) , (17)where we have made use of γ ( u ) ≡ α ( u ) d . Furthermore, equating it to zero fixes u m = u ∗ and we obtain the UV/IR relation in this approximation, ℓ ( u ∗ ) = 2 d β ( u ∗ ) p h ( u ∗ ) γ ( u ∗ ) γ ′ ( u ∗ ) , (18)Eq. (18) shows all the qualitative features discussed previously on the basis of (14). Inparticular, we see that saturation, i.e. ℓ ( u ∗ → u ) → ∞ occurs whenever h ( u ) = 0 or γ ′ ( u ) = 0, whereas γ ( u ) = 0 ensures that there is a branch of solutions with ℓ → γ ′ > ℓ satisfies dS/du m > u m = u , so that theentropy is locally minimized by the capped cylinder at u m = u ∗ = u provided ℓ is largeenough.Hence, the entropy at very large ℓ is well approximated by the induced volume ofthe capped-cylinder in all cases. The contribution from the cylinder contains the cutoffdependence and it is proportional to | ∂ A | , i.e. it gives an area law. On the other hand,at large ℓ it is asymptotically independent of ℓ , so that it drops from the calculation of C A ( ℓ ). The remaining finite term is the volume of the cap at u ∗ ≈ u = M , appropriatelyredshifted to that radial position, i.e. S cap ≈ R d γ | A | G d +2 . (19)For a model asymptotic to AdS with curvature radius R , we can multiply and divide by( u ) d to obtain the extensive law C A ( ℓ ) ≈ S cap ∝ η M N eff M d | A | , (20)with N eff the effective number of degrees of freedom at the UV fixed point CFT and M = u , the value of the capping coordinate, determining the scale of the mass gap. Thenew parameter η M = γ ( u ) d (21)keeps track of the ‘flow’ of relevant parameters from the UV fixed point at u = ∞ down tothe IR mass scale u = M . For example, for a model obtained by Kaluza–Klein reductionon some internal manifold K , we have η M = Vol( K )Vol( K ∞ ) , (22)10here all volumes must be computed in Einstein-frame conventions, in the case of stringbackgrounds. In fact, we could define an effective ‘infrared’ number of degrees of freedomby the product N eff , = η M N eff .In many situations the flow parameter η M is not qualitatively important. This isfor example the case of black-hole backgrounds, with a dual interpretation in terms ofthermal states of the QFT. In general γ = 0 at such a black hole horizon, and we obtainan extensive component of the entropy, in accordance with general expectations (c.f. [3]) S A ( T ) ∝ N eff T d | A | , (23)where u ∼ T for a large-temperature black hole in AdS. In the next section we introducea very interesting system in which magnetic fields or charge condensates are capableof supporting a zero-temperature horizon with the formal properties of a thermal wall,namely supporting an nontrivial extensive law for the entanglement entropy.Finally, the most notable exception to the extensive behavior is the case of resolvedwalls [6], with γ = 0, which automatically yield η M = 0 by the vanishing of Vol( K ) in(22). We have stressed that the ‘disconnected surfaces’ invoked in [6] are in fact standardconnected surfaces in which the ‘endcap’ contributes zero volume due to the degenerateinduced metric at the wall. Indeed, there is a continuous family of surfaces of standardtopology that approximate arbitrarily well the volume of the so-called disconnected sur-face. Exactly the same situation occurs in the two-point function of Polyakov loops, usingin that case the vanishing circle Euclidean black hole backgrounds (cf. for example thediscussion in [13]). Invoking disconnected surfaces has the additional problem that they would naturallyminimize the large ℓ entropy in the presence of thermal walls, a situation in which we wouldlose the understanding of the extensivity of thermal entanglement entropy. We believe itmore likely that disconnected surfaces are related to a different observable, namely theso-called mutual information (cf. for example [14] for a discussion of its properties anduses in quantum information). Mutual information between a bipartite partition of asystem is defined as I [A , B] ≡ S A + S B − S A ∪ B , (24)where the last term is the total von Neumann entropy of the whole system, and it vanishesfor a pure state. On the other hand, for a thermal state the extensive terms cancel outof (24) and one is left with the area-law terms sensitive to the UV cutoff. Applying thisprescription to the holographic ansatz in a black hole background one finds that the ‘caps’ The term ‘disconnected surface’ strictly refers to the case of the strip, in which ∂ A is the union oftwo R d − planes. More generally, the ‘disconnected surface’ is regarded as the cylinder [ u , ∞ ] × ∂ A. We stress once more that the surfaces computing entanglement entropy are always orthogonal tothermal circles, somewhat like spatial
Wilson lines and strictly unlike timelike Wilson lines, also knownas Polyakov lines.
11n the saturation surfaces are subtracted out by the standard Bekenstein–Hawking entropyand what remains is twice the volume of the uncapped cylinder, scaling with an area law.This prediction is in agreement with the work of Ref. [15] where they demonstrate forlattice systems that the mutual information follows an area law. In this respect we proposeto reinterpret the work in [16] as a calculation of the mutual information rather than thestrict entanglement entropy at finite temperature.
In the previous subsections we have discussed the different qualitative possibilitiescorresponding to different types of walls. With little more effort one can extend theanalysis further and consider combined situations, where for instance a soft wall, with γ ′ = 0, develops a degenerate metric at the wall position, γ →
0. In this case, one muststudy the relative rates at work, and concludes that the behavior of the prefactor γ (1)dominates, producing a situation akin to that of resolved walls.Another interesting complication is the consideration of systems with IR walls and finite temperature. In this case one must compare the competing effects of the warp factor γ ( u ) versus the Schwarzschild factor h ( u ). One must also consider phase transitions ofthe background, or Hawking–Page type. Roughly speaking the dominating backgroundat a given temperature is the one with the largest value of the wall coordinate (up totransient effects that depend on the precise value of the free energies). For T ≪ M thesystem is characterized by a zero-temperature wall and h ( u ) = 1 (the ‘confined phase’),whereas for T ≫ M the system is always in the plasma phase, and the relevant wall isthe thermal one, determined by the zero of h ( u ) = 0.We may then consider the behavior of the function s A ( ℓ, T, M ). For ℓ − ≫ T, M theentanglement entropy shows conformal behavior. On the other hand, for ℓ − < max( T, M )the minimal surface will be well-approximated by the capped cylinder, and the finite partof the entanglement entropy will scale ‘extensively’ with the law (23) for T ≫ M or thelaw (20) for T ≪ M . In the cold phase the actual behavior will depend on whether η M vanishes or not. If η M = 0, such as the case of resolved walls, the entanglement entropyat very large ℓ makes a O ( N eff ) jump across the thermal phase transition. On the otherhand, for soft or hard walls with η M ∼ ℓ entropy is extensive at both lowand high temperatures and only the numerical coefficient, proportional to η M , may jumpdiscontinuously across the deconfining phase transition. The characteristic example of an extensive renormalized geometric entropy is that ofa highly thermalized state, which is modeled by a large AdS black hole in the holographicmap. In this picture, the condition for extensivity is the occurrence of an effective wall at u = u which saturates the growth of the minimal hypersurface as ℓ → ∞ . In the zero-temperature limit, u ∼ T → , studied recently in [17]. This black hole has a zero-temperature horizonwith finite entropy density, supported by magnetic and electric fluxes. The holographicdual corresponds to a maximally supersymmetric conformal field theory in 2 + 1 dimen-sions, the worldvolume theory of a stack of N M2-branes, in the presence of an externalmagnetic field and an electric charge chemical potential, both of them associated to agauged U (1) subgroup of the SO (8) R-symmetry. The dyonic black hole is a solutionof the Einstein–Maxwell theory on AdS emerging as a consistent truncation of elevendimensional supergravity on AdS × S [18].The geometry is given by ds = R u (cid:0) h ( u ) dτ + dx + dx (cid:1) + R du u h ( u ) , (25)with h ( u ) = 1 + ( h + q ) u u − (1 + h + q ) u u , (26)and electromagnetic field F = B dx ∧ dx + µ u dt ∧ du − , (27)where the dimensionless electric and magnetic parameters h, q are related to the magneticfield B and the charge chemical potential µ in the dual CFT by the relations B = h u , µ = − q u . (28)The Hawking temperature of this black hole is given by T = u π (cid:0) − h − q (cid:1) . (29)The crucial property for our purposes is that these black holes admit a zero-temperaturelimit at finite u , provided h + q →
3. Note that the restriction h + q = 3 is perfectlycompatible with general independent values of µ and B . In this limit, the horizon positionbecomes a function of the magnetic field and the chemical potential through u ≡ M = 16 h µ + p µ + 12 B i . (30)The resulting model satisfies the conditions laid down in the previous section. There isan effective mass scale determined by the charge and magnetic condensate, M eff , whichmarks the threshold separating the conformal behavior of the entropy, s A ∼ N eff (cid:18) ε − ℓ (cid:19) (31)13rom the extensive behavior s A ∼ N eff (cid:18) ε + M ℓ (cid:19) . (32)One can check the preceding statements by calculating numerically the entanglemententropy per unit length in the direction parallel to the strip. More precisely we find s A = √ N (cid:26) ε + f ( ℓ ) (cid:27) , (33)where f ( ℓ ) is cutoff independent and interpolates between the two extreme limits (31)and (32), as shown in Fig. 5. The explicit form of f ( ℓ ) is given parametrically by f ( u ∗ ) = u ∗ ( − Z dv v p h ( v ) √ − v − v !) , ℓ ( u ∗ ) = 2 u ∗ Z v dv p h ( v ) √ − v (34)where h ( v ) = 1 + 3 u u ∗ v − u u ∗ v . These results can be generalized to arbitrary CFTs in dimension d + 1, with ‘cold’charge condensates dual to extremal charged black holes in AdS d +2 (magnetic fields appeartogether with the charge only for d + 2 = 4). The corresponding supergravity solutionsare studied in Ref. [19]. In these cases, the large volume asymptotics of the geometricentropy is controlled by the chemical potential: S A ∼ N eff M d eff | A | , (35)with M eff = µ .In the preceding discussion we saw how the radius of the black hole vanishes as B and µ are taken to zero, Eq. (30). Therefore, the entanglement entropy s A ( ℓ, µ ) at fixed ℓ also interpolates smoothly between the two asymptotic laws (31) and (32) as µ is turnedon. One may wonder whether this smooth transition is maintained in other situations, inparticular in the case where the CFT lives on a sphere S d of radius R S , dual to asymptoticAdS geometries in global coordinates [19]. The main effect of the finite radius is to set aminimal value of chemical potential, µ c ∼ /R S , below which no extremal black holes arefound. This transition is smooth in the sense that the black hole, present for µ > µ c , stillgrows from zero size , according to the formula u ( µ ) = M eff = √ d − d + 1) R S s µ µ c − , The absolute normalization follows from the relation G = 3 R / √ N which gives us in this case N eff = √ N / igure 5: Numerical plot of the function f ( ℓ ), as defined in Eq. (33), in units where u = M eff = 1. The red continuous line corresponds to the case of a CFT in 2 + 1 dimensions inthe presence of an external magnetic field and/or charge condensate, with a bulk dual given bythe dyonic black hole background. The blue dashed line corresponds to the same CFT withoutany magnetic field or charge condensate, as given by the AdS dual background. We see howextensive behavior, linear in ℓ , sets it at M eff ℓ ≫ µ c = d/ ( d − R S . The phase transition between pure AdS and charged blackholes becomes more severe as soon as we depart from zero temperature. Switching on thetemperature parameter we generate a critical line in the ( T, µ ) plane, (cid:18) TT c (cid:19) + (cid:18) µµ c (cid:19) = 1 , (36)interpolating between our smooth µ = µ c critical point at T = 0 and the standard µ = 0Hawking–Page transition [20] at T = T c = d/ πR S . All T > µ at fixed T , the black hole nucleateswith a finite size u ( T ) = 2 πT /d . Hence, a discontinuous behavior of s A as a function of µ will only occur at T >
0, at least within the classical approximation, to order O ( N eff ). Our main observation in this note is the recognition of the extensivity as a generic prop-erty of renormalized entanglement entropy at zero temperature, when calculated using theAdS/CFT ansatz of Ref. [3]. Whenever the holographic geometry shows an infrared ‘wall’that effectively puts an end to the spacetime, the very large volume asymptotics of thegeometric entropy shows extensive behavior in principle . In practice, many IR walls inconcrete holographic models are associated to vanishing cycles in extra dimensions. In allthese cases, as explicitly pointed out in [6], the extensive term in the entropy vanishes toleading order in the 1 /N eff expansion (this was characterized in the text as η M = 0). Itwas suggested in [6] that the vanishing of the renormalized entanglement entropy can beregarded as an order parameter for confinement, as all of these models hold that property.Confinement, or finite string tension, arises in a rather similar fashion from the geomet-rical point of view, namely the saturation of Wilson loops at an IR wall is very similar tothe saturation of minimal hypersurfaces of [3]. There are subtle differences though, sinceWilson loops are orthogonal to vanishing cycles in extra dimensions, directly responsiblefor η M = 0. On the other hand, in high-temperature plasma phases the extensive behaviorof the minimal hypersurfaces resembles that of spatial Wilson loops.In the better-defined models, namely the resolved walls and the thermal walls, one cansay that the extensivity is associated to the breaking of conformal symmetry without thegeneration of a mass gap, since the plasma has O ( N eff ) massless degrees of freedom while,in contrast, the resolved walls have a mass gap and the extensive term in the entropyvanishes.It is worth stressing that these comments only apply to leading order in the large N eff expansion. There are many examples of confinement models without strict massgap, having Goldstone bosons in the spectrum from some spontaneously broken globalsymmetry (cf. for example [21]). However, in all these cases, the number of masslessmodes is O (1) in the large N eff limit, and thus their effects are invisible in the classicalapproximation to the geometric entropy that is being used here.16he typical example of extensive renormalized entanglement entropy is that of highlythermalized states. We have shown in this paper that one can find systems where theextensivity holds down to zero temperature, supported by magnetic field and/or chargecondensates. This is an interesting prediction of the AdS/CFT ansatz for entanglemententropy, although the microscopic interpretation of this property in weak coupling remainsan open problem. Indeed, it would be very interesting to check the conditions for thisextensivity using purely QFT methods. Finally, the identification of systems with peculiarentropy behavior at zero temperature opens up possible applications to the emerging fieldof quantum phase transitions [22]. Acknowledgments
We are indebted to J. I. Cirac, C. G´omez and S. Sachdev for discussions. This workwas partially supported by MEC and FEDER under grant FPA2006-05485, CAM undergrant HEPHACOS P-ESP-00346 and the European Union Marie Curie RTN networkunder contract MRTN-CT-2004-005104. C.A.F. enjoys a FPU fellowship from MEC undergrant AP2005-0134.