Holographic entanglement entropy in the nonconformal medium 5 1 0 2 Chanyong Parka∗ r a M a Center for Quantum Spacetime (CQUeST), Sogang University, Seoul 121-742, Korea 5 ] h t - p e h ABSTRACT [ 2 v 8 0 We investigate holographically the entanglement entropy of a nonconformal medium whose dual 9 geometry is described by an Einstein-Maxwell-dilaton theory. Due to an additional conserved charge 2 0 corresponding to the number operator, its thermodynamics can be represented in a grandcanonical or . 1 canonical ensemble. We study thermodynamics in both ensembles by using the holographic renormal- 0 5 ization and the entanglement entropy of a nonconformal medium. After defining the entanglement 1 : chemical potential which unlike the entanglement temperature has a nontrivial size dependence, we v i find that the entanglement entropy of a small subsystem satisfies the relation resembling the first X law of thermodynamics in a medium. Furthermore, we study the entanglement entropy change in r a the nonconformal medium caused by the excitation of the ground state and by the global quench corresponding to the insertion of particles. ∗ e-mail : [email protected] 1 Introduction For the last decade, there have been a huge amount of efforts to understand strongly interacting systems via the AdS/CFT correspondence [1]. This new concept allowed us to study various micro- scopic as well as macroscopic properties of a conformal field theory (CFT) in the strong coupling regime, for example, 4-dimensional = 4 super Yang-Mills theory [2, 3, 4] or 3-dimensional = 6 N N Chern-Simons gauge theory [5, 6]. Its dual geometry is usually represented as an asymptotic anti de- Sitter (AdS) space with a proper compact manifold. These researches have been further generalized to the relativistic nonconformal and nonrelativistic field theory [8]-[26]. Accumulating knowledges on those nonconformal examples would be important in understanding the underlying structure of the gauge/gravity dualityindepthandinapplyingthemtomorerealistic physicalphenomenaofquantum chromodynamics and condensed matter system. Moreover, in order to figure out the quantum aspect of such systems, the entanglement entropy becomes an important issue[27]-[45]. In this paper, we will study the entanglement entropy and its thermodynamics-like behavior in a medium holographically. For regarding a nonconformal theory, we should violate a scaling symmetry of the dual geometry which can be realized by adding a scalar field called dilaton. Then, the resulting geometry does not allow an asymptotic AdS space as a solution. Let us take into account an Einstein-Maxwell-dilaton theory. This gravity theory permits two different types of the generalized black brane solution. One is a charged dilatonic black brane whose dual theory is mapped to a relativistic nonconformal theory with matter, while the other is an uncharged black brane dual to a generalized Lifshitz geometry where the Lifshitz scaling symmetry is broken [13]. The first corresponds to the deformed Reinssner- Nordstro¨m AdS black brane with a nontrivial dilaton profile. The gauge field plays a different role in those two examples. For a charged dilatonic black brane, the bulk gauge field provides an additional conserved charge representing one of the black brane hairs, so that its dual is clearly interpreted as the numberdensity operator of the matter. In the generalized Lifshitz theory, the bulk field is not free and does not provide a new black brane hair. Instead, it breaks the boost symmetry and generates the anisotropy between time and spatial coordinates. This is why the Lifshitz-type uncharged black brane appears [8, 9, 11, 13]. Anyway, since we are interested in the nonconformal medium, we focus on a charged dilatonic black brane from now on. In general, it is not easy to calculate the entanglement entropy of an interacting quantum field theory(QFT).However,thegauge/gravitydualitycanshedlightonstudyingtheentanglemententropy even in a strong coupling regime. In [27, 28, 29], it was shown that the holographic entanglement entropy proportional to the area of the minimal surface exactly reproduces the known results in a 2-dimensional CFT [46]. This work was further generalized to the higher dimensional cases. In an IR limit, the holographic entanglement entropy in a black hole geometry reduces to the well-known Bekenstein-Hawkingentropy. Meanwhile,inasmallsubsystemcorrespondingtoaUVlimititdescribes theentanglement entropy of excited states [47]. Inspiteofthefactthattheentanglement temperature 1 is different from the real temperature of the system, the holographic entanglement entropy satisfies the thermodynamics-like relation. In a dual CFT, the entanglement temperature is proportional to the inverse of a subsystem size, T 1/l. This is also true for a relativistic nonconformal QFT E ∼ dual to a hyperscaling violation geometry [47]-[55]. This fact implies that the size dependence of the entanglement temperature is independent of details of the theory and the entangling surface. From now on, we say that the entanglement temperature is universal because it always has the same form in a relativistic dual QFT1. Is this still true in a medium? To answer this question, let us first think of thermodynamics. In a medium, there exists an additional conserved quantity corresponding to the number of particles. So the first law of thermodynamics is modified into the form including the particle number. For the entanglement entropy to satisfy such a modified thermodynamic relation, one should define a new variable representing the chemical potential which we will call the entanglement chemical potential. Like the entanglement temperature, it is different from the chemical potential defined in thermody- namics. Due to the modification of the thermodynamic relation and the new conserved charge in the medium, we cannot easily answer the previous question and furthermore new issues appear. Does the entanglement entropy in a medium follow the modified thermodynamics-like relation? If so, does the entanglement temperature still show the same universality? Lastly, does the newly defined entangle- ment chemical potential have a universal form independent of the details of the theory? One of goals in this paper is to clarify them. We findthat the entanglement entropy in a medium follows the modi- fied first thermodynamics-like relation and that the entanglement temperature still remains universal. However, we show that the size dependence of the entanglement chemical potential nontrivially relys on the nonconformality. Finally, we consider the uniform insertion of particles at zero temperature which can be regarded as a global quench deforming the original theory. Since this global quench modifies the quantum states of the system, the entanglement entropy changes. Under such a global quench, we calculate the change of the entanglement entropy quantitatively. The rest of paper is organized as follows. In Sec. 2, we study a charged black brane solution of an Einstein-Maxwell-dilaton gravity which is dual to a nonconformal medium. From this solution, we calculate the thermodynamic properties in a grandcanonical and canonical ensemble. In Sec. 3, its entanglement entropy in a small subsystem is taken into account. Due to an additional conserved quantity,theentanglementchemicalpotentialisnewlydefined. Usingitweshowthattheentanglement entropy follows the first thermodynamics-like relation and that the entanglement chemical potential has a nontrivial size dependence relying on the nonconformality. In addition, we also investigate the change of the entanglement entropy under the global quench corresponding to the uniform insertion of particles. We finish our work with some concluding remarks in Sec. 4. 1Inmoregeneralnonrelativisticcaseswithadynamicalexponent,thesizedependenceoftheentanglementtemperature is furthergeneralized toTE ∼1/lz [47]. 2 2 Charged dilatonic black brane in the Einstein-Maxwell-dilaton theory Let us consider the following Einstein-Maxwell-dilaton gravity [13, 14, 16] 1 e2αφ S = d4x√ g R 2(∂φ)2 F Fµν 2Λeηφ , (1) 2κ2 − − − 4 µν − Z (cid:20) (cid:21) where Λ denotes a negative cosmological constant. This theory provides several different geometric solutions. If φ is a constant and F = 0, the simplest solution is given by a 4-dimensional AdS space. µν It can be generalized to a Schwarzschild AdS (SAdS) black brane in the Poincare patch (or SAdS black hole in the global patch). The SAdS black brane is characterized by one parameter called the black brane mass. Turning on the gauge flux, SAdS black brane is further generalized to a Reissner- Nordstro¨m AdS (RNAdS) black brane with two hairs, mass and charge. The asymptote of all these solutions is described by an AdS geometry. When φ has a nontrivial profile, the previous geometries are not solutions anymore. In this case, the solutions of the Einstein-Maxwell-dilaton theory are classified as follows. If F = 0, the µν Einstein-Maxwell-dilaton theory reduces to an Einstein-dilaton theory, which allows a hyperscaling violation geometry [14]-[26]. Since the overall factor of the hyperscaling violation metric breaks the scaling symmetry, its asymptote is not an AdS space. In spite of breaking of the scaling symmetry, the rotation and translation symmetries of the boundary space still survive. This fact implies that its dualfieldtheorycorrespondstoarelativisticnonconformalQFT.Ingeneral,thehyperscalingviolation geometry has a naked singularity at the center which may indicates the instability or incompleteness of the theory. This fact indicates that the dual QFT is IR incomplete. To avoid this problem, one can regard the black brane geometry. Analogous to a SAdS black brane, the hyperscaling violation geometry can beeasily generalized toan unchargedblack branewherethesingularity is hiddenbehind the horizon. In the dual field theory at finite temperature, there is no IR incompleteness because the Hawking temperature plays an effective IR cutoff. Even in this case, the zero temperature limit still remains problematic. Another way to get rid of the IR incompleteness is to take into account a medium which is dual of a charged black brane geometry. In this case, the dual QFT has an IR fixed point at which the dual theory effectively becomes a 1 + 1-dimensional CFT. As a result, a QFT with matter dual to a charged dilatonic black brane is free from the IR incompleteness even at zero temperature. In order to obtain a charged dilatonic black brane solution, let’s turn on the gauge flux with an appropriate parameter α. Then, we can expect that an uncharged black brane solution of the Einstein-dialton gravity [13] is modified into a charged one with two black brane hairs. It is true only for a specific value of α. For general α, intriguingly, there exists another uncharged black brane solution in which the boost symmetry as well as the scaling symmetry are broken. Thus the time and spatial coordinate behave differently [12, 13]. This is the generalization of the well-known Lifshitz geometry [8]. In this paper, we concentrate on a QFT dua to a charged dilatonic black brane and 3 investigate its quantum aspects described by a holographic entanglement entropy. The equations of motion fo the Einstein-Maxwell-dilaton theory are 1 e2αφ e2αφ R Rg +g Λeηφ = 2∂ φ∂ φ g (∂φ)2 + F F λ g F2, (2) µν µν µν µ ν µν µλ ν µν − 2 − 2 − 8 1 ηΛ α e2αφ ∂ (√ g∂µφ) = eηφ + F2, (3) µ √ g − 2 8 − 0 = ∂ (√ ge2αφFµν). (4) µ − In order to solve these equations, we take a logarithmic dilaton profile φ(r) = φ φ logr, (5) 0 1 − where φ and φ are two integration constants. Since φ can be absorbed into the cosmological 0 1 0 constant, we can set φ = 0 without loss of generality. Now, we consider the following metric ansatz 0 for a charged dilatonic black brane dr2 ds2 = g(r)2f(r)dt2+ +h(r)2(dx2 +dy2), (6) − g(r)2f(r) with g(r) = g rg1, h(r) = h rh1. (7) 0 0 Here the diffeomorphism allows us to set g = h = 1. When we turn on a time-component gauge 0 0 field A only, the electric field satisfying (4) is given by t q F = e−2αφ. (8) rt h(r)2 Note that a charged black brane we consider is the generalization of an uncharged black brane studied in the Einstein-dilaton theory [25], which preserves the boundary Lorentz symmetry. Since the bulk gauge field, following the gauge/gravity duality, is related to the matter, the dual field theory of a charged black brane corresponds to a relativistic nonconformal QFT with the matter. In order to preserve the boundary Lorentz symmetry, g = h should be satisfied. Furthermore, only when 1 1 α= η/2, there exists a charged black brane solution satisfying all equations of motion. In this case, − the integration constants are determined as 4 2η 4(12 η2) g = , φ = and Λ= − , (9) 1 4+η2 1 4+η2 − (4+η2)2 and the black brane factor is given by m b f(r)= 1 + , (10) − ra rc with 12 η2 4+η2 16 a = − , b = Q2 and c = a+1 = , (11) 4+η2 16 4+η2 4 where m and Q denote two black brane hairs. In the η = 0 limit, this charged dilatonic black brane reduces to an RNAdS black brane and the scaling symmetry is restored [25]. The near horizon geometry of the charged dilatonic black brane reduces to AdS R2, which is independent of the 2 × nonconformality, η, and shows the existence of an IR fixed point effectively described by a 1 + 1- dimensional CFT. Above we used the nonconformality parameter to clarify the nonconformal effect, which is also related to the hyperscaling violation exponent [26] 2η2 θ = . (12) −4 η2 − As shown in [56], the holographic renormalization together with regularity conditions of bulk fields provide a boundary stress tensor consistent with the black brane thermodynamics. The regularity of the metric requires that there is no conical singularity at the horizon and yields the Hawking temperature 12 η2 (4+η2)2 Q2 4−η2 T = − 1 r4+η2. (13) H 4(4+η2)π  − 16(12 η2) 16  h − r4+η2 h   From the Maxwell equation, the time component of the vector field A is determined as t Q A =2κ2µ , (14) t − r where µ is an integration constant interpreted as a chemical potential. The regularity of the vector field norm at the horizon gives rise to the relation between the chemical potential and the particle number N = 2κ2V µr , (15) 2 h where N = QV . 2 2.1 Holographic renormalization of the grandcanonical ensemble LetusconsidertheholographicrenormalizationoftheEinstein-Maxwell-dilaton theory,whichprovides direct interpretation of the boundary energy-momentum tensor as thermodynamic quantities. With an Euclidean signature, the Einstein-Maxwell-dilaton action is rewritten as S = d4x , (16) E D L Z with 1 e2αφ = √g R 2(∂φ)2 F Fµν 2Λeηφ , (17) LD −2κ2 − − 4 µν − (cid:20) (cid:21) where the Euclidean metric is given by dr2 ds2 = r2g1f(r)dt2+ +r2g1(dx2 +dy2), (18) E r2g1f(r) 5 and the Euclidean vector field becomes Q A = i 2κ2µ . (19) τ − − r (cid:18) (cid:19) These metric and vector field together with the dilaton field in (5) satisfy the Eucidean equations of motion. In order to evaluate the on-shell gravity action, we should add several boundary terms. The first is the Gibbons-Hawking term which is required to define the metric variation well 1 S = d3x √γ Θ, (20) GH κ2 Z∂M where γ indicates an induced metric on the boundary and the extrinsic curvature is given, in terms ij of the unit normal vector n , by µ 1 Θ = ( n + n ). (21) µν µ ν ν µ −2 ∇ ∇ The second boundary term is a local counter term which is needed to make the on-shell action finite at the boundary. The correct counter term is 8 S = d3x √γ eηφ/2. (22) ct (4+η2)κ2 Z∂M This term is the same as the one used in the holographic renormalization of the Einstein-dilaton theory [26]. Since the vector field of the charged black brane does not generate new divergence at the UV regime, no additional counter is required [56]. If we impose a Dirichlet boundary condition on the vector field at the asymptotic boundary, it fixes the chemical potential. In this case, all physical quantities should berepresented as functions of the chemical potential and the on-shell gravity action, inthedualQFTpointofview,isproportionaltothegrandpotentialofagrandcanonicalensemble. On the other hand, imposing a Neumann boundary condition instead of a Dirichlet boundary condition is related to choose a canonical ensemble and requires an additional boundary term corresponding to the Legendre transformation. The grand potential with a Dirichlet boundary condition leads to Ω(T ,µ,V ) = T (S +S +S ) H 2 H E GH ct 4 η2 V2 (12−η2)/(4+η2) (4+η2)κ4 µ2 = − r 1+ , (23) −2(cid:0)(4+η2(cid:1))κ2 h 4 r2(4−η2)/(4+η2)! h where V denotes the spatial volume at the boundaryand the horizon r becomes an implicit function 2 h of T , µ and V from (13) and (15). The boundary energy-momentum tensor, which is obtained by H 2 varying the on-shell gravity action with respect to the boundary metric δ Ti lim 2 γik d3x LD , (24) j ≡ r0→∞(cid:18)− Z δγkj(cid:19) 6 reads E = T0 0 4V2 (12−η2)/(4+η2) (4+η2)κ4 µ2 = r 1+ , (4+η2)κ2 h 4 2(4−η2)/(4+η2) r ! h T1 T2 1 2 P = = − V − V 2 2 4 η2 (12−η2)/(4+η2) (4+η2)κ4 µ2 = − r 1+ . (25) 2(4+η2)κ2 h 4 2(4−η2)/(4+η2) (cid:0) (cid:1) r ! h From this result, one can see that the grand potential is related to pressure Ω = PV . (26) 2 − Fromtheexact differentialrelation, thecanonical conjugatevariablesof thefundamentalvariables, T , µ and V , are evaluated to H 2 ∂Ω 2πV2 8/(4+η2) S = = r , (27) − ∂T κ2 h H(cid:12)µ,V2 (cid:12) ∂Ω (cid:12) N = (cid:12) = 2κ2V2rhµ, (28) − ∂µ (cid:12)TH,V2 (cid:12) P = ∂Ω(cid:12)(cid:12) = 4−η2 r(12−η2)/(4+η2) 1+ (4+η2)κ4 µ2 . (29) − ∂V2(cid:12)TH,µ 2((cid:0)4+η2)(cid:1)κ2 h 4 rh2(4−η2)/(4+η2)! (cid:12) (cid:12) Since the conjugate var(cid:12)iable of temperature is the entropy, S denotes the renormalized thermal en- tropy derived from the renormalized grand potential. Intriguingly, this renormalized thermal entropy coincides with the Bekenstein-Hawking entropy. The particle number in (28) is in agreement with the regularity of the vector field. As a consequence, the holographic renormalization results are per- fectly matched to those of the charged black brane thermodynamics in (A.11). The equation of state parameter of this system is given by PV 1 η2 2 ω = = . (30) E 2 − 8 Since the equation of parameter is independent of the chemical potential, it is the same as that obtained in the Einstein-dilaton theory [26]. Here η indicates the nonconformality representing the deviation from the CFT. According to the AdS/CFT correspondence, the conformal dimension of the dual operator for a bulk p-form field in AdS is determined by [2, 3] d+1 (∆+p)(∆+p d) = m2, (31) − where the largest value of ∆ corresponds to the conformal dimension of the dual operator. This relation says that for η = 0 the dual operator of a massless bulk vector field (p = 1 and d = 3) has a 7 conformal dimension 2. In general, we may consider many different conformal dimension 2 operators composed of scalars or fermions. One example we are interested in is the operator composed of two fermions, = ψ¯γ ψ. Since a fermion in a 2+1-dimensional conformal field theory has a conformal µ µ O dimension 1, can bea dualoperator. In this case, similar to the 5-dimensional RNAdS black brane µ O [56], we can regards the boundary value of a time-component gauge field A (z = 0) as the chemical t potential. For a general η, the relation between the bulk field and boundary operator is not clear. However, if it is regarded as the nonconformal deformation from the conformal field theory, it may be possible to generalize the AdS/CFT correspondence to the noncoformal case. Here, we just assume that there exists such a generalization. The action form we consider openly appears in the string theory with specific value of η [58, 59, 60, 61]. In this case, the bulk scalar field appears as a dilaton in the string theory and the boundary value of the dilaton field is identified with the gauge coupling of the dual QFT. In the holographic point of view, the nontrivial dilaton profile implies the nontrivial gauge coupling depending on the energy scale. For η = 0, since the dual theory is conformal, the dilaton field becomes trivial. On the other hand, (30) shows an explicit nonconformality for a general η. It would be interpreted as the effectofanirrelevantdeformationorinteraction becauseitaffects ontheUVbehavior. Moreprecisely, the deviation from the conformal theory can be read from the trace of the energy-momentum tensor. Taking the trace of the stress tensor in (25), yields Ti = E 2P = η2 r(12−η2)/(4+η2)+ η2κ2 µ2r . (32) i − (4+η2)κ2 h 4 h The first term shows the effect of a nonconformal interaction, whereas the second is the effect of the matter. These effects disappear in the conformal limit, η 0, as expected. Intriguingly, for η2 = 4 → the energy and pressure are reduced to V E = 2 1+2κ4µ2 r and P = 0, (33) 2κ2 h (cid:0) (cid:1) which implies that the dual system is composed of pressureless particles, the so-called dust. Now let usconsider the zero temperaturebehavior. Intheextremal case (T = 0) except thedust, H the horizon in terms of the chemical potential is given by (4+η2)/(4−η2) (4+η2)κ2µ r = . (34) h 2 12 η2 ! − p If the dual matter is fermionic, we may regard a Fermi surface. If there exists such a Fermi surface even in the strong coupling regime, the Fermi surface energy at zero temperature can be identified withthechemical potential, ǫ = µ. From(28)theFermisurfaceenergyisproportionaltothefermion F number density, n = N/V , 2 ǫ n(4−η2)/8. (35) F ∼ 8 For the conformal case, the Fermi surface energy is proportional to √n which is similar to that of 2+1-dimensional free fermions, although we cannot directly compare them. In the dust case with η2 = 4, (34) becomes singular so that we cannot apply it directly. Instead, we should look at the metric which becomes simple for the dust Q2 g = r m+ . (36) tt − − 2r (cid:18) (cid:19) For m √2Q, it has two horizons ≥ m m2 2Q2 r± = ± − , (37) 2 p and the case saturating m = √2Q gives rise to the extremal limit corresponding to the zero tempera- ture. The Hawking temperature reads in terms of the chemical potential 1 T = 1 2κ4µ2 . (38) H 4π − (cid:0) (cid:1) At zero temperature, the chemical potential is given by 1 µ = . (39) √2κ2 Using this relation, the horizon is determined only by m m r = , (40) h 2 where m still remains as a free parameter. The number density of the dust at zero temperature yields n = √2m, (41) where (39) was used. This result shows that there is no direct relation between Fermi surface energy and momentum. 2.2 Holographic renormalization of the canonical ensemble Asmentioned previously,ifoneimposesaNeumannboundaryconditionon thevector field,thecharge density Q is fixed. In this case, the dual system is described by a canonical ensemble. To see this, let us vary the action with respect to the vector field. Then, it generally generates a nontrivial boundary term, the so-called Neumannizing term [10], 1 δS = d3x √g e2αφgrrgττF δA . (42) E 2κ2 rτ τ Z∂M WhenimposingtheDirichletboundarycondition,thistermautomaticallyvanishesbecauseofδA = 0. τ However, the Neumann boundary condition cannot get rid of the Neumannizing term. In order to remove it, we should add a new boundary term 1 S = d3x A Jµ, (43) bd 2κ2 µ Z∂M 9