# Holographic Renormalization in Dense Medium

Authors: Chanyong Park
Journal: Advances in High Energy Physics
 (2014)

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## Abstract

The holographic renormalization of a charged black brane with or without a dilaton field, whose dual field theory
describes a dense medium at finite temperature, is investigated in this paper. In a dense medium, two different thermodynamic descriptions are possible due to an additional conserved charge. These two different thermodynamic ensembles are classified by the asymptotic boundary condition of the bulk gauge field. It is also shown that in the holographic renormalization regularity of all bulk fields can reproduce consistent thermodynamic quantities and that the Bekenstein-Hawking entropy is nothing but the renormalized thermal entropy of the dual field theory. Furthermore, we find that the Reissner-Nordström AdS black brane is dual to a theory with conformal matter as expected, whereas a charged black brane with a nontrivial dilaton profile is mapped to a theory with nonconformal matter although its leading asymptotic geometry still remains as AdS space.


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## Body

## 1. Introduction

The AdS/CFT correspondence [1–5] is a fascinating tool in studying the strongly interacting system like quantum chromodynamics (QCD) or condensed matter system. Although there is no direct proof for the AdS/CFT correspondence, it is conceptually supported from the same symmetry between a d+2-dimensional Anti de Sitter (AdS) gravity and d+1-dimensional conformal field theory (CFT). The key relation for this duality is that the semiclassical partition function of AdS gravity corresponds to the generating functional of a dual CFT [1]:(1)e-Sgrav=〈e-S〉CFT.
This relation has been widely used and checked in various works related to integrability and localization [6–8]. For more realistic systems, it is important to encode the matter effect properly which has been considered by considering the dual geometry in [9–11]. In this paper, we will investigate such a matter theory by using the holographic renormalization technique [12–22].In general, the partition function of a quantum field theory suffers from the UV divergence which should be renormalized by adding appropriate counter terms. This is also true in the dual gravity where the corresponding IR divergence is caused due to infinite volume of the background geometry. In [12, 13], it was shown that adding local counter terms to the AdS boundary makes the on-shell gravity action finite and that quantities derived from it can be reinterpreted as those of the dual CFT. This holographic renormalization has been generalized even to the non-AdS geometries like the Lifshitz geometry [23–28] and Einstein-dilaton gravity [29–39]. In these non-AdS examples, the renormalized gravity action, after assuming the gauge/gravity duality, was used to understand thermodynamics and hydrodynamics of the dual nonrelativisitic or nonconformal theory. In this procedure, the counter terms can not only get rid of the UV divergence but also yield additional finite contributions, so finding the correct counter terms is crucial for obtaining correct physical quantities. This is also true in the renormalization of a quantum field theory.In holography, a conformal field theory matches to asymptotic AdS geometry. Introducing matter may break the conformal symmetry depending on the properties of matter. In the strong coupling regime, it is hard to investigate such matter effects through the traditional quantum field theory method. Although many things still remain to be clarified, recent numerous works based on the AdS/CFT correspondence provided some clues for understanding a strongly interacting system. In the holographic set-up, matter of the dual CFT can be realized as a field in AdS space. In particular, a massless vector field in AdS is mapped to quark or hadronic number operator in the deconfining or confining phase, respectively [9–11, 40, 41]. Note that the black brane geometry is dual to a deconfining phase, while the nonblack brane geometries, thermal AdS, and thermal charged AdS in the hard wall model describe a confining phase. From now on, we concentrate on a charged black brane geometry.The holographic renormalization of thed+2-dimensional gravity theory leads to the stress tensor of the d+1-dimensional boundary field theory. For a charged black brane, the dual theory contains quark matter. In the thermodynamic point of view, the existence of matter makes two different thermodynamic interpretations possible [42, 43]. On the gravity side, these two different ensembles can be represented by the asymptotic boundary condition of the vector field. For example, the grand canonical ensemble described by the chemical potential corresponds to the Dirichlet boundary condition, while the Neumann boundary condition is dual to the canonical ensemble. These two thermodynamic systems are connected to each other through the Legendre transformation. In a charged black brane, there exists a natural boundary condition for a vector field at the horizon. The regularity of a vector field at the horizon is related to the Legendre transformation. As mentioned before, the holographic renormalization requires appropriate counter therms. In addition, in the canonical ensemble, one more boundary term, the so-called Neumannizing term, is needed to describe the Neumann boundary condition which is crucial to obtain the correct thermodynamics. We show that physical quantities derived by the holographic renormalization perfectly satisfy the thermodynamic law in both thermodynamic ensembles. Intriguingly, in the extremal limit of a Reissner-Nordström AdS (RNAdS) black brane we find that the Fermi surface energy of the dual field theory is proportional to N1/3 in the UV limit, where N denotes the number of quark. If there exists asymptotic freedom in the dual CFT, one can identify the matter with the free relativistic fermion because it gives rise to ϵF~N1/3. However, since there is no asymptotic freedom in the holographic dual CFT, one can not directly interpret it as free relativistic fermions. We also consider another charged black brane in the Einstein-Maxwell-dilaton theory in which the holographic renormalization also leads to the consistent physical quantities satisfying the thermodynamic law. Although the extremal limit is not well defined in this case, one can still think of the Fermi surface in the low temperature limit. The Fermi surface energy of the Einstein-Maxwell-dilaton theory is also proportional to ϵF~N1/3 similar to the RNAdS black brane case. The holographic renormalization results show that dual matter of the RNAdS black brane does not break the conformal symmetry at the UV fixed point, whereas dual matter of the Einstein-Maxwell-dilaton theory breaks the conformal symmetry in spite of the asymptotic AdS geometry.The rest of paper is organized as follows. In Section2, after applying the holographic renormalization to the Einstein-Maxwell theory, we calculate the on-shell actions and boundary stress tensors of two thermodynamic ensembles. From these result, we derive various thermodynamic quantities and show that they satisfy the thermodynamic law. In Section 3, the holographic renormalization technique is applied to another asymptotic charged AdS black brane of the Einstein-Maxwell-dilaton theory. Finally, we finish this work with some concluding remarks.


## 2. Holographic Renormalization of the RNAdS Black Brane

Let us first take into account a 5-dimensional RNAdS black brane. A charged black brane solution usually has two hairs, charge, and mass, so that its thermodynamics can be represented as two different thermodynamic systems depending on the choice of the fundamental thermodynamic variables [42]. On the gravity theory side, the choice of ensemble is closely related to the asymptotic boundary condition of a bulk gauge field. For example, imposing the Dirichlet boundary condition on the gauge field corresponds to taking a grand canonical ensemble, whereas one should impose the Neumann boundary condition in order to describe a canonical ensemble. In general, those two thermodynamic systems are related to each other by the Legendre transformation (see the details in the Appendix). According to the AdS/CFT correspondence, the black brane thermodynamics can be identified with that of the dual field theory [5, 13, 25–27, 30, 31, 38, 43]. In this section, by using the holographic renormalization we will investigate thermodynamics of the field theory dual to the RNAdS black brane. As will be shown, the holographic renormalization results are perfectly matched to the black brane thermodynamics.For the direct comparison with the black brane thermodynamics, we regard an Euclidean Einstein-Maxwell action with a negative cosmological constant. To obtain the Euclidean action from the Lorentzian one in (A.1), one can apply Wick rotation which describes rotation of time coordinate in a complex plane:
(2)xt⟶-ixτ,xt⟶ixτ.
Under this Wick rotation, the field strength Frt with a Lorentzian signature transforms to
(3)Frt⟶∂rAt-i∂τAr.
In order to preserve the usual definition form of the field strength FMN=∂MAN-∂NAM up to an overall factor, we should also rotate the time component of the gauge field as At→iAτ. Then, the Lorentzian field strength changes to the Euclidean one, iFrτ, with AM={Aτ,A→}. These Wick rotations lead to the following Euclidean action:
(4)SE=∫‍d5xg[-12κ2(R-2Λ)+14g2FMNFMN],
where the Euclidean metric is used. Since the Euclidean action is the same as the Lorentzian one up to an overall minus sign, all equations of motion are the same only except the fact that the Euclidean metric should be used. For example, the Euclidean Einstein and Maxwell equations have the same forms as the Lorentzian ones in (A.2), which allows the following Euclidean RNAdS black brane solution [9]:
(5)Aτ=-iAt=-i(g2μ-Q2r2),ds2=r2f(r)dτ2+dr2r2f(r)+r2(dx2+dy2+dz2),
with
(6)f(r)=1-mr4+κ26g2Q2r6,
where m and μ are a black brane mass and chemical potential, respectively, and we define its electric charge as (κ2/6g2)Q for later convenience. As mentioned before, a charged black brane has two thermodynamic interpretations due to the conserved charge. In the thermodynamic point of view, one corresponds to the grand canonical ensemble described by temperature, volume, and chemical potential. If one uses the charge as a fundamental variable instead of the chemical potential, the black brane thermodynamics must be represented as the canonical ensemble.The requirement of the regular metric at the horizon generally leads to specific Euclidean time periodicityβ corresponding to the inverse Hawking temperature. Due to the existence of the well-defined temperature, one can expect that the black brane may be regarded as a thermal system. This is true if the entropy is given by the Bekenstein-Hawking entropy proportional to the horizon area. This thermodynamic interpretation causes a big issue, why the entropy is proportional to the area but not to volume. There have been many attempts to account for this area law. One of them is the holographic principle in which people have tried to map the black brane thermodynamics to that of the boundary theory. Recently, the concept of the holography has been further developed to the gauge/gravity duality of the string theory and becomes a fascinating tool in studying a strongly interacting system [1–5, 20]. Following this holographic interpretation the area related to the Bekenstein-Hawking entropy is identified with the volume of the boundary theory, where the Bekenstein-Hawking entropy behaves as an extensive quantity expected in ordinary thermodynamics. However, even in this case it is still mysterious why the Bekenstein-Hawking entropy is defined at the horizon not the boundary. In general, the Hawking temperature is interpreted as temperature detected by an observer living at the asymptotic boundary. Similarly other conserved quantities are also defined at the asymptotic boundary. In order to get well-defined thermodynamics of the boundary theory, it must be understood why the Bekenstein-Hawking entropy appears as the thermal entropy of the dual theory. The holographic renormalization together with natural assumptions, as will be shown, indicates that the renormalized thermal entropy of the boundary theory is exactly given by the Bekenstein-Hawking entropy.Physically, it is quite natural to assume that all fields are regular in the entire space. If not so, the theory becomes singular. In the Einstein-Maxwell theory, there exist two bulk fields, metric and vector fields. At the horizon, the absence of the conical singularity in the metric determines the time periodicity. The Hawking temperature is represented as the inverse of the time periodicity:(7)TH=rhπ-κ2Q212πg21rh5,
where the horizon rh satisfies f(rh)=0. Although the Hawking temperature is evaluated at the horizon, as mentioned before, it should be regarded as temperature at the asymptotic boundary. Under the coordinate transformation, the invariant combination of the vector field is given by gμνAμAν which reduces to gττAτAτ on the RNAdS black brane geometry. Since gττ diverges at the horizon, the regularity requires that Aτ should vanish at the horizon. This fact yields a relation between μ and Q:
(8)μ=Q2g2rh2.
In the holographic renormalization, these two consequences of natural assumptions can derive all desired thermodynamic quantities as will be shown. It is worth to note that two parameters of the black brane, m and Q, can be represented as functions of TH and μ (or Q depending on the boundary condition of the vector field) [9]. From now on, we take into account TH and μ (or Q) as fundamental parameters in which rh becomes a function of them.In general, the gravity action without additional boundary terms is not well defined because its variation becomes problematic at the boundary. Therefore, the following two boundary terms should be added:(9)Sb=1κ2∫∂M‍ddxγΘ+ζg2∫∂M‍d4xγnrFrτAτ,
where γab and Θ denote the induced metric and the extrinsic curvature. The unit normal vector is given by nr=grr. The first is the Gibbons-Hawking term and the last is called the Neumannizing term which determines the boundary condition of Aτ, for example, ζ=0 or 1 for the Dirichlet or Neumann boundary condition, respectively [23]. As a consequence, the action having the well defined variations is given by the sum of the bulk in (4) and boundary actions in (9).The useful and crucial relation in the AdS/CFT correspondence is that the on-shell gravity action is proportional to the free energy (or the generating functional) of the boundary (or dual) field theory:(10)e-Son⁡=〈e-S〉CFT=e-βF,
where F means the boundary free energy [1]. Despite the previous regularity assumption at the asymptotic boundary, the on-shell gravity action still suffers from some divergences caused by infinity in r. Following the holographic principle, these divergences can be reinterpreted as the UV divergences of the dual field theory and removed by appropriate local counter terms analogous to quantum field theory. This method is called the holographic renormalization [13]. In this procedure counter terms not only remove the UV divergences but also provide finite contributions. Since physics crucially depends on these finite contributions finding the correct counter terms is really an important issue.The correct counter terms should satisfy the following prescriptions.(i)
The on-shell action should be finite.(ii)
The variation of the on-shell action with respect to the leading boundary value of all bulk fields should also be finite.As mentioned before, the first prescription corresponds to finiteness of the free energy. The second implies the physical quantities of the dual theory should be also finite. In order to understand this, it should be noticed that, in general, the bulk fields denoted byΦ collectively have the following expansion form at the asymptotic boundary [2–4]:
(11)Φ=Jrm+(1+⋯)+〈O-〉rm-(1+⋯),
where ellipsis denotes small corrections. If m+>m-, on the dual field theory side, J and O- correspond to a source and vev of its dual density operator, respectively and O- becomes a conjugate variable of J. The on-shell gravity action, Son⁡, then includes the term related to the generating functional:
(12)Son⁡=⋯+∫‍dτ∫‍d3xJ〈O-〉+⋯,
where the Euclidean time integral runs from 0 to β. Defining 〈O〉=∫‍d3x〈O-〉, the one point correlation function, when the source J is uniform, is given by
(13)β〈O〉=-∂Son⁡∂J,
where the inverse temperature, β, comes from the Euclidean time integral. Therefore, the second prescription implies the one point function obtained by the holographic renormalization should be finite:
(14)〈O〉=-1β∂Son⁡∂J=finite.
It is consistent with our intuition. Since 〈O〉 is a physical observable, it must be finite. In the Einstein-Maxwell theory there are two bulk fields, so the second prescription leads to
(15)〈Tab〉=-2β∂Son⁡∂γab,〈N〉=-1β∂Son⁡∂μ,
where the factor 2 is introduced for later convenience. In the AdS/CFT correspondence, the electric charge of the RNAdS black brane is related to the quark number density operator of the dual field theory. So the quark number operator becomes N=QV3, where V3 is the regularized volume at the asymptotic boundary.


### 2.1. Grand Canonical Ensemble at the UV Fixed Point

In the charged black brane geometry, the regularity conditions at the horizon in (7) and (8) determine two integration constants of the Einstein and Maxwell equations. After requiring the asymptotic AdS geometry, however, there still remains one integration constant in the vector field which should be fixed by another boundary condition. One can usually choose a Dirichlet or Neumann boundary condition at the asymptotic boundary.Let us first impose the Dirichlet boundary condition on the gauge field, which fixes the chemical potential corresponding to the boundary value of the gauge field. Then, the dual theory is described by the grand potential of a grand canonical ensemble. In this case, since the variation of the renormalized action with respect to the gauge field is well defined, an additional boundary term is not needed. When the Neumann boundary condition instead of the Dirichlet boundary condition is imposed, this is not true anymore as will be shown in the next section. The resulting renormalized action with the Dirichlet boundary condition becomes(16)Srengrand=SE+SGH+Sct,
where Sct means the counter terms. For the asymptotic AdS space, the correct counter terms have been found in [13]. In the Poincare patch of a five-dimensional AdS space, the counter term is given by
(17)Sct=3κ2∫∂M‍d4xγ.
In the RNAdS black brane the vector field depending only on the radial coordinate does not generate a new divergence, so an additional counter term is not required.From the renormalized action, one can easily read the thermodynamic quantities by evaluating the boundary stress tensor. At the UV fixed point (r0→∞), the grand potential from the on-shell gravity action yields
(18)Ω≡lim⁡r0→∞Srengrandβ,=-V32κ2rh4-13g2μ2V3rh2,
where (8) is used. This grand potential is consistent with the result obtained from the black brane thermodynamics (A.21). The renormalized boundary energy-momentum tensor is then defined as [12]
(19)〈Tμν〉≡lim⁡r0→∞(-2γμρβδSrengrandδγρν)=lim⁡r0→∞(1κ2∫‍d3xγ(Θμν-δμνΘ)-2γμρδSctδγρν),
where the integral implies the boundary spatial volume integral. The explicit internal energy and pressure read
(20)E=T00=3V32κ2rh4+g2V3μ2rh2,P=-TiiV3=12κ2rh4+13g2μ2rh2,
where i is not summed. From the previous second prescription the vev of the number operator becomes
(21)〈N〉≡lim⁡r0→∞(-1β∂Srengrand∂μ)=2g2V3rh2μ.
When imposing the Dirichlet boundary condition on the vector field, all physical quantities must be represented as functions of TH, V3, and μ which are fundamental variables describing the dual field theory. Since the grand potential of this system is also a function of those variables, the following identity naturally appears:
(22)dΩ=∂Ω∂THdTH+∂Ω∂μdμ+∂Ω∂V3dV3.
Since the grand potential in (18) is linearly proportional to V3, its derivative with respect to V3 satisfies
(23)∂Ω∂V3=-P.
Comparing ∂Ω/∂μ with (21), we can easily see
(24)∂Ω∂μ=-N,
where 〈N〉 is denoted by N. These relations indicate that the conjugate variables of V3 and μ are given by P and N, respectively. Now, let us define
(25)S≡-∂Ω∂TH,
and then S becomes the conjugate variable of TH. If the system we consider is a thermal system, S can be naturally identified with the thermal entropy. Before checking the first law of thermodynamics, however, we can not say yet that our system is in thermal equilibrium. Using the fact that rh is given by a function of TH and μ, S at fixed μ and V3 reduces to
(26)S=-∂Ω∂TH|μ,V3=2πV3κ2rh3.
The values obtained here automatically satisfy the following relation:
(27)Ω=E-THS-μN,
which together with the exact differential in (22) leads to the first law of thermodynamics
(28)dE=THdS+μdN-PdV3.
These results show that the dual field theory is really a thermal system, as one expected. Since the grand potential is the renormalized quantity of the dual field theory, the derived S from it is also the renormalized one. As shown before, it satisfies all thermodynamic relations so that it should be identified with the thermal entropy of the dual field theory. Intriguingly, the thermal entropy derived here is exactly the same as the Bekenstein-Hawking entropy in (A.9). As a result, the holographic renormalization only with the quite natural regularity conditions leads to thermodynamics of the dual theory which is coincident with the black brane thermodynamics. In this case, all thermodynamic quantities have appropriate dual theory interpretations. In particular, the Bekenstein-Hawking entropy appears as the renormalized thermal entropy.Note that the chemical potentialμ and quark number operator N=QV3 are conjugate variables and, following the AdS/CFT correspondence, μ is a source of N (or Q) in the dual field theory. Since the grand potential is described by the chemical potential, one can easily evaluate the correlation function of the number operator in the grand canonical ensemble. At the UV fixed point (r0→∞), the one-point function of the number operator is given by (21), which on the gravity side corresponds to the electric charge of the black brane. One can easily check that this one point function coincides with the regularity of the vector field in (8). Moreover, as will be seen in the next section, it is also related to the Legendre transformation of the thermodynamic system. From (7) and (8), rh can be rewritten in terms of μ and TH as
(29)rh=πTH+π2TH2+(4/3)κ2g2μ22.
Using this, the expectation value of the number operator can be rewritten, in terms of the fundamental thermodynamic variables of the grand canonical ensemble, as
(30)〈N〉=g2V32μ(πTH+π2TH2+43κ2g2μ2)2.
In particular, at zero temperature (rh=κgμ/3) this correlator simply reduces to
(31)〈N〉=23g4V3κ2μ3.
Interestingly, this zero temperature expectation value of the number operator shows that the Fermi surface energy (ϵF=μ), if it exists [44], is given by
(32)ϵF=(32κ2g4)1/3(NV3)1/3,
where N denotes the expectation value of the number operator. In the general statistical mechanics, the Fermi surface energy of a nonrelativistic free fermions, due to the nonrelativistic dispersion relation ϵ=p2/2m, yields
(33)ϵF~(NV3)2/3.
On the other hand, for the relativistic free fermions with the dispersion relation, ϵ=p, the Fermi surface energy in the natural unit is given by
(34)ϵF=(3π2)1/3(NV3)1/3,
which looks very similar to the dual field theory result in (32). If the holographic dual theory of the asymptotic AdS geometry has asymptotic freedom, one can easily identify the holographic dual matter with the relativistic free fermion. Unfortunately, because of the absence of an asymptotic freedom in the dual theory of the asymptotic AdS space, one can not naively interpret the holographic dual matter as free fermion. So it would be interesting to construct the dual geometry of the field theory with the asymptotic freedom and study it further. Although the dual matter of the RNAdS geometry is not a relativistic free fermion, it shows the relativistic free fermion-like Fermi surface energy which might be related to physical property of the non-Fermi liquid [44].Before concluding this section, it is worth to note that the trace of the boundary stress tensor vanishes. Since the second terms in (20) are contributions from the holographic matter, this result says that the holographic dual matter of the RNAdS black brane geometry is also conformal. This is not always true in the asymptotic AdS geometry. As will be shown later, a different asymptotic charged AdS black brane solution shows that its dual matter can break the boundary conformal symmetry in spite of the asymptotic AdS geometry.


### 2.2. Canonical Ensemble

As shown in [9], imposing the Neumann boundary condition instead of the Dirichlet boundary condition corresponds to the Legendre transformation in the dual theory, which describes a canonical ensemble instead of a grand canonical ensemble. Although there is no required additional counter term similar to the grand canonical ensemble case, a new boundary term should be added to satisfy the Neumann boundary condition. Varying the gauge kinetic term with respect to the gauge field leads to a Neumannizing term [23]:
(35)δSN=1g2∫∂M‍d4xggrrgττFrτδAτ.
This term, when the Dirichlet boundary condition is imposed, automatically vanishes due to δAτ=0, whereas it does not automatically vanish when the Neumann boundary condition is imposed. In order to get a well defined gauge field variation with the Neumann boundary condition, one should add an additional boundary term
(36)Sbd=1g2∫∂M‍d4xAμJμ,
where Aμ={Aτ,0,0,0} and Jμ={iQ,0,0,0}. Then, the variation of this new boundary term together with (35) gives rise to
(37)δS=δSN+δSbd,=1g2∫∂M‍d4x(ggrrgττFrτ+iQ)δAτ.
The vanishing of the action variation leads to the following Neumann boundary condition:
(38)ggrrgττFrτ=-iQ,
which is nothing but the Euclidean version of the Nuemann boundary condition in (A.6). Therefore, the resulting renormalized action for the canonical ensemble becomes
(39)Srencan=SE+SGH+Sbd+Sct.
From this renormalized action, the free energy of the dual canonical ensemble is given by
(40)F≡lim⁡r0→∞Srencanβ,=-V32κ2rh4+5N212g2V31rh2,
where rh should be regarded as a function of TH and N. Comparing the free energy with the grand potential in (18) says that on the dual field theory side the new boundary term reduces to Sbd/β=μN and that the free energy and the grand potential are related by the Legendre transformation as they should do. After evaluating the boundary stress tensor similar to the previous case, one can easily read the internal energy and pressure in terms of N:
(41)E=T00≡lim⁡r0→∞(-2γ0ρβδSrencanδγρ0),=3V32κ2rh4+N24g2V31rh2,Pi=-TiiV3≡lim⁡r0→∞(2γiρβV3δSrencanδγρi),=12κ2rh4+N212g2V321rh2.
These results exactly coincide with the RNAdS black brane thermodynamics of the canonical ensemble in (A.13) and (A.16).


## 2.1. Grand Canonical Ensemble at the UV Fixed Point

In the charged black brane geometry, the regularity conditions at the horizon in (7) and (8) determine two integration constants of the Einstein and Maxwell equations. After requiring the asymptotic AdS geometry, however, there still remains one integration constant in the vector field which should be fixed by another boundary condition. One can usually choose a Dirichlet or Neumann boundary condition at the asymptotic boundary.Let us first impose the Dirichlet boundary condition on the gauge field, which fixes the chemical potential corresponding to the boundary value of the gauge field. Then, the dual theory is described by the grand potential of a grand canonical ensemble. In this case, since the variation of the renormalized action with respect to the gauge field is well defined, an additional boundary term is not needed. When the Neumann boundary condition instead of the Dirichlet boundary condition is imposed, this is not true anymore as will be shown in the next section. The resulting renormalized action with the Dirichlet boundary condition becomes(16)Srengrand=SE+SGH+Sct,
where Sct means the counter terms. For the asymptotic AdS space, the correct counter terms have been found in [13]. In the Poincare patch of a five-dimensional AdS space, the counter term is given by
(17)Sct=3κ2∫∂M‍d4xγ.
In the RNAdS black brane the vector field depending only on the radial coordinate does not generate a new divergence, so an additional counter term is not required.From the renormalized action, one can easily read the thermodynamic quantities by evaluating the boundary stress tensor. At the UV fixed point (r0→∞), the grand potential from the on-shell gravity action yields
(18)Ω≡lim⁡r0→∞Srengrandβ,=-V32κ2rh4-13g2μ2V3rh2,
where (8) is used. This grand potential is consistent with the result obtained from the black brane thermodynamics (A.21). The renormalized boundary energy-momentum tensor is then defined as [12]
(19)〈Tμν〉≡lim⁡r0→∞(-2γμρβδSrengrandδγρν)=lim⁡r0→∞(1κ2∫‍d3xγ(Θμν-δμνΘ)-2γμρδSctδγρν),
where the integral implies the boundary spatial volume integral. The explicit internal energy and pressure read
(20)E=T00=3V32κ2rh4+g2V3μ2rh2,P=-TiiV3=12κ2rh4+13g2μ2rh2,
where i is not summed. From the previous second prescription the vev of the number operator becomes
(21)〈N〉≡lim⁡r0→∞(-1β∂Srengrand∂μ)=2g2V3rh2μ.
When imposing the Dirichlet boundary condition on the vector field, all physical quantities must be represented as functions of TH, V3, and μ which are fundamental variables describing the dual field theory. Since the grand potential of this system is also a function of those variables, the following identity naturally appears:
(22)dΩ=∂Ω∂THdTH+∂Ω∂μdμ+∂Ω∂V3dV3.
Since the grand potential in (18) is linearly proportional to V3, its derivative with respect to V3 satisfies
(23)∂Ω∂V3=-P.
Comparing ∂Ω/∂μ with (21), we can easily see
(24)∂Ω∂μ=-N,
where 〈N〉 is denoted by N. These relations indicate that the conjugate variables of V3 and μ are given by P and N, respectively. Now, let us define
(25)S≡-∂Ω∂TH,
and then S becomes the conjugate variable of TH. If the system we consider is a thermal system, S can be naturally identified with the thermal entropy. Before checking the first law of thermodynamics, however, we can not say yet that our system is in thermal equilibrium. Using the fact that rh is given by a function of TH and μ, S at fixed μ and V3 reduces to
(26)S=-∂Ω∂TH|μ,V3=2πV3κ2rh3.
The values obtained here automatically satisfy the following relation:
(27)Ω=E-THS-μN,
which together with the exact differential in (22) leads to the first law of thermodynamics
(28)dE=THdS+μdN-PdV3.
These results show that the dual field theory is really a thermal system, as one expected. Since the grand potential is the renormalized quantity of the dual field theory, the derived S from it is also the renormalized one. As shown before, it satisfies all thermodynamic relations so that it should be identified with the thermal entropy of the dual field theory. Intriguingly, the thermal entropy derived here is exactly the same as the Bekenstein-Hawking entropy in (A.9). As a result, the holographic renormalization only with the quite natural regularity conditions leads to thermodynamics of the dual theory which is coincident with the black brane thermodynamics. In this case, all thermodynamic quantities have appropriate dual theory interpretations. In particular, the Bekenstein-Hawking entropy appears as the renormalized thermal entropy.Note that the chemical potentialμ and quark number operator N=QV3 are conjugate variables and, following the AdS/CFT correspondence, μ is a source of N (or Q) in the dual field theory. Since the grand potential is described by the chemical potential, one can easily evaluate the correlation function of the number operator in the grand canonical ensemble. At the UV fixed point (r0→∞), the one-point function of the number operator is given by (21), which on the gravity side corresponds to the electric charge of the black brane. One can easily check that this one point function coincides with the regularity of the vector field in (8). Moreover, as will be seen in the next section, it is also related to the Legendre transformation of the thermodynamic system. From (7) and (8), rh can be rewritten in terms of μ and TH as
(29)rh=πTH+π2TH2+(4/3)κ2g2μ22.
Using this, the expectation value of the number operator can be rewritten, in terms of the fundamental thermodynamic variables of the grand canonical ensemble, as
(30)〈N〉=g2V32μ(πTH+π2TH2+43κ2g2μ2)2.
In particular, at zero temperature (rh=κgμ/3) this correlator simply reduces to
(31)〈N〉=23g4V3κ2μ3.
Interestingly, this zero temperature expectation value of the number operator shows that the Fermi surface energy (ϵF=μ), if it exists [44], is given by
(32)ϵF=(32κ2g4)1/3(NV3)1/3,
where N denotes the expectation value of the number operator. In the general statistical mechanics, the Fermi surface energy of a nonrelativistic free fermions, due to the nonrelativistic dispersion relation ϵ=p2/2m, yields
(33)ϵF~(NV3)2/3.
On the other hand, for the relativistic free fermions with the dispersion relation, ϵ=p, the Fermi surface energy in the natural unit is given by
(34)ϵF=(3π2)1/3(NV3)1/3,
which looks very similar to the dual field theory result in (32). If the holographic dual theory of the asymptotic AdS geometry has asymptotic freedom, one can easily identify the holographic dual matter with the relativistic free fermion. Unfortunately, because of the absence of an asymptotic freedom in the dual theory of the asymptotic AdS space, one can not naively interpret the holographic dual matter as free fermion. So it would be interesting to construct the dual geometry of the field theory with the asymptotic freedom and study it further. Although the dual matter of the RNAdS geometry is not a relativistic free fermion, it shows the relativistic free fermion-like Fermi surface energy which might be related to physical property of the non-Fermi liquid [44].Before concluding this section, it is worth to note that the trace of the boundary stress tensor vanishes. Since the second terms in (20) are contributions from the holographic matter, this result says that the holographic dual matter of the RNAdS black brane geometry is also conformal. This is not always true in the asymptotic AdS geometry. As will be shown later, a different asymptotic charged AdS black brane solution shows that its dual matter can break the boundary conformal symmetry in spite of the asymptotic AdS geometry.


## 2.2. Canonical Ensemble

As shown in [9], imposing the Neumann boundary condition instead of the Dirichlet boundary condition corresponds to the Legendre transformation in the dual theory, which describes a canonical ensemble instead of a grand canonical ensemble. Although there is no required additional counter term similar to the grand canonical ensemble case, a new boundary term should be added to satisfy the Neumann boundary condition. Varying the gauge kinetic term with respect to the gauge field leads to a Neumannizing term [23]:
(35)δSN=1g2∫∂M‍d4xggrrgττFrτδAτ.
This term, when the Dirichlet boundary condition is imposed, automatically vanishes due to δAτ=0, whereas it does not automatically vanish when the Neumann boundary condition is imposed. In order to get a well defined gauge field variation with the Neumann boundary condition, one should add an additional boundary term
(36)Sbd=1g2∫∂M‍d4xAμJμ,
where Aμ={Aτ,0,0,0} and Jμ={iQ,0,0,0}. Then, the variation of this new boundary term together with (35) gives rise to
(37)δS=δSN+δSbd,=1g2∫∂M‍d4x(ggrrgττFrτ+iQ)δAτ.
The vanishing of the action variation leads to the following Neumann boundary condition:
(38)ggrrgττFrτ=-iQ,
which is nothing but the Euclidean version of the Nuemann boundary condition in (A.6). Therefore, the resulting renormalized action for the canonical ensemble becomes
(39)Srencan=SE+SGH+Sbd+Sct.
From this renormalized action, the free energy of the dual canonical ensemble is given by
(40)F≡lim⁡r0→∞Srencanβ,=-V32κ2rh4+5N212g2V31rh2,
where rh should be regarded as a function of TH and N. Comparing the free energy with the grand potential in (18) says that on the dual field theory side the new boundary term reduces to Sbd/β=μN and that the free energy and the grand potential are related by the Legendre transformation as they should do. After evaluating the boundary stress tensor similar to the previous case, one can easily read the internal energy and pressure in terms of N:
(41)E=T00≡lim⁡r0→∞(-2γ0ρβδSrencanδγρ0),=3V32κ2rh4+N24g2V31rh2,Pi=-TiiV3≡lim⁡r0→∞(2γiρβV3δSrencanδγρi),=12κ2rh4+N212g2V321rh2.
These results exactly coincide with the RNAdS black brane thermodynamics of the canonical ensemble in (A.13) and (A.16).


## 3. Holographic Nonconformal Matter in the Asymptotic AdS Geometry

One can also take into account another deformed charged black brane which still has an asymptotic AdS geometry. One of the known examples is the deformed RNAdS black brane with two different Liouville scalar potentials, where the scalar field was identified with a dilaton because it controls the physical coupling [45–48]. In general, the dilaton field is related to the running coupling of the dual field theory, so the existence of the nontrivial dilaton field implies that the dual field theory is deviated from the UV conformal fixed point if the dual operator of the dilaton is relevant. In the deformed charged black brane, the dilaton is coupled to the gauge field dual to the matter field so that this nontrivial dilaton coupling can change the property of the dual matter field. By using the the holographic renormalization studied in the previous section, we will investigate the dual field theory of the deformed charged black brane and the property of the dual matter.With the Lorentzian signature, the action describing the deformed charged black brane is given by(42)S=12κ2∫‍d5x-g×[R-14e4ϕFμνFμν-12∂μϕ∂μϕ+1R2(8e2ϕ+4e-4ϕ)],
where R denotes the AdS radius. From now on, we set R=1 for convenience. Then, the known solutions are
(43)ds2=r2e2A(r)(-f(r)dt2+dx→2)+e2B(r)r2f(r)dr2,Aμdxm=Atdt,
with
(44)A(r)=13log⁡(1+Q28mr2),B(r)=-23log⁡(1+Q28mr2),f(r)=1-m(r2+Q2/8m)2,At=2κ2μ-Q2(r2+Q2/8m),ϕ(r)=16log⁡(1+Q28mr2),
where we used slightly different notations from [45] for more concrete thermodynamic interpretation. In the above, free parameters, m, Q, and μ, correspond to the mass, charge, and chemical potential of the charged black brane, respectively. More precisely, if one expands the gauge field near the asymptotic boundary to
(45)At=2κ2μ-Q2r2+O(1r4),
two integration constants μ and Q appear. Furthermore, following the AdS/CFT correspondence, these values can be reinterpreted, similar to the previous case, as the quark chemical potential μ and number density Q(46)Q=lim⁡r→0-ge4ϕgttgrr∂rAt.
Note that, for Q→0, the above solution simply reduces to the usual Schwarzschild AdS black brane.The absence of a conical singularity at the horizon gives rise to Hawking temperature which is associated with the horizonrh:
(47)rh=πTH,
and the Bekenstein-Hawking entropy leads to
(48)SBH=2πV3κ2(rh3+Q28mrh).
From the vanishing of the black brane factor and the gauge potential at the horizon, one can easily find two relations between parameters:
(49)Q=4κ2μm,m=rh2+2κ4μ2,
which are convenient in describing the grand canonical ensemble because all other parameters are given by functions of TH and μ. In the canonical ensemble, m and μ should be considered as functions of TH and Q, which have complicated forms:
(50)μ=Q4κ2m,m=13rh2+24/3rh43(16rh6+27Q2+3Q96rh6+81Q2)1/3m+(16rh6+27Q2+3Q96rh6+81Q2)1/3324/3.
From the thermodynamic point of view, since m is given by a very complicated function of TH and Q in the canonical ensemble, it is not easy to find analytic thermodynamic quantities directly from the thermodynamic laws. From now on, we try to evaluate the thermodynamic quantities through the holographic renormalization scheme instead of the thermodynamic laws.With the Euclidean signature, the action for the grand canonical ensemble is(51)Sgrand=-12κ2∫‍d5x-g×[R-14e4ϕFμνFμν-12∂μϕ∂μϕm+1R2(8e2ϕ+4e-4ϕ)].
In the holographic renormalization, one should add the Gibbons-Hawking and counter terms in (9). If we impose the Dirichlet boundary condition on Aτ, the dual thermodynamics represents the grand canonical ensemble in which the chemical potential instead of the number operator becomes a basic thermodynamic variable. Then, the grand potential proportional to the on-shell gravity action leads at the asymptotic boundary to(52)Ω(TH,μ,V3)≡SEβ=-π4V32κ2TH4-2π2κ2V3TH2μ2-103κ6V3μ4.
Furthermore, from the boundary stress tensor, the internal energy and pressure are represented as functions of TH and μ:
(53)E≡T00E=3π4V32κ2TH4+6π2κ2V3TH2μ2+143κ6V3μ4,P≡-TiiV3E=π42κ2TH4+2π2κ2TH2μ2+103κ6μ4.
The one-point function of the number operator is(54)N≡lim⁡r0→∞(-1β∂Srengrand∂μ)=43κ2μV3(3π2TH2+10κ4μ2).
Similar to the holographic renormalization of the RNAdS black brane, the renormalized thermal entropy S appears as the conjugate variable of TH(55)S≡-∂Ω∂TH|μ,V3=2π2V3THκ2(π2TH2+2κ4μ2)
and satisfies the thermodynamic relation
(56)Ω=E-THS-μN.
Comparing it to the Bekenstein-Hawking entropy in (48), one can easily see that the derived thermal entropy is exactly the same as the Bekenstein-Hawking entropy and all thermodynamic quantities derived here are consistent with those obtained from the black brane geometry.Forμ=0, the thermodynamics of the Schwarzschild AdS black brane is reproduced. When Q=23/2m3/4, the charged black brane reaches the extremal limit, where TH and SBH reduce to zero. It should be however noted that the extremal limit has naked singularity at the center (r=0) so that the zero temperature is not well defined. In the very low temperature limit (TH≪κ2μ), the expectation value of the number operator is proportional to μ3. If there exists a Fermi surface, the Fermi surface energy leads to
(57)ϵF~N1/3,
which is similar to the RNAdS black brane case. Unlike the RNAdS black brane case, however, the trace of the boundary stress tensor shows a nonzero value depending on the chemical potential
(58)〈Taa〉μ=E-3PV=-163κ6V3μ4,
where the subscript μ implies a nontrivial vacuum state |0〉μ. This result indicates that the dual matter breaks the conformal symmetry. Why does not the trace of the boundary stress tensor vanish in spite of the asymptotic AdS geometry? The reason is as follows. Although the deformation of the asymptotic AdS geometry by the dilaton and gauge field does not change the leading behavior of the metric, it can contribute some subleading corrections to the metric at r-2 order. Since these subleading terms provide nonzero finite contribution to both the on-shell action and boundary stress tensor even at the UV cutoff, the dual conformal symmetry is broken without changing the leading asymptotic geometry.In order to describe the canonical ensemble instead of the grand canonical ensemble, we should impose the Neumann boundary condition on the vector field which requires one more boundary term(59)Scan=Sgrand+1g2∫∂M‍d4xAμJμ,
where Aμ={-iAt,0,0,0} and Jμ={iQ,0,0,0}. Since the new additional boundary term cancels Neumannizing term, the vector field variation is well defined. In the dual theory point of view, inserting this boundary term corresponds to the Legendre transformation because the new boundary term reduces to
(60)1g2∫∂M‍d4xAμJμ=βμN.
Therefore, (59) becomes
(61)F=Ω+μN,
where F≡Scan/β and Ω≡Sgrand/β denote the free energy and grand potential, respectively. As mentioned before, it represents the Legendre transformation between the grand canonical ensemble and canonical ensemble. The on-shell action and boundary energy-momentum tensor in the canonical ensemble yield the following free energy, internal energy and pressure:
(62)F=-π4V32κ2TH4+2π2κ2V3TH2μ2+143κ6V3μ4,E=3π4V32κ2TH4+6π2κ2V3TH2μ2+143κ6V3μ4,P=π42κ2TH4+2π2κ2TH2μ2+103κ6μ4,
where since the basic variables of the canonical ensemble are TH, N, and V3, the chemical potential is given by a function of those variables:
(63)μ=(96π6κ18TH6+81κ16(N2/V32)-9κ8NV3)2/3431/3π2κ6TH2-21/396π6κ18TH6+81κ16(N2/V32)×(96π6κ18TH6+81κ16(N2/V32)-9κ8NV3)2/396π6κ18TH6+81κ16(N2/V32)hhhhh96π6κ18TH6+81κ16(N2/V32)-9κ8NV3)2/3)×(96π6κ18TH6+81κ16(N2/V32)-9κ8NV3)1/3262/3κ496π6κ18TH6+81κ16(N2/V32)hhhhhl×(96π6κ18TH6+81κ16(N2/V32)-9κ8NV3)1/396π6κ18TH6+81κ16(N2/V32)hhhhhhhhh96π6κ18TH6+81κ16(N2/V32)-9κ8NV3)1/3)-1.
The holographic renormalization results in (62) coincide with the black brane thermodynamics and satisfies all thermodynamic relations.


## 4. Discussion

We have studied the holographic renormalization of the charged black branes whose asymptotic geometry is given by an AdS space. In the general renormalization scheme of the field theory, finding correct counter terms is very important because they cancel the UV divergences and at same time provide some finite contributions. Therefore, the renormalized physical quantities usually depend on the finite contributions of the counter terms. This is also true in the holographic renormalization. Following the AdS/CFT correspondence and using the holographic renormalization, in this paper, we have investigated the physical quantities of the dual field theory with matter. As mentioned before, for obtaining the correct physical results one should take into account the correct counter terms. In the charged black brane with an asymptotic AdS geometry, the bulk gauge field dual to matter does not require new divergent terms in the on-shell gravity action so that one need to introduce more counter terms except one used in the Schwarzschild (neutral) AdS black brane. Although there are no additional counter terms, the change of the metric caused by the bulk gauge field leads to a new finite contribution to the on-shell gravity action which depends on the physical quantities of matter, the chemical potential, or number density.In the RNAdS black brane case, the boundary stress tensor generated by the on-shell gravity action becomes traceless. As expected usually, the dual matter does not break the conformal symmetry of the asymptotic AdS geometry. Furthermore, the expectation value of the number operator shows that the Fermi surface energy at zero temperature is proportional toN1/3 which intriguingly resembles that of the free relativistic fermion. In spite of the resemblance, however, one can not identify the dual matter of the RNAdS black brane with a free relativistic fermion due to the absence of the asymptotic freedom. So it is more preferable to identify the dual matter with the non-Fermi liquid, strongly interacting fermion.In the RNAdS black brane, there is no nontrivial dilaton field associated with the running coupling constant of the dual field theory. We further regarded the charged black brane in the Einstein-Maxwell-dilaton theory. The asymptotic geometry of it is also given by an AdS space, which is the leading part of the metric, and the next to leading part gives rise to the nontrivial finite contribution to both the on-shell gravity action and the boundary stress tensor. Unlike the RNAdS black brane, the boundary stress tensor of the charged dilatonic black brane is not traceless which implies that the dual matter breaks the conformal symmetry although the leading part of the asymptotic metric is not changed.Finally, we also showed that the holographic renormalization results can be reinterpreted as two different ways depending on the asymptotic boundary condition of the gauge field. These results are exactly matched with the thermodynamic laws of the charged black brane geometries that we considered. All above results are derived at the UV fixed point. So it is interesting to investigate the renormalizaiton group flow of them.



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*Source: 565219-2014-09-02.xml*