Holographic phase transition and Quasinormal modes in Lovelock gravity
aa r X i v : . [ h e p - t h ] J a n Holographic phase transition and quasinormal modes in Lovelockgravity
Kai Lin a , ∗ Jeferson de Oliveira b , † and Elcio Abdalla a ‡ a Instituto de F´ısica, Universidade de S˜ao Paulo,C.P. 66318, cep 05315-970, S˜ao Paulo, SP, Brazil and b Instituto de F´ısica, Universidade Federal de Mato Grosso,Cep 78060-900, Cuiab´a, MT, Brazil (Dated: August 8, 2018)
Abstract
In this work we aim at discussing the effects of the higher order curvature terms on the LovelockAdS black holes quasinormal spectrum and, in the context of gauge/gravity correspondence, theirconsequences for the formation of holographic superconductors. We also explore the hydrodynamiclimit of the U (1) gauge field perturbations in d dimensions. PACS numbers: 04.50.Gh, 04.70.Bw ∗ Electronic address: [email protected] † Electronic address: jeferson@fisica.ufmt.br ‡ Electronic address: [email protected] . INTRODUCTION The AdS/CFT relation, discovered in the framework of string theory [1, 2], has surpassedits natural cradle to spread into the realm of condensed matter by means of the holographicconstruction [3–5]. In such a case, it does not matter much what is the type of the blackhole in the AdS space as a physical object, but rather what is the CFT theory describedin the process. Indeed, the CFT theory is the backbone of the construction and classicalperturbations of the gravity set up may lead, changing the black hole, or generally speakingchanging the AdS set up, to valuable information about the CFT counterpart. We move tonew condensed matter systems, thus to new physics.Recently, a series of models have been considered, with various degrees of success toobtain models concerning condensed matter systems, see [6, 7] for a partial and incompletelist. Several different physical situations have been touched, such as superconductivity,for perturbations of Reissner-Nordstrom solutions [3] and superfluidity [7], as well as whendealing with time dependent solutions [8], density waves [9]. Applications in high energyphysics have been particularly important [10],[11]. Higher order corrections to the gravitycounterpart have been often used, but a general discussion is still missing [12–15].Here, we are going to discuss details of the higher order corrections to gravity and theirconsequences for the holographic field theory. In particular, we consider Lagrangians whosefield equations are at most of second order[16] which, in the case of generalizations of gravitylead us to the Lovelock Lagrangians[17]. The paper is organized as follows. Section II pro-vides a review of the d -dimensional Lovelock gravity and the black hole solutions consideredin this work. In Sec. III we obtain the quasinormal spectrum of d -dimensional charged Love-lock black holes due to a scalar probe field. In Secs. IV and V we explore in the probe limitthe formation of holographic superconductors in the presence of higher order corrections tothe curvature. Also we compute the real time R -current correlators due to electromagneticperturbations due to electromagnetic field. In Sec. VI we discuss the results and some finalcomments are given. 2 I. THE LOVELOCK GRAVITY
String theory brought the idea that higher dimensional curvature terms in the gravityaction are basically mandatory to cope with quantum corrections at the Planck scale [18].On the other hand, field equations with higher time derivatives are unstable. Such a result,originally relying upon Ostrogradsky [19] long ago on very general grounds has been rederivedin simple terms [20]. Nevertheless, there are theories with complex dynamics involving higherorder terms in the Lagrangian but whose equations of motion are at most second order intime [16, 17, 21]. We discuss here the cases of Lovelock gravity as discussed in [22], where,in several space-time dimensions we have Einstein gravity corrected by higher order termsbut with second order differential equations for the fundamental metric fields. The solutionsof the field equations we are considering are those of Refs. [22] with nonvanishing charge,that is, the gravity sector in d dimensions is described by the action S = − πG Z d d x k X p =0 L ( p ) , (1)where k is an integer strictly smaller than d +12 labeling inequivalent theories, the individualLagrange densities are L ( p ) = l p − k d − k (cid:18) kp (cid:19) ǫ µ · µ d ǫ a ··· a d R µ µ a a · · · R µ p − µ p a p − a p e µ p +1 a p +1 · · · e µ d a d , with l denoting the d -dimensional AdS radius related to the cosmological constant Λ byΛ = − ( d − d − l and the curvature is R µνab = R µνab + 1 l e µa e νb , where e µa is the vielbein.It is an established result that the equations of motion are second order in the timederivatives. The Einstein equations have been solved [22] and a series of AdS black holesolutions emerge from these actions, the most important result used in the present work.Solutions are labeled by the space-time dimension d , the integer k defined above.In order to consider charged solutions, the gravity action (1) has to be supplemented bythe Maxwell action S M = − ǫ Z d d x √− gg µρ g νσ F µν F ρσ , (2)3here ǫ is related to the vacuum permeability ǫ .Solutions to the Einstein-Maxwell system are labeled by the space-time dimension d , theinteger k defined above and the charge Q . From [22, 23], those solutions read ds = − (cid:18) η + r l − g k ( r ) (cid:19) dt + dr η + r l − g k ( r ) + r d Σ d − , (3)where η = − , , η = 1, d Σ d − is the angular measure on thesphere, η = 0 implies a flat black hole where d Σ d − is the flat metric and η = − Q of the black hole by meansof the expression g k = GM + δ d − k, r d − k − − ǫ ˆ Gd − Q r d − k − ! /k , (4)where ˆ G is the gravitational constant, δ d − k, is the standard Kronecker delta and M is theblack hole mass. For a generic value of k , the black holes described by the line element(3) have two horizons ( r − , r + ) located at the zeros of g tt , satisfying r − < r + . The blackhole family describe by (3) include the d -dimensional Reissner-Nordstr¨om AdS-black holesfor k = 1 and the charged AdS-Gauss-Bonnet black holes for k = 2 and d >
5. For adiscussion on the causal structure of Gauss-Bonnet gravity, see [24]. Also, the line element(3) is asymptotically AdS for all values of k and d . III. LOVELOCK SCALAR QUASINORMAL MODES
In this section we are going to explore the quasinormal spectrum of charged AdS-Lovelockblack holes considering a probe scalar field evolving at such geometry.The black hole quasinormal modes (QNM) of asymptotically AdS black holes is obtainedby considering probe fields dynamics supplemented by ingoing boundary conditions at theevent horizon and Dirichlet boundary conditions at spatial infinity [25, 26]. In the context ofAdS/CFT correspondence the decay of QNM is interpreted as the return to equilibrium of athermal state in the quantum field theory at finite temperature living at the AdS boundary[27]. For a recent review on the subject see [28], [29] and the references therein. In particular,scalar fluctuations on the bulk geometry are related to the poles of thermal retarded Greenfunction [30] and the electromagnetic perturbations are associated to the poles of retardedGreen functions of R − symmetry currents at the boundary.4he next procedure is standard but new. We consider the scalar perturbations of theabove system. Scalar perturbations are easily obtained. We rewrite the metric in a formthat we use in the numerical analysis below, that is, ds = − f ( r ) dt + dr f ( r ) + r d Σ d − , (5)with f ( r ) = η + r l − (cid:20) Mr d − k − − Q r d − k − (cid:21) k . (6)Depending on the curvature (thus on η ), the angular part of (5) changes accordingly. Wecan rewrite the parameters in terms of the inner horizon r − and the event horizon r + as M = 1 r d − − r d − − " r d − − k + (cid:18) η + r l (cid:19) k − r d − − k − (cid:18) η + r − l (cid:19) k ,Q = 1 r − d − − r − d + " r d − − k + (cid:18) η + r l (cid:19) k − r d − − k − (cid:18) η + r − l (cid:19) k . (7)In this paper, we study the planar black hole case, namely η = 0, and without loss thegenerality, we set l = 1. Now, it is standard to compute the scalar modes. Because it is ananti-de Sitter spacetime, we should use Horowitz-Hubeny method [27] to calculate the scalarquasinormal modes of this black hole. According to this method, we set v = t + R drf ( r ) , sothe metric is rewritten as ds = − f ( r ) dv + 2 dvdr + r d Σ d − , (8)and then the scalar equation is given by f ( r ) φ ′′ + ( f ′ − iω ) φ ′ − V ( r ) φ = 0 , (9)where V = ( d − f ′ r + ( d − d − f r + ( d − L r . The transformation z = r is introduced,so that the region of variable becomes 0 ≤ z ≤ h ( h = r + ). The boundary condition atevent horizon require φ ( r + ) = 1 but φ should vanish at infinity. Therefore, the scalar fieldequation is given by s ( z ) d φdz + t ( z ) z − h dφdz + u ( z ) φ ( z − h ) = 0 , (10)5here s ( z ) = − z fz − h ,t ( z ) = − z (cid:18) z − dfdz + 2 zf + 2 iω (cid:19) ,u ( z ) = ( z − h ) V. (11)we can expand s ( z ) = P s i ( x − h ) i , t ( z ) = P t i ( x − h ) i and u ( z ) = P u i ( x − h ) i and φ ( z ) = P a i ( x − h ) i at event horizon z = h . Considering the boundary condition at horizon,we have a = 1, and substitute s , u , t and φ into Eq.(11), we obtain the recursion relation a n = − P n n − X i =0 [ i ( i − s n − i + it n − i + u n − i ] a i . (12)where P n = n ( n − s + nt , so all the a i can be obtained. Finally, according to anotherboundary condition φ (0) = 0, we always can get the value of ω from equation P i a i = 0.It is very convenient to use Wolfram Mathematica to realize the above process, so we usethis software to calculate the quasinormal modes of this black hole. We find a sequence ofquasinormal frequencies as function of the temperature of the black hole. It is a tedious butstraightforward procedure. Nevertheless, we find some interesting results.The first noteworthy result with angular quantum number l = 0 is the fact that spaceswith higher values of k are stiffer, namely have larger values for the imaginary part of thefrequency, as shown in Figs. 1 through 4. Also, we clearly see that the real and imaginarypart of frequencies scales linearly with the Hawking temperature, which is expected of AdSblack holes [27]. We have the results for various values of the temperature as given in TableI, where b is the value of black hole charge Q . TABLE I: Various values of quasinormal frequencies, where T is the Hawking temperature b d = 4 d = 5, k = 1 d = 5, k = 2 d = 6, k = 1 d = 6, k = 20 (7 . − . i ) T (9 . − . i ) T ( − . i ) T (10 . − . i ) T (8 . − . i ) T . − . i ) T (9 . − . i ) T ( − . i ) T (10 . − . i ) T (9 . − . i ) T . − . i ) T (11 . − . i ) T ( − . i ) T (11 . − . i ) T (9 . − . i ) T As we see in Fig. 3, the scalar quasinormal frequencies for a five and seven-dimensionalLovelock black hole, with k = 2 and k = 3, respectively, are purely damped, namely there6
10 15 20 T H H Ω L b = = = = = T H - Im H Ω L b = = = = = FIG. 1: Real(left) and imaginary(right) quasinormal modes behavior in terms of the Hawkingtemperature for d = 4 , k = 1. T H H Ω L b = = = = = T H - Im H Ω L b = = = = = FIG. 2: Real(left) and imaginary(right) quasinormal modes behavior in terms of the Hawkingtemperature for d = 5 , k = 1. T H - Im H Ω L b = = = = = T H - Im H Ω L b = = = = = FIG. 3: Purely damped quasinormal modes behavior in terms of the Hawking temperature for d = 5 , k = 2 and d = 7 , k = 3 is no oscillatory phase for the perturbation. Such a result seems to be a general feature of7 T H H Ω L b = = = = = T H - Im H Ω L b = = = = =
15 10 15 T H H Ω L b = = = = = T H - Im H Ω L b = = = = = FIG. 4: Effect on the real part(left) and imaginary part(right)of quasinormal frequencies by addingcharge to the six-dimensional case for k = 1, k = 2. Lovelock theories with d = 2 k + 1 since the gravity theory reduces to Chern-Simons gravityin these cases. The purely damped frequencies are not new in literature. The same resulthas been found in the behavior of scalar quasinormal modes of the three dimensional Lifshitzblack hole [31], whose gravity theory is the new massive gravity (NMG). The correspondingaction, as in the Lovelock case, contains higher order corrections in the curvature. Thedynamics of probe scalar fields in higher dimensional Lifshitz black holes (d=5,...,10) donot show an oscillating phase either [32]. Thus, at least in case of Lovelock gravity with d = 2 k + 1 and Lifshitz black holes, the purely damped modes are related to the highercurvature terms.Also, we observe that the effect of adding charge to the black hole is to increase thequasinormal frequency value. There is also a quite important increase in the imaginarypart of the frequencies for k = 2, when the models seems to get stiffer. This effect is lesspronounced for the six-dimensional case, see Fig. 4.8 V. THE PHASE TRANSITION AND CONDUCTIVITY
According to the AdS/CFT dictionary, the scalar perturbation ψ corresponds, at the AdSborder, to the order parameter of the conformal field theory. The gauge field perturbationgives rise to the border source and to the current. We can thus analyze whether we canhave a superconducting phase and compute the conductivity. As it turns out, we have theconductivity as a function of the frequency, what is a physically relevant object to studythe properties of the conformal field theory at the border (or else, of the condensed mattersystem at the border). For later purposes it is going to be useful to write the above functionin terms of the event horizon radius r + and the Cauchy horizon r c ≡ γr + . From this pointon we shall work on flat topology, η = 0 and l = 1. We have M = γ d − − b d − − r d − , Q = γ d − − γ d − γ d − − r d − , (13)where γ = r − r + . In terms of the new parameters we have f ( r ) = r − r d − k + r − dk ) (cid:20) − γ d − − γ d − r d − − γ d − − γ d − − γ d − (cid:21) k , (14)and the Hawking temperature is given in terms of the local gravity at the black hole eventhorizon, T c = d − − γ d − [2 d − − γ d − ( d − k (1 − γ d − ) π r + . (15)In order to obtain the phase transition, we consider the Lovelock gravity action (1)coupled to a classical charged scalar field ψ and the electromagnetic gauge field A µ , whoseaction is S c = Z d d x √− g (cid:20) − ǫ F µν F µν − |∇ ψ − iqAψ | − m | ψ | (cid:21) , (16)where ∇ is the covariant derivative, q and m are the scalar field charge and mass respectively.We consider the equation of motion of the matter and gauge fields, in such a way thatscalars are functions of the radial variable in order to define an order parameter at theborder. The scalar potential corresponding to the gauge field ( φ ≡ A ) is, consequently, afunction of the radial variable. The vectorial components of the gauge field are functions of t and r . Without loss of generality we consider only A x , whose time dependence is harmonic,that is, A x ( ~x, t ) = A x ( ~x ) e − iωt . 9e are going to consider the equations of motion of the electric potential A = Φ( r ), ofthe scalar field Ψ( r ) and the x -component of the vector potential, A x ( r ). Moreover, it isuseful to change variables from r to z = r . Also, foreseeing the asymptotic behavior of thefields, we redefine them as φ ( z ) = Φ(1 /z ), ψ ( z ) = Ψ(1 /z ) z − λ f , A x ( z ) = A x (1 /z ). We in thispaper use the shooting method to calculate numerically the holographic superconductor andthe conductivity. The Maxwell-Klein Gordon equations in the probe limit for k = 1 and d = 5 read ψ ′′ , + 3( γ + b ) z − ( γ + b + 1) z − z − z )[( γ + γ ) z − z − ψ ′ , + m (1 − z )[( γ + γ ) z − z −
1] + z φ , ( z − z ) [( γ + γ ) z − z − ψ , = 0 ,φ ′′ , − ψ ′ , z − ψ , φ , z ( z − γ + γ ) z − z −
1] = 0 ,A ′′ x, , + 5( γ + γ ) z − γ + γ + 1) z − z − z )[( γ + γ ) z − z − A ′ x, , + z ω − z − γ + γ ) z − z − ψ , ( z − z ) [( γ + γ ) z − z − A x, , = 0 . By the shooting method, we choose the value of the fields near the horizon, solve thedifferential equations to the spatial infinity and compare with the boundary condition. Forsolving the differential equation we choose the functions as power series of z −
1. We thusobtain the value of the function at the boundary and compare with the boundary condition.We subsequently consider the cases d = 5 , k = 2, d = 6 , k = 1 and d = 6 , k = 2, whoseequations of motion are given in the appendix. A. Numerical analysis
Let us first concentrate on the 5-dimensional case, where the function f ( r ) defining themetric is given by f ( r ) = η + r R − r k − e r − k ( b − k b " b ( r − r e ) (cid:18) b r e R + η (cid:19) k − b k ( r − r e b ) (cid:18) r e R + η (cid:19) k k . (17)Here, η = +1 , , − b is a measure of the charge of the black hole, R is the inverse of thecosmological constant and r e the event horizon.We search for static solutions for the electric potential and for the scalar field seekingat the order parameter at the border. Moreover, we look for a vector potential at a given10requency (as above) in order to test the Ohm’s law. The fields obey the coupled differentialequations ψ ′′ ( r ) + (cid:18) f ′ ( r ) f ( r ) + d − r (cid:19) ψ ′ + (cid:20) φ ( r ) f ( r ) − m f ( r ) (cid:21) ψ ( r ) = 0 , (18) φ ′′ + d − r φ ′ − q ψ ( r ) f ( r ) φ ( r ) = 0 , (19) A ′′ x ( r ) + (cid:18) f ′ ( r ) f ( r ) + d − r (cid:19) A ′ x ( r ) + (cid:18) ω f ( r ) − ψ ( r ) f ( r ) (cid:19) A x ( r ) = 0 . (20) B. Results for phase transition
According to the usual AdS/CFT dictionary, when we approach the AdS boundary theexpansion of the perturbations near the boundary leads to CFT fields with well-definedphysical interpretation [2, 3]. For the scalar field, in particular, we have the expansion ψ ( r ) = ψ (1) r + ψ (2) r + higher order in (cid:18) r (cid:19) . (21)The expansion coefficients hO i i = √ ψ ( i ) are, according to the above mentioned dictionary,order parameters of the boundary theory, as long as we choose appropriated boundaryconditions, that is, if ψ (1) = 0 we define hO i and for ψ (2) = 0 we define hO i . Thecomputaion of either field uses the shooting method found by [4].We considered various choice of parameters. Generally speaking, the order parameter hO i is larger than hO i , and the one corresponding to k = 2 larger than the one corre-sponding to k = 1, see Fig. 5 and 6. This result about k means that the nonlinearityenhances the order parameter, but strangely enough the conductivity goes the other way(see next subsection), namely the conductivity (both real and imaginary part) are smallerfor k = 2. Thus, order does not mean, in this case, better conductivity properties.The effect of dimensionality upon the phase transition is to lower the value of the con-densate as the number of spatial dimensions increase. Such an effect is present in bothcondensates hO i and hO i , see Fig. 7 for an example. C. Results for conductivity
Now, we are going to compute the conductivity for each boundary operator hO i and hO i following the standar AdS/CFT recipe [3]. Solving numerically the equation (20),11 .2 0.4 0.6 0.8 1.00.000.050.100.150.200.25 TT c < O > Λ T c k = = = = TT c < O > Λ T c k = = = = FIG. 5: Condensation of operators hO i and hO i for the five-dimensional uncharged case ( b = 0). TT c < O > Λ T c k = = = = TT c < O > Λ T c k = = = = FIG. 6: Condensation of operators hO i and hO i for the five-dimensional charged case ( b = 0 . TT c < O > Λ T c d = = = = TT c < O > Λ T c d = = = = FIG. 7: Condensation dependence on the dimensionality and the values of k imposing ingoing wave boundary conditions at the black hole event horizon and consideringthe asymptotic behavior of A x for large r , we have that the leading term is the current h J µ i and the sub leading one the dual source A (0) x , both defined at the AdS border.12aving these two quantities, we compute the conductivity σ ( ω ) through the Ohm’s law σ ( ω ) = − i h J µ i ωA x . (22)We present in Fig.(8)-Fig.(11) the real end imaginary part of conductivity σ ( ω ) of hO i and hO i for five dimensional black hole in charged and uncharged cases. The conductivityphenomena is qualitatively very similar in both cases k = 1 and k = 2 for the two operators,but the k = 2 corrections to the curvature seems to lower the conductivity comparing to the k = 1 case. Ω T c Σ k = = = = Ω T c - Im Σ k = = = = FIG. 8: Real(left) and imaginary(right) parts of hO i conductivity for zero black hole charge infive dimensions and varying k . Ω T c Σ k = = = = Ω T c - - - - Im Σ k = = = = FIG. 9: Real(left) and imaginary(right) parts of hO i conductivity for zero black hole charge infive dimensions and varying k .
10 15 20 25 30 Ω T c Σ k = = = = Ω T c - Im Σ k = = = = FIG. 10: Real(left) and imaginary(right) parts of hO i conductivity for charged black hole in fivedimensions and varying k . Ω T c Σ k = = = = Ω T c - - - Im Σ k = = = = FIG. 11: Real(left) and imaginary(right) parts of hO i conductivity for charged black hole in fivedimensions and varying k . V. R -CURRENT CORRELATORS AND HYDRODYNAMICAL QUASINORMALMODES In this section we are going to apply the AdS/CFT correspondence [2] [33] in orderto compute the real time R − current correlators, which can be expressed in terms of theboundary value of the gauge invariant quantities such as the electric field at the spatialinfinity. As one knows, the electromagnetic fluctuations, in the AdS/CFT context, give riseto the correlators associated to the R − symmetry at the boundary field theory.Following the procedure outlined in [33] [34], the imposition of Dirichlet boundary con-ditions on the gauge invariant variables lead to the poles of the field theory correlationfunctions and, according to [35], the quasinormal frequency spectra of the asymptotic AdSblack hole considered. Moreover, a consequence of applying the approach [35] is that the14lectromagnetic quasinormal spectra presents a set of modes which behaves like a diffusionwave in the long wavelength and low frequency limit, such a limit is called hydrodynamiclimit of perturbations. The two main results of the section is the explicit form of correlatorsin the field theory defined at the boundary of Lovelock black holes and the frequency ofdiffusion quasinormal modes for dimension d ≥ A. Correlators due to Electromagnetic Field
We are going to consider as our bulk geometry the uncharged d − dimensional planarLovelock black hole, represented by the following line element ds = r l u / ( d − " − g ( u ) dt + d − X i dx i + r u / ( d − dφ + l ( d − u g ( u ) du . (23)The function g ( u ) is the horizon function given by g ( u ) = 1 − u γ , γ = ( d − k ( d − . The event horizon is located at r = r or u = 1, the radial coordinate r ∈ [ r , + ∞ ] ismapped to u ∈ [1 ,
0] through u = r /r . In order to have a black hole with planar topology, atleast one of the extra dimensions has to be compact [23], so in the above metric φ ∈ [0 , π ],and the remaining directions have the domain x i ∈ [ −∞ , + ∞ ], where i = 1 · · · d − A µ ,whose equations governing its dynamics are the Maxwell equations, ∂ µ (cid:0) √− gF µν (cid:1) = 0 , (24)where F µν = ∂ µ A ν − ∂ ν A µ , and the metric components which enter in Maxwell equation arethose given by (23). Using the isometries of black hole spacetime, we can decompose thegauge field A µ in Fourier as following A µ ( t, x i , φ, u ) = 1(2 π ) d − Z ( dw )( dm )( dq i ) d − e − iωt + imφ + iq i x i A µ ( ω, m, q i , u ) . (25)It is possible, without loss of generality, choose a ( d − − dimensional wave vector ~p =( − ω, Q a )(with a = 1 , · · · , d − Q a = ( m, q i ) = (0 , q, ~ A µ propagating in one of the planar directions x i = ( x, ~ A φ and the even perturbations A t , A x , A u .Our gauge choice is the radial gauge where A r = A u = 0, and the fundamental gaugeinvariant variables for the two classes of perturbations are the transverse component ofelectric field E φ for the odd perturbations and the component E x for the even perturbations.The equations governing the dynamics are obtained from the Maxwell equations (24) writtenon the spacetime (23): E ′′ φ + g ( u ) ′ g ( u ) E ′ φ + w − q g ( u )( d − g ( u ) u d − d − E φ = 0 , (26) E ′′ x + g ( u ) ′ w g ( u ) [ w − q g ( u )] E ′ x + + w − q g ( u )( d − g ( u ) u d − d − E x = 0 , (27)where the primes refers to derivatives with respect to u direction. For convenience, we havenormalized the quantities w and q in terms of black hole Hawking temperature T = ( d − πl k r , namely, w = ( d − πk ωT , q = ( d − πk qT . Following the AdS/CFT recipe [30], the current-current two point correlators are givenby the field E µ ( µ = φ, x ) near the AdS boundary, which in our case, is obtained throughthe solution of equations (26) and (27) at u ≈ E φ = a φ ( w , q ) + b φ ( w , q ) u , (28) E x = a x ( w , q ) + b x ( w , q ) u , (29)furthermore, close to the event horizon E µ = g ( u ) ± ikd − w , where the positive exponent corre-sponds to outgoing waves and the negative exponent to ingoing waves at the event horizon,also the choice of sign means the electric field at AdS boundary is taken as classical source ofretarded (negative) or advanced (positive) current-current two point correlators. The ingo-ing waves at the event horizon are physically motivated boundary conditions for a classicalblack hole, thus, we are going to adopt the negative exponent for the electrical field E µ meaning that we are considering the retarded correlators of the holographic field theory.16he next step is to consider the electromagnetic action at the AdS boundary ( u ≈ S = ( d − r ( d − η l d − Z dωdq (2 π ) (cid:20) g ( u ) q g ( u ) − w E x ( u, − ~p ) E ′ x ( u, ~p ) − g ( u ) w E φ ( u, − ~p ) E ′ φ ( u, ~p ) (cid:21) . (30)where η is the normalization of the action, from [37] one finds1 η = ( d − d ]2 ( d − π d Γ[ d ] ( N c − , with N c representing the number of D − branes. Also, we can rewrite the electric field atthe AdS boundary in terms of the gauge field in the same region A µ ( ~p ) = A µ ( u → , ~p ) andapplying the Lorentzian prescription [30], C µν ( ω, ~p ) = 2 δ SδA µ ( ~p ) δA ν ( − ~p ) , (31)one finds the current-current correlators C tt ( ω, q ) = ( d − r d − η l d − b x ( w , q ) a x ( w , q ) q w − q , (32) C xx ( ω, q ) = ( d − r d − η l d − b x ( w , q ) a x ( w , q ) w w − q , (33) C φφ ( ω, q ) = ( d − r d − η l d − b φ ( w , q ) a φ ( w , q ) , (34) C tx ( ω, q ) = ( d − r d − η l d − b x ( w , q ) a x ( w , q ) wqw − q . (35)Using the components of C µν it is possible to express the transversal Π T ( ω, q ) and longitu-dinal Π L ( ω, q ) self-energies of the ( d −
1) holographic thermal field theoryΠ T ( ω, q ) = ( d − r d − η l d − b φ ( w , q ) a φ ( w , q ) , (36)Π L ( ω, q ) = ( d − r d − η l d − b x ( w , q ) a x ( w , q ) . (37)Therefore, the electromagnetic correlation functions are fully determined by the relations b φ ( w , q ) /a φ ( w , q ) and b x ( w , q ) /a x ( w , q ) and the poles of the correlators are the same asthe zeros of a φ ( w , q ) and a x ( w , q ) [35]. To find the poles, we impose Dirichlet boundaryconditions on the electric field at AdS boundary and ingoing wave conditions at the blackhole event horizon. 17 . Diffusion Quasinormal modes To determine the self-energies found in the preceding computation, we have to solve thedifferential equations for E x and E φ . Analytical solutions are unknown, unless in the so-called hydrodynamical limit of the perturbations. Such a limit is achieved by considering aset of perturbations with small frequencies and small wave numbers, w ≪ , q ≪ . From the point of view of the thermal field theory, at least one of the electromagneticquasinormal frequencies has to behave as a diffusion mode in the hydrodynamical limit. So,if we impose Dirichlet and ingoing-wave boundary conditions to the differential equations(26) and (27), we found that there is not a transversal diffusion mode, namely, does not exista value of ω that is compatible with E φ = 0 at AdS boundary. Such a result is independenton the dimensionality of the bulk and the flavor of the Lovelock theory, in other words,independent on d and k . However, we found that for the longitudinal mode, there is ahydrodynamical mode given by w = − i q ( d − ⇒ ω = − i ( d − π ( d − kT q , (38)whose diffusion coefficient can be read off D = ( d − π ( d − kT . (39)This is the main result of the section. We found that the diffusion coefficient dependscrucially on the flavor of Lovelock gravity. As we increase the corrections to the curvaturein Lovelock Lagrangian the diffusion coefficient tend to zero, so the charge diffusion inlongitudinal direction in thermal field theory is diminished in gravity duals with correctionsto the curvature. VI. CONCLUDING REMARKS
In this work we have studied the effects of higher order corrections to the gravity upon thescalar and hydrodynamical quasinormal modes spectrum, the condensation of holographicoperators and their conductivity. 18egarding to the scalar quasinormal modes, we found that the corrections to the curvaturediminish the quasinormal modes oscillating phase, it is similar to the dynamics of a perturba-tion in a very dense material medium. We see from Fig.3 the case where the real part of thefrequencies are zero, so these modes are purely damped. Moreover, we found in the hydro-dynamical limit a purely damped diffusive quasinormal mode ω = − i ( d − q / π ( d − kT ,which depends strongly on the k parameter.We obtained explicitly the phase transition giving the condensation of operators hO i and hO i . The influence of curvature corrections of the Lovelock gravity is to increase the valueof the condensate, in both charged and uncharged cases. Also, we compute the conductivity,where we found that the considered gravity bulk diminish the real part and imaginary partof σ ( ω ) as we add more corrections to curvature.As an extension of this work, it would be interesting to consider charged fermions fieldsevolving on the gravity bulk given by the family of Lovelock black holes in order to inves-tigate if purely damped quasinormal frequencies are allowed in this case. Another problemwhich will be address in a future work is the question of gravitational stability of Lovelockblack holes and the computation of holographic stress-energy tensor of field theory on thespacetime AdS boundary. Appendix A: Equations of motion for ( d = 5 , k = 2) , ( d = 6 , k = 1) and ( d = 6 , k = 2) The general equations of motion of the scalar and gauge fields are ψ ′′ , + (cid:16) ( b + b + 1) z − p − b ( b + 1) ( z − (cid:17) z p − b ( b + 1) ( z −
1) + ( b + b ) z − ( b + b + 1) z ψ ′ , + (cid:16) m (cid:16) z p − b ( b + 1) ( z − − (cid:17) + z ϕ , (cid:17) z (cid:16) z p − b ( b + 1) ( z − − (cid:17) ψ , = 0 ,ϕ ′′ , − ϕ ′ , z + 2 ϕ , ψ , z (cid:16) z p − b ( b + 1) ( z − − (cid:17) = 0 ,A ′′ x, , + (cid:18) ( b + b ) z √ − b ( b +1)( z − − z (cid:19) z − p − b ( b + 1) ( z −
1) + 1 z A ′ x, ,
19 2 (cid:16) z p − b ( b + 1) ( z − − (cid:17) ψ , + ω z z (cid:16) z p − b ( b + 1) ( z − − (cid:17) A x, , = 0 .ψ ′′ , + − b + b + 1) − b + 1) ( b + 1) ( b + 1) z + 6 b ( b + b + b + b + 1) z z ( b − ( b + 1) ( b + 1) ( b + 1) z + b ( b + b + b + b + 1) z + b + 1) − z ψ ′ , + ϕ , − m h z − z (cid:16) b − z ( b − + b − b b − (cid:17)i z h z − z (cid:16) b − z ( b − + b − b b − (cid:17)i ψ , = 0 ,ϕ ′′ , − z ϕ ′ , − b − ϕ , ψ , ( b − z − ( b − z + ( b − b ) z = 0 ,A ′′ x, , + ( − b ) z b − + b ( b + b + b + b +1 ) z b + b +1 − z z − z (cid:16) b − z ( b − + b − b b − (cid:17) A ′ x, , ω − h z − z (cid:16) b − z ( b − + b − b b − (cid:17)i ψ ( z ) z h z − z (cid:16) b − z ( b − + b − b b − (cid:17)i A x, , = 0 .ψ ′′ , − b − ( b − ) b z − √ b − √ z ( b − ( b − b z − + z z − q z ( b − ( b − b z − b − + 2 z ψ ′ , m z q z ( b − ( b − b z − b − − m + z ϕ , z (cid:18) z q z ( b − ( b − b z − b − − (cid:19) ψ , = 0 ,ϕ ′′ , − z ϕ ′ , + 2 ϕ , ψ , z q z ( b − ( b − b z − b − − z = 0 ,A ′′ x, , + − b − ( b − ) b z − b − r z ( b − ( b − ) b z − ) b − − z z − q z ( b − ( b − b z − b − A ′ x, , z q z ( b − ( b − b z − b − ψ , + ω z − ψ , z (cid:18) z q z ( b − ( b − b z − b − − (cid:19) A x, , = 0 (A1)We get the results for different values of b (representing the charge) k and the dimension d . Acknowledgements
This work was supported by FAPESP No. 2012/08934-0 and CNPq, Brazil. [1] J. Maldacena
Adv. Theor. Math. Phys. (1998) 231-252.[2] E. Witten Adv. Theor. Math. Phys. (1998) 253-291.[3] Sean A. Hartnoll, Christopher P. Herzog, Gary T. Horowitz JHEP (2008) 015.
4] Sean A. Hartnoll, Christopher P. Herzog, Gary T. Horowitz,
Phys. Rev. Lett. (2008)031601.[5] S. S. Gubser,
Phys. Rev.
D78 (2008) 065034.[6] G. T. Horowitz
Lect. Notes Phys. (2011) 313-347; S. S. Gubser, S. S. Pufu
JHEP (2008) 033; G. T. Horowitz, M. M. Roberts
Phys. Rev.
D78 (2008) 126008.[7] Hong Liu, J. McGreevy, D. Vegh
Phys. Rev.
D83 (2011) 065029; M. Cubrovic, J. Zaanen,K. Schalm
Science (2009) 439-444; Sung-Sik Lee
Phys. Rev.
D79 (2009) 086006; C.Charmousis, B. Gouteraux, B.S. Kim, E. Kiritsis, R. Meyer
JHEP (2010) 151; C.P.Herzog, P.K. Kovtun, D.T. Son
Phys. Rev.
D79 (2009) 066002; R. B. Mann
JHEP (2009) 075; S. Bhattacharyya, V. E. Hubeny, S. Minwalla, M. Rangamani
JHEP (2008)045.[8] M.J. Bhaseen, J. P. Gauntlett, B.D. Simons, J. Sonner, T. Wiseman
Phys. Rev. Lett. (2013) 015301.[9] A. Aperis, P. Kotetes, E. Papantonopoulos, G. Siopsis, P. Skamagoulis, G. Varelogiannis
Phys.Lett.
B702 (2011) 181-185.[10] P. Kovtun, D. T. Son and A. O. Starinets,
Phys. Rev. Lett. , 111601 (2005).[11] G. Policastro, D. T. Son and A. O. Starinets, Phys. Rev. Lett. , 081601 (2001).[12] R. Gregory, S. Kanno and J. Soda, JHEP , 010 (2009).[13] Q. Pan, B. Wang, E. Papantonopoulos, J. Oliveira and A. B. Pavan,
Phys. Rev.
D81 , 106007(2010).[14] D. C. Zou, S. J. Zhang and B. Wang,
Phys. Rev.
D 87 , 084032 (2013).[15] X. M. Kuang, E. Papantonopoulos, G. Siopsis and B. Wang,
Phys. Rev.
D 88 , 086008 (2013).[16] Gregory Walter Horndeski
Int. J. Theor. Phys. (1974) 363-384[17] D. Lovelock, J. Math. Phys. , 498 (1971).[18] Curtis G. Callan, Jr., C. Lovelace, C.R. Nappi, S.A. Yost (Princeton U.). Nov 1986. 36 pp.Published in Nucl.Phys. B288 (1987) 525[19] M. Ostrogradski, Mem. Ac. St. Petersbourg VI 4, 385 (1850). 406, 407, 430[20] R. Woodard Lect. Notes Phys. (2007) 403-433, astro-ph/0601672[21] Miguel Zumalacarregui, Tomi S. Koivisto, David F. Mota
Phys. Rev.
D87 (2013) 083010.[22] Juan Crisostomo, Ricardo Troncoso, Jorge Zanelli
Phys. Rev.
D62 (2000) 084013.[23] Rodrigo Aros, Ricardo Troncoso, Jorge Zanelli
Phys. Rev.
D63 (2001) 084015.
24] K. Izumi,
Phys. Rev.
D 90 , 044037 (2014).[25] V. Cardoso and J. P. S. Lemos,
Phys. Rev.
D63 , 124015 (2001).[26] V. Cardoso and J. P. S. Lemos, Phys. Rev. D , 084017 (2001).[27] G. T. Horowitz and V. E. Hubeny, Phys. Rev. D , 024027 (2000).[28] E. Berti, V. Cardoso and A. O. Starinets, Class. Quant. Grav. (2009) 163001.[29] R. A. Konoplya and A. Zhidenko, Rev. Mod. Phys. (2011) 793.[30] D. T. Son and A. O. Starinets, JHEP , 042 (2002).[31] B. Cuadros-Melgar, J. de Oliveira and C. E. Pellicer, Phys. Rev. D , 024014 (2012).[32] E. Abdalla, O. P. F. Piedra, F. S. Nuez and J. de Oliveira, Phys. Rev. D , no. 6, 064035(2013).[33] P. K. Kovtun and A. O. Starinets, Phys. Rev. D , 086009 (2005).[34] A. S. Miranda, J. Morgan and V. T. Zanchin, JHEP , 030 (2008).[35] A. Nunez and A. O. Starinets, Phys. Rev. D , 124013 (2003).[36] S. Chandrashekar, The Mathematical Thoery of Black Holes , New York, Oxford UniversityPress, (1983).[37] D. Z. Freedman, S. D. Mathur, A. Matusis and L. Rastelli, Nucl. Phys. B , 96 (1999)., 96 (1999).