Viscosity bound violation in holographic solids and the viscoelastic response

Journal of High Energy Physics, Jul 2016

Abstract We argue that the Kovtun-Son-Starinets (KSS) lower bound on the viscosity to entropy density ratio holds in fluid systems but is violated in solid materials with a nonzero shear elastic modulus. We construct explicit examples of this by applying the standard gauge/gravity duality methods to massive gravity and show that the KSS bound is clearly violated in black brane solutions whenever the massive gravity theories are of solid type. We argue that the physical reason for the bound violation relies on the viscoelastic nature of the mechanical response in these materials. We speculate on whether any real-world materials can violate the bound and discuss a possible generalization of the bound that involves the ratio of the shear elastic modulus to the pressure.

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Viscosity bound violation in holographic solids and the viscoelastic response

Revised: June Viscosity bound violation in holographic solids and the Lasma Alberte 0 1 2 4 5 6 Matteo Baggioli 0 1 2 3 5 6 Oriol Pujolas 0 1 2 5 6 0 Campus UAB , 08193 Bellaterra (Barcelona) Spain 1 The Barcelona Institute of Science and Technology , BIST 2 Strada Costiera 11 , 34151, Trieste , Italy 3 Department of Physics, Institute for Condensed Matter Theory, University of Illinois , USA 4 Abdus Salam International Centre for Theoretical Physics , ICTP 5 respondence , Space-Time Symmetries 6 1110 W. Green Street, Urbana, IL 61801 , U.S.A We argue that the Kovtun-Son-Starinets (KSS) lower bound on the viscosity to entropy density ratio holds in uid systems but is violated in solid materials with a nonzero shear elastic modulus. We construct explicit examples of this by applying the standard gauge/gravity duality methods to massive gravity and show that the KSS bound is clearly violated in black brane solutions whenever the massive gravity theories are of solid type. We argue that the physical reason for the bound violation relies on the viscoelastic nature of the mechanical response in these materials. We speculate on whether any real-world materials can violate the bound and discuss a possible generalization of the bound that involves the ratio of the shear elastic modulus to the pressure. Holography and condensed matter physics (AdS/CMT); Gauge-gravity cor- 1 Introduction 2 3 4 Viscoelasticity Holographic solids 4.1 4.2 Numerical results Analytic estimate 4.2.1 4.2.2 The real part: elasticity The imaginary part: viscosity 5 Discussion A Holographic renormalization for = 2 1 Introduction Viscoelastic properties of the holographic solids measured, including examples like super uid helium [6] and the QCD quark gluon plasma (see e.g. [ 7 ]). By now it is well established that the KSS bound is violated by higher curvature corrections to the Einstein theory. In particular, the violation of the bound was observed 1We work in the units where ~ = kB = 8 G s = 1 4 : { 1 { in Einstein gravity supplemented by the quadratic Gauss-Bonnet term [8]. In terms of the Gauss-Bonnet coupling GB the viscosity to entropy density ratio was found to be s = 1 4 [1 in the dual eld theory allow for superluminal propagation velocities for GB > 9=100, thus imposing a new lower bound on the viscosity to entropy ratio [ 9 ]. In the light of these results it is at present not clear whether a universal fundamental bound on the shear viscosity to entropy ratio exists. For a review on the bound violation in higher derivative theories of gravity, see [ 10 ] and references therein. Another example of violation of the bound has been found in anisotropic theories [11{14]. In this work we shall study the possibility of violating the KSS bound in the eld theories dual to massive gravitational theories. In distinction from Gauss-Bonnet gravity, introducing a non-vanishing graviton mass modi es gravity in the infrared and therefore this can have a large impact on the homogeneous and static response. In holographic context, massive gravity has already turned out to provide a useful mechanism of incorporating momentum dissipation [15, 16]. It was also shown that although the energy-momentum tensor is not conserved in eld theories dual to massive gravity, it nevertheless admits an e ective hydrodynamic description at su ciently high temperatures and small values of graviton mass [17, 18]. Here we shall use the most general massive gravity models that can be written in terms of two Stuckelberg scalar elds rst introduced in [19] (for earlier related works, see [20, 21]). Basing on the analogy to the at space e ective eld theory (EFT) description of solids and uids [22{26] it has been argued recently that these two elds models of massive gravity can also be broadly divided into solids and uids [ 27 ]. The claim was further supported by studying the response of the massive gravity black branes to external shear strain deformations: massive gravities with the symmetries of the solid EFTs are endowed with a non-zero shear elastic modulus. The uid massive gravities, however, exhibit zero elastic response. A precise de nition of solid and uid massive gravities will be given in section 3. The purpose of this work is to analyze the impact of the graviton mass term on the viscosity and on the KSS bound in the case of solid massive gravity black branes. These are holographically dual to solids that we shall refer to as holographic solids henceforth. Naively, one might expect that solids correspond to the large viscosity limit, therefore does not exhibit any reduction in the =s ratio. However, this is not the case. The characteristic response under mechanical deformations of solids is very di erent from that of uids, especially under static and homogeneous deformations. A material that exhibits an elastic response, i.e. a solid, counteracts an applied constant stress with a constant in time deformation characterized by a displacement vector, ui, and a constant strain uid, in turn, a constant applied stress results in a constant ow | a constant velocity gradient or strain rate, @(iu_ j). The response under mechanical ! 1, and { 2 { deformations can actually be more complex and exhibit both types of behaviour. This is what happens in viscoelastic materials, a more precise de nition of which we defer for the next section. Heuristically, it is clear that if certain massive gravity black branes are endowed with a notion of elasticity then they must be viscoelastic at least in the limit of small graviton mass where one recovers the Einstein gravity with viscosity given by (1.1). Below we investigate how turning on a nite elastic response a ects the viscosity and show that the KSS bound is violated in theories dual to solid massive gravities. 2 Viscoelasticity In the standard mechanical linear response theory [ 28, 29 ] the internal stress of a homogeneous and isotropic material due to a constant shear deformation described by the displacement vector ui can be expressed via the linear relation HJEP07(216)4 between the traceless stress tensor and the traceless part of the linear displacement tensor Ti(jT ) = G ui(jT ) 1 2 uij = Ti(jT ) = u_ i(jT ) : Ti(jT ) = G ui(jT ) + u_ i(jT ) : = G : 1 lim !!0 ! R Im GTij Tij { 3 { The proportionality coe cient G in the shear stress/strain ratio is the modulus of rigidity and is non-zero only for solid materials, which are said to exhibit an elastic response. Fluid materials, instead, show a viscous response whereby the constant internal stress is due to a deformation with a constant shear rate, The shear stress/strain rate ratio de nes the viscosity of the material. Materials that exhibit a viscoelastic response display both elastic and viscous uid behaviour. The simplest way to introduce such materials is by considering a general time dependence in uij (and in Ti(jT )). The most common presentation of viscoelasticity then consists of a low frequency response relation of the form [28] Some of the typical viscoelastic phenomena such as creep and stress relaxation follow easily from (2.4): the system relaxes to an equilibrium con guration within a `Maxwell' relaxation time given by It is clear that (2.4) can be understood as a truncated expansion in time derivaties of a kind of damped oscillator, and that the most important parameters that encode the viscoelastic response at low frequencies are the modulus of rigidity, G, and the shear viscosity, . For both the modulus of rigidity and the shear viscosity, one can de ne convenient notions of these transport coe cients from Green-Kubo (GK) formulas that relate them to the correlators of the stress tensor [30]. The GK shear viscosity is thus given by [2, 3] (2.1) (2.2) (2.3) (2.4) (2.5) can similarly be de ned as [ 27 ] R is the retarded Green's function of the stress tensor. The GK modulus of rigidity (2.6) In terms of the two parameters de ned in (2.5) and (2.6), the static mechanical response of generic isotropic materials can be depicted in the fG; g plane. The G = 0 axis corresponds to uids. The = 0 axis to non-dissipative (e.g. at zero temperature) solids. The rest of the two dimensional space is spanned by viscoelastic materials. As we shall see, solids dual to massive gravity black branes do lie inside this plane. An important technical comment is now in order. The holographic computation of R the Green's function GTij Tij involves the evaluation of the boundary on-shell action which is in general divergent. Hence, the modulus of rigidity de ned as (2.6) is a UV-sensitive R parameter, and in fact the real part of GTij Tij at low frequencies can be a ected by local counterterms (`contact terms'). We will show this explicity for the holographic models in the computations below. From a physical point of view, this is not at all surprising. In most solids, the modulus of rigidity does scale with the atomic spacing, hence the UV sensitivity is actually a physical property. This does not mean, however, that one cannot compute this quantity in a meaningful way in our holographic models. Instead, it requires that one performs a careful renormalization procedure using appropriate local counterterms that remove the divergences and respect the symmetries of the theory/material. As we will see below, for the materials dual to uid and solid massive gravities in AdS, there are indeed various renormalization prescriptions that one can use, which can be mapped to the existence of various counterterms that give a nite contribution to the modulus of rigidity G. As our guiding principle we shall use the known fact that in a translationally invariant boundary theory the retarded Green's function vanishes for zero momentum [31{33]. According to the de nition in (2.6) this gives G = 0. Since translationally invariant theories admit an e ective low-energy hydrodynamical description we associate a vanishing elasticity as a property of uids without momentum dissipation. On the other hand, in the case of solids the translational symmetry is broken and we expect a non-zero low freqency response, i.e. G 6= 0. Importantly, we nd below that there is a simple renormalization scheme that correctly reproduces this expectation and gives a vanishing shear modulus in the ` uid limit' | in the limit where the volume preserving spatial di eomorphisms are unbroken. This de nition of the uid limit includes both uids that are translationally invariant, arising from Einstein gravity duals, and uids in which the translational invariance is broken, arising from massive gravity theories. The existence of such renormalization scheme is non-trivial and translates into the existence of local counterterms that respect the appropriate symmetries. One of the results of the analysis below is to exhibit that these counterterms exist as standard covariant countertems that can be used in the holographic renormalization of the solid and uid type theories. { 4 { 3 Holographic solids We consider a 3 + 1 dimensional gravity theory S = Z d4x p g 1 2 R + 6 L2 where L is the AdS radius, m is a dimensionless mass parameter, and m2 L2 V (X; Z) + Z r!0 d x 3 p K ; (3.1) HJEP07(216)4 L2 4 Z F F X 1 2 tr[IIJ ] ; det[IIJ ] ; I IJ and the indices I; J = fx; yg are contracted with IJ . In (3.1), we have included the Gibbons-Hawking boundary term where is the induced metric on the AdS boundary, and K = r n is the extrinsic curvature with n vector to the boundary. Around the scalar elds background ^I = iI xi the metric admits | an outward pointing unit normal the black brane background solution Additionally, the choice of counterterms giving G = 0 in the uid limit reproduces the results obtained by the standard prescription for nding the retarded Green's function as a ratio between the subleading to leading modes. We use the latter prescription in our numerical analysis and it correctly gives a nonzero rigidity G only for holographic solids. In any case, let us emphasize that the de nition of the viscosity is completely una ected by the details of this renormalization procedure | the counterterms do not a ect the R imaginary part of the GTij Tij correlator, which is the main object of this article. (3.2) (3.3) (3.4) (3.5) (3.6) with the emblackening factor given in terms of the background value of the mass potential: ds2 = L 2 dr2 f (r)r2 + f (r)dt2 + dx2 + dy2 r2 ; encoded in the transverse traceless tensor mode of the metric perturbations.2 In 3 + 1 dimensional bulk, tensor modes exist only for perturbations homogenous in the transverse directions and its two helicity-two components can be parametrised as h+ and h Lr22 hxy. The quadratic action for the homogeneous tensor mode takes the form 21 Lr22 hxx hyy S = Z d x where h stands for any of the two components h+; h and we have de ned a mass function 2We de ne the metric perturbations as g = g^ +h so that the inverse perturbations g = g^ +h are given by h = g^ g^ h + O(h2). M 2(r) 21r2 V^X (r) : { 5 { The subscript X denotes the partial derivative VX Fourier mode, de ned through h(t; r) = R d2! h!(r)e i!t, becomes that of a massive scalar @@XV . The equation of motion for the eld living on the black brane background (3.3): f 0 2 f r r 2 h! = 0 : (3.7) It is very important to emphasize here that the mass of the tensor mode is only due to the X dependence of the potential V (X; Z). Hence, in the case when V is only a function of Z the graviton remains massless. In our previous work we have argued that in the case when uids, whereas the presence of an X dependence, i.e. when V = V (X; Z), indicates that the material is a solid [ 27 ]. We have also shown that there is no elastic response in the case of uids. Moreover, since for uids the graviton mass is zero, the universality proof [5] for the viscosity to entropy ratio based on the membrane paradigm is applicable and we expect no violation of the KSS bound. Without loss of generality we therefore only consider the theories describing solids with graviton mass terms of the form V (X) = Xn : Here we are allowing for general values of n in order to see what is the impact of this parameter on the elasticity and viscosity. There is a naturally preferred value, n = 1, where the model enjoys enhanced consistency properties in the sense that the Stuckelberg sector does not introduce a low strong coupling scale (see [ 19, 27, 35 ] for more consistency On the background solution X^ = (r=L)2, and the graviton mass potential (3.8) yields ; r L +4 = to the energy density " in the renormalized dual eld theory, i.e. " = M L2, and reads: can be found as usual to be T = jf 0(rh)j = 4 1 4 rh 3 m2 rh L +4 { 6 { We see that there is an extremal value of the graviton mass m at which the temperature of the black brane vanishes: vergent contributions due to the graviton mass term of the form Sbdy Hence, for + 1 < 0 additional counterterms that remove the divergences are needed. In order to nd the covariant counterterms expressed in terms of the bulk elds the full m2(r=L) +1. procedure of holographic renormalization has to be carried out. However, the nite action can be found by simply removing the divergences with counterterms of the form Scmount = m2 R d3x p P1 n=1 cn(r=L)n, where is the metric induced on the boundary and cn are constant coe cients. They can be xed in such a way that all the divergent terms cancel out. In practice, only several of the counterterms are needed and most of the coe cients cn are equal to zero.4 We note that the coe cients cn depend neither on the temperature nor the chemical potential and, thus, do not a ect the thermodynamics. We nd the nite Euclidean on-shell boundary action and identify it with the thermodynamic potential = T SbEdy + ScEount = V L2 2 1 r 3 + h 2 2rh m2( + 3) rh L3( + 1) L +1 + 0(m2; cn) ; (3.14) where ScEount = Scmount + 12 R d3x p (4=L), V is the area of the spatial boundary, and we denote by 0 a constant contribution due to the renormalization procedure. Given the free energy, we nd the pressure of the boundary theory as p = =V . We see then, that the null energy condition is always satis ed in the boundary theory since L2 2 rh rh " + p = T + 2 > 0 : We also nd that the charge density and entropy density are given by the usual expressions: = 2L2=rh and s = 2 L2=rh2. Together with the energy density given in (3.11) they satisfy the rst law of thermodynamics " + p = sT + . We note that the pressure de ned from the free energy as p = =V does not coincide with the hydrodynamical pressure P appearing in the constitutive relation for the hydrodynamic stress energy tensor T uid and satisfying P = "=2. Instead, " 2 P T xuxid = = p m2q ; q 1 ( + 4) rh L ( + 1) L +1 : This is very similar to what happens in the presence of a constant magnetic eld [32] and has been already outlined for the simple case V (X) = X in [20]. 3Upon analytic continuation = it, SE = iSL[gL] = SL[gE] and A = iAt. We integrate the r=0 volume integral as R r=rh dr. There are contributions to the on-shell action from both integration limits. 4Such renormalization procedure for the dRGT massive gravity was carried out in [16]. The full covariant renormalization for the mass Lagrangians with = f 2; 3g was done in [20, 21]. { 7 { (3.15) (3.16) In this section we nd the shear viscosity (2.5) and elastic modulus (2.6) in the eld theory dual to the bulk massive gravity with the mass function r!0 lim h = h0 + By comparing to (3.6) and (3.9) we see that this de nition corresponds to the mass potential V (X) = n1 Xn with n = (4 + )=2. From the equation of motion (3.7) it follows that in the near-boundary region the metric (4.1) (4.2) (4.3) HJEP07(216)4 showing that the scaling dimension of h is = 3 and is independent on the radial dependence of the graviton mass. The gauge/gravity duality prescription then allows one to nd the retarded Green's function as the ratio of the subleading to leading mode of the graviton (see e.g. [33]): R GTij Tij = 2 2L d h3 h0 where d = 3 is the number of spatial dimensions. We numerically solve the equation of motion for the graviton and extract the retarded Green's function by using the above expression. In gure 1 we show the real part of the Green's function and the viscosity to entropy density ratio as a function of the graviton mass for di erent values of the exponent . We rst observe that the =s ratio goes below the universal value 1=(4 ) 0:08 for graviton mass parameter values m > 0 and thus violates the KSS bound. As expected, in the uid regime with m = 0 we recover the standard universal value (1.1). The second observation that we make is that the real part of the Green's function becomes negative for all values of apart from = Although, negative modulus of elasticity can, in principle, be observed in nature it is always associated with instabilities (see e.g. [37]). From the holographic perspective, the fact that there is an instability is not so surprising because the kinetic terms for the Stuckelberg elds are non-canonical for V (X) = Xn with n > 3=2 and n = 1=2 (corresponding to > 1 and = 3 respectively). Both the numerical and analytical results give a positive rigidity modulus for the canonical Stuckelberg case, n = 1 5More precisely, the asymptotic UV expansion of the metric perturbations h reads: rli!m0 h = h0 1 + hp r p L r 3 L + : : : + h3 (1 + : : : ) + : : : For < 1 we have p < 3 (for example, p = 2 for = 2). However, this does not a ect the identi cation of h0 as the source and h3 as the v.e.v. of the operator associated to the bulk eld h and the consequent de nition of the Green's function given in (4.3). { 8 { 1.0 0.5 -0.5 -1.0 =s ratio, the horizontal dashed line shows the value 2) with V = X, which can therefore be singled out as the most reasonable model from the phenomenological point of view. We would like to point out that the fact that the KSS bound can be violated in theories with massive gravity duals was also noticed in [18] for the case = 2 corresponding to V = X=2 and graviton mass m = p2rh 1 for which the energy density (3.11) vanishes6 and m=T = p = 0. However, this observation was made only at one particular value of the It was then argued by the authors that this result is irrelevant for the physical viscosity due to the fact that for graviton masses of order m=T & 1 the dual eld theory does not admit a coherent hydrodynamic description. Instead a crossover from the coherent hydrodynamic phase of the system to an incoherent regime occurs for graviton mass that is comparable to the black brane temperature. In the results presented in this paper we see the violation of the KSS bound also at arbitrary small values of the graviton mass where the hydrodynamic description applies. We therefore believe that our ndings are physically signi cant and suggest that the KSS bound can be violated in materials with non-zero elastic response. In general, however, we nd that the question of whether or not the black branes are close to having a hydrodynamic description is not particularly relevant in the context of holographic solids. In these systems we do not expect the dynamics to be understood in terms of hydrodynamics while there does exist a well de ned low energy e ective eld theory description of solids de ned as an expansion at low frequencies and momenta. 4.2 Analytic estimate In this section we shall provide an analytic derivation for the elasticity and viscosity. For this we shall solve the equation of motion (3.7) for the metric perturbation h in the zero frequency limit by (i) imposing ingoing boundary conditions at the horizon and (ii) requiring that near the AdS boundary the bulk eld hjr!0 = h0. The on-shell boundary density vanishes at m = p2rh 1. 6The expression for the energy density (3.11) is given for the models V = Xn. To account for the factor 1=2 in the model considered in [18] we need to replace m2 ! m2=2 in (3.11). It then follows that the energy { 9 { action can then be brought in the form Sbdy = Srh + S = Z d x 2 Z d! 2 h0 F (!; r)h0 r=rh r= : be extracted from the on-shell boundary action as R GTijTij = from the horizon is neglected. following ansatz where the function F (!; r) is only evaluated at the AdS boundary while the contribution HJEP07(216)4 The equation of motion (3.7) for h in the zero frequency limit can be solved by the h(r) = h0 e 4i !T log f 0(r) + 1(r) + : : : The ingoing boundary conditions are satis ed by the exponential ansatz while the functions 0 and 1 are required to be regular at the horizon. Near the boundary we demand that 0(0) = 1 ; 1(0) = 0 : The equations for 0 and 1 are obtained by solving the original equation of motion (3.7) order by order in the frequency !: 2 z f z 2 f f f z2 00(z) z2 01(z) 0 0 4 m2 rh4 z2 M 2(z) L2f (z) 4 m2 rh4 z2 M 2(z) L2f (z) 0(z) = 0 ; 1(z) f 2 f 0 00(z) + 0(z) 2 f 0 z f f 00 f = 0 ; (4.9) change to the probe limit results. and use an ansatz for the functions graviton mass parameter m2: where we have introduced the variable z = r=rh and the primes denote derivatives with respect to z. We note that the last summand in the equation for 1 vanishes on the Schwarzschildde Sitter background f (z) = 1 z3. This simpli es the calculations and corresponds to the probe limit of the full massive gravity theory where we neglect the backreaction of the graviton mass and of the U (1) eld on the background metric. We shall perform the calculations in this limit. A comparison between the numerical results obtained in the proble limit with the corresponding results obtained in the full model including the backreaction is shown in gure 2 for = 2. As expected, we see that the e ect of backreaction is negligible for small values of the graviton mass and introduces no qualitative We consider the mass function (4.1) which now becomes M 2(z) = z (rh=L) =(2L2) 0 and 1 that is perturbative in the dimensionless 0 = 1 X m2n n ; n=0 1 = 1 X m2n n n=0 (4.10) (4.4) (4.5) (4.6) (4.7) (4.8) 0.08 0.06 2) and the probe limit results (dashed lines) for rh = 1 and zero chemical potential = 0. Left: the real part of the Green's function; right: =s ratio. with 0 = 1 and 0 = 0. The equations of motion for 1 and 1 follow from (4.8) and (4.9) and read: f We emphasize that the equation (4.11) is exact for any form of the emblackening factor f (z) whereas the equation (4.12) is only valid for the emblackening factor f (z) = 1 The quadratic on-shell boundary action up to real contact terms is given by Sbdy = L2 Z 4rh3 d3x f z2 h(z) h0(z) z=1 z= : The contribution from the AdS boundary, z = with mass squared and frequency reads ! 0, up to rst order in the graviton S = As we shall see below the last term has a non-zero contribution only in the case when 4 which corresponds to the mass potential V (X) = V0 equivalent to a cosmological constant. Since we do not want to modify the cosmological constant and the corresponding radius of the asymtptotically AdS we shall not consider the case = 4. 4.2.1 The real part: elasticity Up to rst order in m2 the real part of the Green's function (4.5) is determined solely by the derivative of the function 1 as: G = Re G = L2 2rh3 l!im0 m2 f (4.11) (4.12) z3. (4.13) (4.14) For any given M 2(z), the equation of motion (4.11) can be integrated as For the mass function (4.1) this gives G = 2m2rh lim xed by demanding that the function 1(z) is regular at the horizon z = 1 (as assumed implicitly in (4.15)): We note that in the near boundary limit ! 0 the expression (4.16) is divergent for < 1. This signals the breakdown of the perturbative method, and one has to invoke a proper regularization method in order to reproduce the nite results that are obtained numerically. There are several ways to do that including the matching asymptotic series approach used in [2], the holographic renormalization along the lines of [41, 42], and others. Here we exploit the fact that the expression (4.16) is nite in the and obtain the result for < 1 by analytic continuation in . In other words, we take the ! 0 limit for > 1 ! 0 limit assuming > 1 and then continue analytically the result in . This method is apparently equivalent to a regularization method where the divergent terms are simply discarded. The real part of the Green's function thus becomes G = L2 2 rh3 m2 c ; 6 = The comparison of the expression (4.18) with the numerical results for = 2 is given in gure 3. It shows a good agreement with the numerical results thus further validating our regularization approach. For the sake of completeness we perform the full holographic renormalization of the action (3.1) with = 2 in the appendix A. As expected we nd that upon addition of proper counterterms the divergent contributions to the Green's function cancel. However, the nal expression for the real part of the Green's function obtained by the analytic continuation does not exactly coincide with the results obtained by using the covariant holographic renormalization approach | the two results di er by nite counterterms. Since the analytic continuation method agrees well with the numerical results from the prescription (4.3), we nd that this gives a good con rmation that the Re G at ! = 0 extracted in this way has an unambiguous physical meaning. In particular, it can be interpreted as the rigidity modulus. We leave for the future a more thorough analysis that shall clarify the most appropriate choice of the counterterms for this purpose. 4.2.2 The imaginary part: viscosity The imaginary part of the Green's function Im G = L2 2rh3 l!im0 4 T i! f 2 10 (4.15) (4.16) (4.17) (4.18) (4.19) 1.4 1.2 results as a function of the dimensionless graviton mass parameter m for the mass function with 2. Solid and dashed lines are the numerical and analytic results respectively. Left: real part of the retarded Green's function for rh = 0:6. Right: =s ratio for rh = 1. can be found by solving equation the (4.12). Given (4.16) this leads to f where we have set the upper boundary of integration so that the right hand side vanishes on the horizon z = 1 ensuring that the function 1(z) is regular. In order to evaluate the Green's function we are only interested in the value of the above expression at the AdS boundary z = 0. By using this result we are now able to nd the viscosity to entropy density ratio s = 1 4 1 + 2 3 c m2 Z 0 f 0 1 f 1 x +1 dx where we have used the relation (2.5) and the fact that the temperature of the SchwarzschildAdS solution with the emblackenig factor f (z) = 1 z3 is simply T = 3=(4 rh) and the entropy density is given by s = 2 (L2=rh2). The integral in the above expression can be written in terms of the Harmonic numbers m (4.20) (4.21) (4.22) (4.23) For the value = 2 this gives the exact expression H 13 ( +1) 1 x +1 : s = 1 4 m2 log(3) 1 p The quantity in the brackets is positive and hence we see that the viscosity to entropy density ratio is less than the universal value 1=(4 ) thus violating the KSS bound. In gure 3 we compare the numerical results for the real part of the Green's function and for the =s ratio with the corresponding analytic expressions (4.18) and (4.23). We see a good agreement for small values of the graviton mass parameter m. In this work we have shown that in massive gravity theories of the solid type, with the mass potential given by (3.8), the asymptotically AdS black branes do not obey the KSS lower bound, =s 1=(4 ) [3]. Additionally, we have seen a clear correlation between the bound violation and the presence of a non-zero shear elastic modulus G.7 We have given both numerical and analytical evidence that are in complete agreement with each other. We have performed a number of consistency checks to ensure that the result is physical. In the regime where the theory is completely under control (i.e. free of instabilities and other pathologies) the result holds and it is unambiguous. Therefore, we conclude that at least in terms of the Green-Kubo de nition of the viscosity (2.5) there is a physical violation of the KSS bound in the solid massive gravity black branes. There are many potential implications of this result. First, it becomes clear that the universal value =s = 1=(4 ) exhibited in Einstein gravity hinges upon a key dynamical property of that theory, namely, that the mass of the spin-two graviton mode, as de ned in equation (3.7), vanishes. In the gravitational theory, the parameter relates directly to the spin-two absorption cross section [2]. Hence, it is not surprising that gets suppressed in the presence of a non-zero mass for the spin-two graviton. Interestingly, the universal result for the viscosity to entropy density ratio holds also in the very large class of the uid type massive gravity theories with mass potential V = V (Z), since they give a zero graviton mass as can be seen from equation (3.6). Thus, in massive gravity theories the =s ratio is very sensitive to the spin-two mass m2M 2(r): once it is non-zero, the value of the =s ratio is not universal. Instead, it also depends on temperature and other parameters of the theory and interpolates between the KSS value at m = 0 and zero at large values of m.8 We would also like to stress that both the uid and the solid types of massive gravity theories are dual to theories with momentum dissipation. However, the conjectured KSS bound seems to hold in the case of uids but is violated in the case of solids. The latter argument therefore shows that there is no direct correlation between the introduction of momentum dissipation (i.e. translational symmetry breaking) and the violation of the viscosity to entropy bound. In the dual eld theory picture, =s is the usual viscosity to entropy ratio of the CFT. From [ 27 ], we know that the spin-two graviton mass m2M 2(r) controls directly the shear elastic modulus or modulus of rigidity, G. In eld theory language, the previous result can then be phrased as follows: once the system exhibits a non-zero modulus of rigidity, G, the KSS bound =s 1=4 is violated. In other words, the KSS bound does not apply to holographic solids, incarnated as solid massive gravity black branes. Let us now discuss whether or not some form of this statement can hold for real solids. In the rst place, one can ask why should =s be allowed to go down once the elasticity G 6= 0 is switched on? How can a solid present a better uidity than the most perfect 7As advanced in [ 19, 27 ], this also implies the presence of dynamical transverse phonon modes, the propagation speed of which should be largely determined by G. 8In the limit T =m 1, when the temperature is the dominant scale in the system the graviton mass becomes irrelevant and the usual results ( =s = 1=4 ) are recovered (see [43] for similar analysis). uids? At this point it is already clear that we are dealing with materials that have both viscosity and elasticity and thus qualify as viscoelastic materials in the sense that we introduced in section 2. As emphasized there, the mechanical response for these materials is more complex than for uids or perfectly elastic solids, and there seems to be a sense in which these materials are capable of owing more easily than viscous uids. Some of the particularities of the viscoelastic response are better understood by looking at the response of the material under a time-dependent applied stress. In this regard, some of the properties of viscoelastic materials include the so-called creep and stress relaxation. Once an applied stress is removed instantaneously, the material relaxes ( ows) back to its equilibrium position. Hence, viscoelastic materials, in a way, are able to ow without any applied stress for some time. This situation formally corresponds to = 0, so perhaps this is the physical reason why =s can be small. The bottom line is that the more complex viscoelastic response might be the physical explanation for the violation of the KSS bound. In view of this, we shall entertain the possibility that the violation of the =s 1=4 bound is physical and occurs in more general and realistic materials than just the holographic solids discussed in this paper. We anticipate that there are three types of systems where this can apply: i) strongly correlated solids that admit an e ective description in terms of a strongly coupled QFTs with non-zero rigidity; ii) materials that are described by weakly coupled QFTs with non-zero rigidity;9 iii) general viscoelastic materials. While we do not have a convincing evidence for the violation of the KSS bound in the cases ii) and iii), the holographic computation gives a strong support that it does indeed occur in the case i). Interestingly, graphene does comply with the two conditions of this case: it has an enormous rigidity modulus and it contains strong correlations. Our results thus seem to suggest that the KSS bound might be violated in graphene, although it might be that this depends on other factors such as the degree of disorder or on temperature, as we have seen in the holographic computation. Earlier theoretical calculations show unexpectedly low values of the =s ratio for graphene, even if satisfying the KSS bound [39]. However, to the best of our knowledge, no experimental measurement of the =s ratio has been done so far. A recent theoretical proposal for methods of measuring this ratio in solid-state devices was put forward in [40] giving a hope for more experimental results in the future. In the spirit of the KSS conjecture, one can also wonder whether or not there is any generalization of it that holds in solid systems. From dimensional analysis it is reasonable to expect that if there does exist a more general bound, it should involve the rigidity to pressure ratio, G=p, in addition to the =s ratio. In gure 4 we plot =s against G=p for the holographic solid with = 2 and see a clear correlation. Keeping the KSS logic [3] that the gravity solutions might represent the least dissipative materials, the gure 4 then suggests that there might be a more general bound in (viscoelastic) solids. At relatively large temperatures this would approximately take the form 4 s G + C p 9It is however known that at weak coupling the =s ratio considerably exceeds the KSS value in the case of G = 0 [3]. It therefore seems not so easy that the bound is violated in these cases. 0.08 0.06 = 1 and di erent values of rh. The value of the graviton mass and the temperature are changing along the solid lines, with m = 0 at the point where =s = 1=4 0:08. Similar plots are obtained by keeping m constant and varying T only. with C being an order-one constant. We further note that at zero temperature, T = 0, the viscosity to entropy ratio becomes zero and the rigidity modulus reaches its maximum value. The existence of a similar universal bound on the thermoelectric transport coe cients has been recently conjectured in [44, 45] and explored in holographic theories featuring momentum dissipation in [46]. Note added. While this work was being completed ref. [47] appeared, where the violation of the KSS bound in massive gravity theories is also discussed. Near the nal stages of our work we learned about similar ndings of yet another collaboration that appeared later in [48] after our work was already made public. Where our results overlap, they agree. Acknowledgments We thank Andrea Amoretti, Danny Brattan, Mikhail Goykhman, Andrei Khmelnitsky, Rene Meyer, Daniele Musso, Nick Poovuttikul, Stephan Roche and Dam Thanh Son for very useful discussions and comments about this work. MB would like to thank University of Illinois, ICMT and Philip Phillips for the warm hospitality during the completion of this work. We acknowledge support by the Spanish Ministry MEC under grant FPA201455613-P and the Severo Ochoa excellence program of MINECO (grant SO-2012-0234), as well as by the Generalitat de Catalunya under grant 2014-SGR-1450. A Holographic renormalization for = 2 In section 4.2 we have seen that the on-shell boundary action (4.13) that follows from the action (3.1) is divergent in the near boundary limit r ! 0. In our analytic calculations In this case the divergent on-shell boundary action that follows from (3.1) should be supplemented by the counterterm action [41, 42]: (A.1) W [ ] = Sren = V L2 2T m2 rhL2 1 r (A.2) where a = ft; x; yg is an index in the boundary theory. This consists of counterterms due to the divergences arising in the Einstein action and in the scalar elds action (the Maxwell action is nite and does not require any additional counterterms). The renormalized boundary action is then a sum of (3.1) together with (A.2). On-shell, it reduces to a sum of the surface terms, Sbredny = Srrhen + Sren. Following the prescription of [36, 38], the Minkowski space Green's function is obtained by using the near-boundary action Sren as the generating functional, i.e. W [ ] = Sren. Given W [ ], the correlators of T can then be found by di erentiating W with respect to . Evaluating the generating functional we nd of the retarded Green's function we have dropped the divergent terms and assumed that the nite results that we derive for the retarded Green's function (coinciding with the numerical procedure results) are not a ected by the renormalization procedure. The main goal of this section is to check that assumption by computing the two-point correlator of the stress-energy tensor from the generating functional W [ ], de ned as the renormalized on-shell boundary action. We shall do this while keeping in mind that, in practice, the Green's function derived from the generating functional does not always coincide with the retarded Green's function de ned as in (4.3). The two di er by nite contact terms that can be xed by the Ward identities of the symmetries of the generating functional [31]. We postpone the analysis of the Ward identities to future work. We consider the massive gravity action (3.1) with n = 1 (or equivalently, = 2). According to (3.9) this corresponds to the mass parameter p = m2 rh PV T 6 = : L V T where = r=rh. The part quadratic in metric perturbations in the above action di ers from the boundary action (4.13) used in section 4.2 by nite contact terms and divergent counterterms. The last term in (A.3) coincides with the near boundary contribution of (4.13). We note that the constant background value of the above action S0 = Sren h=0 = V L2 2T 1 r 3 + h 2 2rh m2 rhL2 = V T =T (A.4) does not coincide with the value found from the Euclidean action given in (3.14). Instead, by evaluating the quantity q de ned in (3.16) for the case 2, we nd q = 1 rh ; p m2q = (A.5) Hence, the Lorentzian prescription used in this section seems to suggest that the thermodynamic potential (i.e. the potential that reaches the minimal value in thermal equilibrium) thermodynamic potential is given by = p V .10 is given by the pressure P V . According to the Euclidean prescription, however, the The di erence between the Euclidean and Lorentzian boundary actions (3.14) and (A.4), respectively, appears due to the mass term contribution. In principle, it could be removed by an additional nite counterterm added to the mass sector of the counterterm action (A.2). Working out the possible counterterms, the only choice with not more than two derivatives and preserving the homogeneity and isotropy in the boundary theory that HJEP07(216)4 is nite in the limit ! 0 is Scsooulindt = p 3=2 = pf ( ) 1 + h2( ) ; (A.6) where the parameter shall be determined from the background value S0 of the total boundary action. We have also factored out some dependence on m and rh for convenience. In fact, this is precisely our goal here since the expression (A.4) contradicts the Euclidean results. It is not clear whether an addition of such a counterterm is viable, but we shall nevertheless explore this possibility here. The results without the addition of this ambiguous counterterm can be readily recovered by setting = 0 in the nal expressions. Let us also note that there exists another ` nite' counterterm (i.e., giving nite contributions both to the classical background action and to the two-point functions), which Scouuidnt = m2 Z rh d3x p (A.7) This counterterm would play a very important role in uid massive gravities | that is, in the case when the scalar eld action is invariant under internal volume preserving diffeomorphisms | because it is the only available counterterm that respects this symmetry and gives nite contributions.11 In the rest of this section, we will focus on the e ect of the counterterm (A.6) only. The analysis of the more general ones will lead to similar results. 10This di erence is analogous to the di erence between the free energy F in the canonical ensemble, with xed charge density , and the thermodynamic potential in the grand canonical ensemble, with xed chemical potential , where F = + V (see, e.g., section 2.3 in [33]). By formally identifying F = S0Lorentz=T = PV and = S0Euclid=T = pV we nd F = + m2qV where q is given in (3.16), (A.5). The quantity q thus has a physical interpretation as the `mass density'. Hence, the Euclidean boundary action allows us to work in the ensemble with xed graviton mass parameter m2. In turn, the Lorentzian boundary action de nes the ensemble with xed `mass density' q. This has already be noted in the case of V (X) = X in [20] and is also analogous to what happens in the presence of a constant magnetic eld [32]. as a power series F (x2=z) = P 11For solids, this also means that there are many more form (z)3=4F (x2=z) with x = Tr[J IJ ], z = det[J IJ ] and J IJ n an(x2=z)n with n including positive and negative integer and non-integer numbers. In particular, this shows that the counterterm that can be used for the background value of the on-shell action is independent from the one for the two-point function. nite counterterms, namely, any function of the . The function F can be expressed By combining (A.6) with the boundary action (A.3) we obtain V L2 2T m2 rhL2 1 r The renormalized action (A.8) when evaluated on the solution (4.6), (4.10), is expected to be nite in the ! 0 limit. We check it explicitly only for the real part of the action since only the solution (4.16) is valid also for the full emblackening factor (3.10) considered here. Upon substituting the solution in the boundary action we obtain: Re W [ ] = S0 V 2T p m2(2 + 1) rh h20 + V m2 2T rh h20 + O(m4; !2) (A.9) The action is nite as expected after the renormalization procedure. The Green's function is then given by G = T =V 2W= h20 and for the real part reads Re G = p m2(2 + 1) rh m2 rh + = P 2 m2 rh + G : (A.10) Only the last term is captured by the minimal renormalization procedure carried out in section 4.2. Indeed, by comparing the last term in the above expression with (4.18) with = 2 we see that they coincide and hence it is what we have de ned as elasticity in section 4.2. The term in the brackets in (A.10) arises from nite contact terms and in the absence of the nite counterterm introduced in (A.6), i.e. for = 0, reduces to the hydrodynamic pressure P 6= p, given in (3.16). For the real part of the Green's function we thus obtain Re G = P + G. As a side remark, let us note that this additional contribution to the low-frequency linear response to a constant shear strain is a characteristic feature for uids with Einstein gravity duals [34]. That this is a sensible result can be seen from the constitutive relation: T uids = " u u + P (g + u u ) r h u i ; where ", P are the equilibrium energy density and pressure, is the shear viscosity, u is the uid three-velocity satisfying u u = h u i denotes the transverse traceless part 1, and r of the tensor r u . This energy-momentum tensor describes the uid response to small uctuations. In particular, under metric perturbations gij = ij + hij (that we identify with strain) the uid remains at rest, i.e. u = (1; 0; 0) and up to rst derivatives: By linear response theory this implies the Green's function T ijuids = Phij h_ij : G TuijidTsij = P + i! + O(!2) : (A.11) (A.12) (A.13) We see that the real part of this Green's function is non-vanishing in the zero frequency limit and is indeed proportional to the hydrodynamic pressure P, as in (A.10) with m2 = 0. Returning back to the discussion of the Green's function (A.10), obtained from the renormalized boundary action, we conclude that there is some ambiguity in the de nition of the real part of the Green's function that depends on the renormalization scheme and the choice of nite counterterms. In particular, without adding any of the counterterms (A.6), (A.7), the Green's function gives a non-zero shear modulus G 6= 0 even for uids. This contradicts to the physical intuition that a uid is naturally de ned by a zero shear modulus. The main outcome of our analysis above is, however, the nding that such that Re G modulus G. in uid and solid massive gravities there are new counterterms that can be used to set HJEP07(216)4 Re G = 0 (and thus G = 0 according to (2.6)) in the uid limit. Technically, these new covariant counterterms given in (A.6) and (A.7) exist only for the massive gravity uids but not for the uids dual to Einstein gravity (without the scalar elds). The existence of the local counterterms that preserve the symmetries of the theory both for solid and uid massive gravities (that is, for solids and uids with momentum relaxation) implies that there is a well de ned renormalization procedure for these theories. In practice, this means that for uids dual to uid massive gravities the real part of the Green's function | the stifness G | can be set to zero consistently. 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Lasma Alberte, Matteo Baggioli, Oriol Pujolàs. Viscosity bound violation in holographic solids and the viscoelastic response, Journal of High Energy Physics, 2016, 74, DOI: 10.1007/JHEP07(2016)074