Hydrodynamics for charge density waves and their holographic duals
HHydrodynamics for charge density waves and their holographic duals
Jay Armas
1, 2, ∗ and Akash Jain † Institute for Theoretical Physics, University of Amsterdam, 1090 GL Amsterdam, The Netherlands Dutch Institute for Emergent Phenomena, The Netherlands Department of Physics & Astronomy, University of Victoria,PO Box 1700 STN CSC, Victoria, BC, V8W 2Y2, Canada.
We formulate a theory of dissipative hydrodynamics with spontaneously broken translations,describing charge density waves in a clean isotropic electronic crystal. We identify a novel lineartransport coefficient, lattice pressure, capturing the effects of background strain and thermal expansionin a crystal. We argue that lattice pressure is a generic feature of systems with spontaneously brokentranslations and must be accounted for while building and interpreting holographic models. We alsoprovide the first calculation of the coefficients of thermal and chemical expansion in a holographicelectronic crystal.
Ever since the discovery of high-temperature supercon-ductivity, cuprates continue to be enigmatic owing totheir intricate phase diagrams exhibiting various inter-twined patterns of symmetry breaking [1, 2]. In particular,the phase diagram of copper oxides contains electronicliquid crystal phases that spontaneously break transla-tions and/or rotations. These include the elastic multi-component charge density wave (CDW) phases [3], smec-tic stripe phases, where the CDW pattern only appearsalong a single direction, or nematic spin density wavephases, where translations are intact but rotations arespontaneously broken. All these phases can potentially ap-pear simultaneously with superconducting phases wherethe global U(1) symmetry is also spontaneously broken(see [1, 2, 4] for a comprehensive review). To note is thefact that CDW ground states are an essential featureacross the phase diagram of copper oxides.Electrons in cuprates, in particular in strange metallicphases, are strongly-correlated. This renders the quasi-particle Fermi liquid crystal theory unreliable for thesesystems, even qualitatively, leaving us with only a handfulof techniques for this plethora of phases [2]. Recently,hydrodynamics has been proposed as a theoretical frame-work for studying aspects of strongly correlated electronsystems [5–7], capable of explaining pinning in the opticalconductivity and predicting the magnitude of viscosity inoptimally doped BSCCO [6]. Another series of efforts hasbeen directed towards holography [8–15], where propertiesof strongly coupled quantum systems are being probedusing classical gravity. In fact, within this setting, hydro-dynamics is directly related to such holographic modelsvia the fluid/gravity correspondence [16].However, all previous treatments of hydrodynamics forcharged lattices (see e.g. [6, 17–19]) have not consideredan essential transport coefficient in their constitutive re-lations, namely the lattice pressure . This coefficient firstappeared in [20] in the context of uncharged viscoelasticmaterials, and models a uniform repulsion/attraction be-tween lattice sites in a material with translational order.However, the thermodynamic variation of lattice pressurecan be understood as carrying information about the ther- mal expansion of the lattice: coefficients of thermal and chemical expansion [21]. As also discussed in [20], latticepressure is generically present in holographic models ofviscoelasticity.The main purpose of this letter is to provide the com-plete hydrodynamic theory for isotropic charged crystals,including contributions from lattice pressure. We derivethe hydrodynamic predictions for linear modes and re-sponse functions. As far as we are aware, the sound anddiffusion modes in the longitudinal sector for charged crys-tals have not been previously worked out in full generalityin the literature. We also comment on the signaturesof lattice pressure in holography using a simple class ofholographic models. Our analysis illustrates that manyprevious works have used an incomplete hydrodynamicframework to interpret holographic results in CDW (e.g.[8–12]), as in viscoelasticity (e.g. [22–26])[27]. We derivean analytic formula for the coefficients of thermal andchemical expansion in these simple models.For clarity of presentation and to effectively focus onthe impact of lattice pressure, we restrict our attention toclean CDW phases. That is, we do not consider the effectsof pinning or momentum dissipation due to interactionswith the ionic lattice, or the presence of topological defectssuch as disclinations and dislocations.
Crystal field theory & lattice pressure. —The fundamen-tal ingredient in an effective theory for crystals is a set ofcrystal fields φ I . They represent the spatial distributionof lattice cores within the crystal [20] and can be under-stood as Goldstones of spontaneously broken translations.The indices I, J, . . . = 1 , . . . , k ≤ d run over the num-ber of broken translations, while µ, ν, . . . = 0 , . . . d runover spacetime indices. Physical distances between thecores are measured by h IJ = g µν e Iµ e Jν , where e Iµ = ∂ µ φ I and g µν is the background metric. The I, J, . . . indicesare raised/lowered using h IJ and h IJ = ( h − ) IJ . Thecrystal also carries a “preferred” reference configuration h IJ = δ IJ /α where α is a constant parametrising the“inverse size” of the crystal. Distortions of the crystalaway from this reference configuration are measured by a r X i v : . [ h e p - t h ] F e b the non-linear strain tensor u IJ = ( h IJ − h IJ ) / .The free energy of a crystal in an isotropic phase, upto quadratic order in small strain expansion, takes theform F = − (cid:82) d d x √− g P with P = P f + P (cid:96) (cid:0) u II + u IJ u IJ (cid:1) − B ( u II ) − G (cid:18) u IJ u IJ − d ( u II ) (cid:19) + O ( u ) . (1)Here P f is the thermodynamic or “fluid” pressure and P (cid:96) is the lattice pressure, while B and G are bulk andshear modulus respectively. Classical elasticity theoryusually describes thermodynamically stable states, requir-ing the free energy to be minimised with respect to strainand setting the linear term P (cid:96) | eq = 0 in equilibrium [28].However, in the context of various holographic models,one finds that P (cid:96) | eq (cid:54) = 0 . As argued in [8], such statescan be relevant for strange metallic regions where quan-tum critical fluctuations of the order parameters do notprovide any stable ordered phase. Furthermore, even instates with P (cid:96) | eq = 0 , thermodynamic derivatives of P (cid:96) are generically nonzero and measure the coefficients ofthermal and chemical expansion (see §6 of [28]) α T = 1 B ∂P (cid:96) ∂T , α µ = 1 B ∂P (cid:96) ∂µ . (2)These derivatives are shown to leave nontrivial signaturesin the hydrodynamic spectrum (see e.g. [20, 29]).
Viscoelastic hydrodynamics. —We are interested in low-energy fluctuations of a charged crystal around thermalequilibrium. In addition to φ I , the dynamics in thisregime is governed by conserved operators: stress tensor T µν and charge/particle current J µ ∇ µ T µν = F νρ J ρ − K ext I e Iν , ∇ µ J µ = 0 . (3)Here A µ and K ext I are background sources coupled to J µ and φ I , while g µν is the source for T µν . F µν = 2 ∂ [ µ A ν ] .Collectively, these determine time-evolution of the hydro-dynamic fields: velocity u µ (with u µ u µ = − ), tempera-ture T , and chemical potential µ . The most generic set ofconstitutive relations for T µν and J µ for an isotropic [30]charged viscoelastic fluid at one-derivative order in Lan-dau frame are given as J µ = qu µ − P Iµ σ qIJ P Jν (cid:16) T ∂ ν µT − E ν (cid:17) − P Iµ γ IJ u ν e Jν ,T µν = ( (cid:15) + P ) u µ u ν + P g µν − r IJ e Iµ e Jν − P I ( µ P Jν ) η IJKL P K ( ρ P Lσ ) ∇ ρ u σ . (4)Here, P is the thermodynamic pressure, (cid:15) , q , and s arethe energy, charge, and entropy densities, and r IJ is theelastic stress tensor. All these quantities are functions of T , µ , and h IJ . They obey the thermodynamic relations: d P = s d T + q d µ + r IJ d h IJ and (cid:15) + P = sT + qµ . We have defined P µν = g µν + u µ u ν , P Iµ = P µν e Iν , E µ = F µν u ν .Furthermore, η IJKL , σ qIK , and γ IK are dissipative trans-port coefficient matrices. In addition, the constitutiverelations have to be supplemented with configurationequations determining the time-evolution of φ I , i.e. σ φIJ u µ ∂ µ φ I + γ (cid:48) JK P Kµ (cid:16) T ∂ µ µT − F µν u ν (cid:17) + ∇ µ (cid:0) r JK e Kµ (cid:1) = K ext J . (5)Here, σ φIK and γ (cid:48) IK are two more matrices of dissipativetransport coefficients. At zeroth order in derivatives, theseequations imply that the crystal fields are constant alongthe fluid flow. Taking φ I = α ( x I − δφ I ) , they turns intotheir more familiar form u t ∂ t δφ I = u I − u i ∂ i δφ I + . . . .Following our discussion in [20], it can be checked thateqs. (4) and (5) above are the most generic set of consti-tutive relations and configuration equations that satisfythe local second law of thermodynamics, ∇ µ S µ ≥ , withthe entropy current S µ = s u µ − µT ( J µ − qu µ ) , providedthat the symmetric parts of η ( IJ ) , ( KL ) , (cid:18) σ qIK γ IK γ (cid:48) IK σ φIK (cid:19) , (6)are positive semidefinite matrices. Linear regime. —We are typically interested in crystalsclose to mechanical equilibrium, where we can expand thehydrodynamic equations in small strain. The pressure P can be expanded as in eq. (1), which determines q , s , (cid:15) , and r IJ up to linear order in strain through ther-modynamics. At one derivative order, we only keep thestrain-independent terms, i.e. σ qIJ = σ q h IJ , σ φIJ = σ φ h IJ , γ IJ = γ h IJ , γ (cid:48) IJ = γ (cid:48) h IJ ,η IJKL = (cid:0) ζ − d η (cid:1) h IJ h KL + 2 η h IK h JL . (7)We can identify η and ζ as shear and bulk viscosities, σ q as charge conductivity, σ φ as crystal diffusivity, while γ , γ ’ as certain mixed conductivities. The second lawconstraints in eq. (6) reduce to η, ζ, σ q , σ φ ≥ , σ q σ φ ≥
14 ( γ + γ (cid:48) ) . (8)Finally, we arrive at the constitutive relations in thesmall-strain regime J µ = (cid:0) q f + q (cid:96) u λλ (cid:1) u µ − σ q P µν (cid:16) T ∂ ν µT − E ν (cid:17) − γP µI u ν e Iν ,T µν = (cid:0) (cid:15) f + (cid:15) (cid:96) u λλ (cid:1) u µ u ν + (cid:0) P f + P (cid:96) u λλ (cid:1) P µν + P (cid:96) h µν − η σ µν − ζP µν ∂ ρ u ρ − Gu µν − (cid:0) B − d G (cid:1) u λλ h µν . (9)Here h µν = h IJ e Iµ e Jν and u µν = u IJ e Iµ e Jν . Similarly theconfiguration equations (5) reduce to σ φ u µ e Iµ − h IJ ∇ µ (cid:0) P (cid:96) e µJ − (cid:0) B − d G (cid:1) u λλ e µJ − Gu µν e Jν (cid:1) + γ (cid:48) P Iµ (cid:16) T ∂ µ µT − E ν u ν (cid:17) = h IJ K ext J . (10)We have defined the fluid thermodynamics d P f = s f d T + q f d µ , (cid:15) f + P f = s f T + q f µ and similarly for the latticepressure d P (cid:96) = s (cid:96) d T + q (cid:96) d µ , (cid:15) (cid:96) + P (cid:96) = s (cid:96) T + q (cid:96) µ . Setting u IJ = 0 , note that the mechanical pressure (cid:104) T xx (cid:105) = P f + P (cid:96) gets contribution from both thermodynamic andlattice pressure. Conformality. —Let us briefly comment on the confor-mal limit of our theory, due to its relevance in holography.Requiring that the stress tensor scales appropriately leadsto the conformality constraints at the non-linear level: (cid:15) = d P − r IJ h IJ and h IJ η IJKL = η IJKL h KL = 0 . Inthe linear regime, they imply (cid:15) f = d ( P f + P (cid:96) ) , (cid:15) (cid:96) = d ( P (cid:96) − B ) , ζ = 0 . (11)Notice that having P (cid:96) or (cid:15) (cid:96) non-zero in the theory (un-like [6]), allows for a non-zero B in a conformal crystal.Furthermore, using the expansion coefficients from eq. (2),we can derive the identity T α T + µα µ = ( d + 1) P (cid:96) B − d. (12)In particular, in a state with no lattice pressure or chemi-cal potential, α T < . This is not surprising, as the sizeof a conformal crystal scales inversely with temperatureat constant µ/T . Linear hydrodynamics and modes. —Consider a chargedcrystal on flat spacetime, g µν = η µν , with trivial externalsources, A µ = µ δ tµ , K ext I = 0 . An equilibrium configu-ration on this background is given by T = T , µ = µ , u µ = δ µt , φ I = α x I . We can expand eqs. (3) and (5)linearly in fields around this configuration to obtain theconstitutive relations of linear hydrodynamics. We recoverthe previously known results of [6, 31] with the identifica-tion ξ = 1 /σ φ , γ = − γ/σ φ , and σ = σ q + γ /σ φ , only ifwe choose γ (cid:48) = − γ and set lattice pressure P (cid:96) and bothits derivatives s (cid:96) , q (cid:96) to zero [32].Solving the linear equations in momentum space, wecan find the complete set of linear modes admitted bythe theory. We find two pairs of sound modes, one eachin transverse and longitudinal sectors, and two diffusivemodes in the longitudinal sector ω = ± v (cid:107) , ⊥ k − i (cid:107) , ⊥ k + . . . , ω = − iD q,φ (cid:107) k + . . . . (13)In the transverse sector, one finds that the modes take asimple form known previously (eg. [6]) v ⊥ = Gχ ππ , Γ ⊥ = w f χ ππ Gσ + ηχ ππ , (14)where χ ππ = (cid:15) f + P f + P (cid:96) is the momentum susceptibilityand w f = (cid:15) f + P f is the enthalpy density. The transversespeed v ⊥ is controlled by the shear modulus G ; in the G =0 case this mode reduces to the well known shear diffusion mode in hydrodynamics. Modes in the longitudinal sectorare considerably more involved. With applications toholography in mind, we present the results for conformalviscoelastic fluids here for simplicity. The general non-conformal results are given in the supplementary material.The longitudinal sound mode simplifies in this limit to v (cid:107) = 1 d + 2 d − d Gχ ππ , Γ (cid:107) = w f (cid:0) d − d G (cid:1) σ φ χ ππ v (cid:107) + 2 d − d ηχ ππ . (15)This is the usual sound mode present in hydrodynamics,but gets modified on a lattice. Longitudinal diffusionmodes, on the other hand, are given by the solutions ofthe quadratic (cid:32) D (cid:107) − w f σ φ d − d G + B − P (cid:96) d χ ππ v (cid:107) ( w f + w (cid:96) ) (cid:33) (cid:18) Ξ D (cid:107) d ( w f + w (cid:96) ) − σ q T (cid:19) = D (cid:107) σ φ (cid:18) s f q (cid:96) − q f s (cid:96) w f + w (cid:96) + γT (cid:19) (cid:18) s f q (cid:96) − q f s (cid:96) w f + w (cid:96) − γ (cid:48) T (cid:19) , (16)where Ξ = ∂s f ∂T ∂q f ∂µ − ∂s f ∂µ ∂q f ∂T and w (cid:96) = (cid:15) (cid:96) + P (cid:96) . Thetwo modes are controlled by the coefficients σ q , σ φ : inthe σ φ → ∞ limit we recover the usual charge diffusionmode D q (cid:107) , but modified on a lattice, while in the σ q → limit we obtain the uncharged crystal diffusion mode D φ (cid:107) characteristic of a lattice [33] (see [18]).We note that, in the conformal case, P (cid:96) appears ex-plicitly only in the diffusion modes (modulo the implicitdependence in χ ππ = (cid:104) T tt (cid:105) + (cid:104) T xx (cid:105) ). Therefore, if wewere to ignore P (cid:96) , for instance as in [6], hydrodynamicswould lead to incorrect predictions for diffusion modes(see [29] for a particular example in holographic massivegravity). For non-conformal theories, however, P (cid:96) infectsall the modes in the longitudinal sector explicitly. Response functions and Onsager’s relations. —We cancompute retarded two-point functions in our model bysolving the hydrodynamic equations (9) and (10) in pres-ence of infinitesimal plain wave sources [34]. Working atzero wave vector, we find in the full non-conformal case G RT xx T xx = χ ππ v (cid:107) − iω (cid:18) ζ + 2 d − d η (cid:19) + (cid:104) T xx (cid:105) ,G RT xy T xy = G − iωη + (cid:104) T xx (cid:105) ,G RJ x J x = q f χ ππ − iω ˜ σ q , G Rφ x φ x = 1 ω χ ππ + ˜ σ φ iωG RJ x φ x = − q f iωχ ππ + ˜ γ, G Rφ x J x = q f iωχ ππ + ˜ γ (cid:48) , (17)where we have defined the dissipative response coefficients ˜ σ q = σ q + 1 σ φ (cid:18) q f P (cid:96) χ ππ − γ (cid:19) (cid:18) q f P (cid:96) χ ππ + γ (cid:48) (cid:19) , ˜ σ φ = w f σ φ χ ππ , ˜ γ = w f σ φ (cid:18) γχ ππ − q f P (cid:96) χ ππ (cid:19) , ˜ γ (cid:48) = w f σ φ (cid:18) γ (cid:48) χ ππ + q f P (cid:96) χ ππ (cid:19) . All the remaining response functions are either zero orrelated to these by isotropy. For P (cid:96) = 0 and γ = − γ (cid:48) ,these results reduce to the expressions reported in [31],up to contact terms.If the system enjoys Θ =
T (time-reversal) or
Θ =
PT(spacetime parity) invariance, Onsager’s relations require G RJ x φ x = − Θ G Rφ x J x , setting γ = − γ (cid:48) . This is the caseassumed in [31]. In the case of Θ =
CPT invariance, how-ever, G RJ x φ x = Θ G Rφ x J x and we instead have γ (cid:48) | µ →− µ = γ (note that q f flips sign under CPT). Holography. —As an application of our hydrodynamictheory, we propose a simple holographic model for cleanCDW phases following the discussion in [20, 35]. We alsocompute the coefficients of thermal and chemical expan-sion in this model. Specialising to four bulk dimensions,the model is described by Einstein-Maxwell gravity in thebulk coupled to two scalars S bulk = 12 (cid:90) d x √− G (cid:18) R + 6 − F − V ( X ) (cid:19) . (18)Here G ab is the bulk metric with a, b, ... being the bulkindices, and F ab = 2 ∂ [ a A b ] is the field strength associatedwith the gauge field A a . Here V ( X ) = X + . . . is anarbitrary potential in X = δ IJ G ab ∂ a Φ I ∂ b Φ J for a setof scalar fields Φ I . To describe a thermal state at theboundary, we consider charged black brane solutions ofthe action (18) of the form d s = 1 r f ( r ) d r + r (cid:0) − f ( r )d t + δ IJ d x I d x J (cid:1) , A µ = µ (cid:16) − r r (cid:17) δ tµ , Φ I = α x I , (19)where r is the horizon radius, r → ∞ is the conformalboundary, µ the chemical potential, and α an arbitraryconstant. The blackening factor is given by f ( r ) = 1 − r r − r µ ( r − r )4 r − α r (cid:90) rr d r (cid:48) V ( X ( r (cid:48) )) X ( r (cid:48) ) , (20)where X ( r ) = α /r . The profile of the scalars breaksthe translational invariance in the boundary theory. Thismodel, with V ( X ) = X , has been considered in thecontext of momentum dissipation in [36] with explicitlybroken translations. However, contrary to [36], we intro-duce alternative boundary conditions for Φ I so as to de-scribe spontaneously broken translations at the boundary.We will also allow for an arbitrary renormalisation scaleparameter M in the boundary conditions breaking theconformal symmetry of the model. The holographic renor-malisation procedure along with the choice of boundarycounter-terms is detailed in the supplementary material.Identifying the onshell action as free-energy for theboundary theory, we can read out the thermodynamicpressure in equilibrium P f = r (cid:16) µ r + 2 U − V (cid:17) − α M . (21) Here U ( X ) = − X / (cid:82) dXX − / V ( X ) and X = X ( r ) ,along with V = V ( X ) , V (cid:48) = V (cid:48) ( X ) , and U = U ( X ) .We can also extract the stress tensor, charge current, andscalar expectation values using the boundary behaviourof the solution and read out (cid:15) f = r (cid:18) µ r − U (cid:19) + α M , q f = µr ,s f = 2 πr , P (cid:96) = r V − U ) + α M , (22)along with φ I = α x I . With temperature defined as T = r f (cid:48) ( r ) / (4 π ) , one can check that the expected ther-modynamic relations given below eq. (10) are satisfied.We can easily obtain the bulk modulus by deforming oursolution from α → α + δα , leading to a uniform strain u IJ = − αδ IJ δα , and using eq. (1). We find B = 3 r V − U ) + X V (cid:48) r (cid:16) − V + µ r (cid:17) − V + 2 X V (cid:48) + µ r + α M . (23)One can check that these expressions satisfy the conformalidentities (11) in the absence of M , confirming that M characterises RG flow away from the conformal fixedpoint. Expressions for G and the dissipative transportcoefficients have to be obtained numerically.Finally, we can use the expressions for B and P (cid:96) to readout the expansion coefficients in eq. (2). In particular, ina conformal model with M = 0 around a state with zerolattice pressure P (cid:96) | eq = 0 , we simply find α T = − s f s f T + q f µ , α µ = − q f s f T + q f µ , (24)irrespective of the model dependent potential V ( X ) . Theyfollow the conformal identity (12). We find both theexpansion coefficients to be negative for our holographicconformal crystal. This behaviour is altered for M (cid:54) = 0 .However, negative thermal expansion is not unusual insolid materials [37].A lattice configuration is thermodynamically stable if itminimises the free energy: ( δP f /δα ) T,µ = − /α P (cid:96) | eq = 0 for some α (cid:54) = 0 . Equivalently, (cid:104) T xx (cid:105) must be equatedto P f [38]. Notice that P (cid:96) (cid:54) = 0 in a generic equilibriumconfiguration in (22). This is also the case for similarholographic models with spontaneously broken transla-tions [22–25]. In fact, simple monomial models with V ( X ) = X N and M = 0 , do not admit any thermo-dynamically stable configurations. Fortunately, one canconsider polynomial models, such as V ( X ) = X + λX or the “higher-derivative model” of [9], that do admitthermodynamically stable configurations, in our case α = r ( r − M ) / (2 λ ) . Though the lattice pressure P (cid:96) iszero in such configurations, its thermodynamic derivatives s (cid:96) , q (cid:96) , (cid:15) (cid:96) are still generically nonzero and have to be takeninto account in the hydrodynamic spectrum. This wasverified for the uncharged case in [29]. Previous holo-graphic models for CDW have not been taking the latticepressure into account, leading to the misinterpretation ofsome of their results. Outlook. —We have provided a complete formulation ofhydrodynamics for clean isotropic CDW phases, takinginto account the new transport coefficient P (cid:96) . We findthat P (cid:96) non-trivially modifies the longitudinal sector oflinear fluctuations. Besides being crucial for correctlyinterpreting the holographic results, including those of[8–12], lattice pressure is also highly relevant for realcondensed matter systems. It can describe parts of thephase diagram for which there are no thermodynamicallystable ordered phases and also accounts for the effectsof thermal expansion of the crystal. We have obtainedan analytic expression for the coefficients of thermal andchemical expansion is a class of simple holographic modelsusing lattice pressure.It will be interesting to further include the effects ofexplicit translation symmetry breaking (momentum dissi-pation and pinning) as well as incorporate spontaneousbreaking of U(1) global symmetry. This would provide amore robust theory for realistic scenarios. In this context,it would be relevant to revisit some of the results andpredictions of [5–7, 39] with our understanding of latticepressure, potentially including weak/strong backgroundmagnetic fields. In particular, it is an open questionwhether the existing data can constrain the magnitudeof P (cid:96) or its gradients for specific materials. It would alsobe interesting to work out an analogous formulation forsmectic and nematic charged liquid crystal phases.In the context of holography, we focused on equilib-rium thermal states dual to planar black hole geometries.However, this work provides the necessary linear trans-port theory for interpreting near-equilibrium states. Bycomputing the quasinormal modes and using the Kuboformulae reported here, one can extract all first ordertransport coefficients and check whether the modes re-ported here reproduce holographic results. 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In this appendix we provide details on the completeset of longitudinal dispersion relations for non-conformalcharged isotropic crystals, as well as the holographic renor-malisation procedure employed in the letter.
Longitudinal modes. —In the main text, we presentedlongitudinal modes in case of a conformal charged vis-coelastic fluid for simplicity. Here we provide the genericnon-conformal expressions. Let us first change the ther-modynamic variables from ( T, µ ) to ( (cid:15) f , q f ) via ∂T∂(cid:15) f = 1 T Ξ ∂q f ∂µ , ∂T∂q f = − T Ξ ∂(cid:15) f ∂µ ,∂µ∂(cid:15) f = − T Ξ ∂q f ∂T , ∂µ∂q f = 1 T Ξ ∂(cid:15) f ∂T , (A.1)where Ξ = ∂s f ∂T ∂q f ∂µ − ∂s f ∂µ ∂q f ∂T . The longitudinal soundvelocity is given as v (cid:107) = ( w f + w (cid:96) ) ∂P m ∂(cid:15) f + ( q f + q (cid:96) ) ∂P m ∂q f χ ππ + B + 2 d − d G − P (cid:96) χ ππ , (A.2)whereas the attenuation is Γ (cid:107) = (cid:16) ∂P m ∂q f (cid:17) σ q + σ φ (cid:16) F + ∂P m ∂q f γ (cid:17) (cid:16) F − ∂P m ∂q f γ (cid:48) (cid:17) v (cid:107) χ ππ + ζ + 2 d − d ηχ ππ . (A.3)Here P m = P f + P (cid:96) is the mechanical pressure. For clarityof presentation, we have defined F = w f (cid:18) v (cid:107) − ∂P m ∂(cid:15) f (cid:19) − q f ∂P m ∂q f , F = T ∂ ( µ/T ) ∂q f . (A.4)The quadratic governing the diffusion modes, on the otherhand, is given by D (cid:107) T F (cid:32) D (cid:107) − w f σ φ d − d G + B − P (cid:96) χ ππ v (cid:107) Ξ T F (cid:33) − σ q T (cid:34)(cid:32) ( w f + w (cid:96) ) χ ππ Ξ T F + 2 d − d G + B − P (cid:96) χ ππ (cid:33) D (cid:107) v (cid:107) − w f σ φ d − d G + B − P (cid:96) χ ππ v (cid:107) Ξ T F (cid:35) = D (cid:107) σ φ v (cid:107) (cid:20) ( w f + w (cid:96) ) χ ππ T Ξ F (cid:18) s f q (cid:96) − q f s (cid:96) w f + w (cid:96) + γT (cid:19) (cid:18) s f q (cid:96) − q f s (cid:96) w f + w (cid:96) − γ (cid:48) T (cid:19) + B + 2 d − d G − P (cid:96) χ ππ (cid:18) ∂P f /∂q f F − γT (cid:19) (cid:18) ∂P f /∂q f F + γ (cid:48) T (cid:19)(cid:35) . (A.5) Holographic renormalisation. —In order for the action (18) to be finite on the class of black brane solutionsin (19), we need an additional Gibbons-Hawking-Yorkcounter-term at the boundary along with a boundarypotential for the scalars. To wit, S counter = (cid:90) r = r c d x √− γ (cid:0) K − V ( ¯ X ) (cid:1) , (A.6)where we have defined the induced metric γ µν at thelocation of the cutoff surface r = r c . We have assumed theboundary to be flat and avoided any curvature dependentterms. Here ¯ X = δ IJ γ µν ∂ µ Φ I ∂ ν Φ J and K = G ab D a n b is the mean extrinsic curvature, where n a is an outwardpointing normal vector to the surface and D a the covariantderivative compatible with the bulk metric G ab . For theonshell action not to have any divergences, the potentialmust take the form ¯ V ( ¯ X ) = 2 (cid:18) − (cid:113) − U ( ¯ X ) (cid:19) − (cid:88) n M n r c ( r c ¯ X ) n . (A.7)In (A.7) we have introduced M n , which are arbitrary cut-off dependent renormalisation scale parameters taken tobe regular in the limit r c → ∞ . Their presence spoils theconformal symmetry of the holographic model by impos-ing non-conformal boundary conditions. Different valuesof M n describe different physical theories, or differentpoints in the RG flow of the same physical theory. In theletter, we have only turned on M = M for simplicity.Assuming the bulk potential falling as V ( X ) ∼ X ∼ r − near the boundary, we can read out the boundaryvalue of the fields g µν = lim r c →∞ r c γ µν , A µ = lim r c →∞ A µ , φ I = lim r c →∞ Φ I . (A.8)For potentials falling off more quickly, the qualitativebehaviour of the scalars is different; see [41] for more de-tails. Fields in eq. (A.8) serve as sources for the respectiveoperators obtained by varying the total onshell action δS onshellbulk+counter = (cid:90) r = r c d x √− g (cid:18) T µν δg µν + J µ δA µ − Π I δφ I (cid:19) , (A.9)where T µν = lim r c →∞ r c (cid:18) Kγ µν − K µν − γ µν + ¯ V ( ¯ X ) γ µν − δ IJ ¯ V (cid:48) ( ¯ X ) γ µρ γ νσ ∂ ρ Φ I ∂ σ Φ J (cid:19) ,J µ = lim r c →∞ r c F ua n a , Π I = lim r c →∞ r c δ IJ (cid:18) V (cid:48) ( X ) n a ∂ a Φ J + 1 √− γ ∂ µ (cid:0) √− γ γ µν ¯ V (cid:48) ( ¯ X ) ∂ ν Φ J (cid:1) (cid:19) . (A.10) We have arrived at the holographic formula for T µν and J µ . In this picture, however, the fields φ I serve as sourcesleading to explicit breaking of translations. In order todescribe spontaneous symmetry breaking, we deform thetheory with a surface action S alternative = (cid:82) d x √− g Π I φ I implementing alternative quantisation and switching theroles of Π I and φ I . In this picture, K ext I ≡ Π I are thebackground sources coupled to the dynamical fields φ I .Finally, the boundary conditions imposed on our holo-graphic model for spontaneously broken translations are g µν = η µν , A µ = µδ tµ , and Π I = 0= 0