Diffusion and universal relaxation of holographic phonons

JHEP10(2019)068 Published for SISSA by Springer Received:July 5, 2019 Revised:September 10, 2019 Accepted:September 15, 2019 Published:October 7, 2019 Diffusion and universal relaxation of holographic phonons Andrea Amoretti,a,b Daniel Are´an,cBlaise Gout´erauxdand Daniele Mussoe aDipartimento di Fisica, Universit`a di Genova and I.N.F.N., Sezione di Genova, via Dodecaneso 33, I-16146, Genova, Italy bPhysique Th´eorique et Math´ematique and International Solvay Institutes, Universit´e Libre de Bruxelles, C.P. 231, 1050 Brussels, Belgium cInstituto de F´ısica Te´orica UAM/CSIC, Calle Nicol´as Cabrera 13-15, Cantoblanco, 28049 Madrid, Spain dCPHT, CNRS, Ecole polytechnique, IP Paris, F-91128 Palaiseau, France eDepartamento de F´ısica de Part´ıculas, Universidade de Santiago de Compostela, and Instituto Galego de F´ısica de Altas Enerx´ıas (IGFAE), R´ua de Xoaqu´ın D´ıaz de R´abago, s/n Campus Vida, 15782 Santiago de Compostela, Spain E-mail: [email protected],[email protected], [email protected],[email protected] Abstract: In phases where translations are spontaneously broken, new gapless degrees of freedom appear in the low energy spectrum (the phonons). At long wavelengths, they couple to small fluctuations of the conserved densities of the system. This mixing is captured by new diffusive transport coefficients, as well as qualitatively different collective modes, such as shear sound modes. We use Gauge/Gravity duality to model such phases and analytically compute the corresponding diffusivities in terms of data of the dual background black hole solution. In holographic quantum critical low temperature phases, we show that these diffusivities are governed by universal relaxation of the phonons into the heat current when the dynamical critical exponent z > 2. Finally, we compute the spectrum of transverse collective modes and show that their dispersion relation matches the dispersion relation of the shear sound modes of the hydrodynamic theory of crystalline solids. Keywords: Holography and condensed matter physics (AdS/CMT), Effective Field Theories, Global Symmetries, Space-Time Symmetries ArXiv ePrint: 1904.11445 Open Access,c The Authors. Article funded by SCOAP3.https://doi.org/10.1007/JHEP10(2019)068 JHEP10(2019)068 Contents 1 Introduction 1 2 Review of Wigner crystal hydrodynamics 3 3 Holographic model 5 3.1 Setup 5 3.2 Holographic renormalization 7 3.3 AC boundary correlators 9 3.4 Shear modulus and shear viscosity 9 4 Diffusivity matrix from horizon data 11 5 Saturation of entropy bound and universal relaxation 13 6 Transverse collective excitations at nonzero wavevector 17 7 Outlook 20 A AC boundary correlators at zero density 22 B Numerics 24 B.1 Black hole geometries 24 B.2 AC fluctuations 25 B.2.1 Computing the correlators 26 B.3 Transverse fluctuations 27 1 Introduction One of the most celebrated results originating from Gauge/Gravity duality [1–4] is the computation of the shear viscosity of the N= 4 super Yang-Mills plasma at infinite N(N is the rank of the gauge group) and ’t Hooft coupling in terms of the entropy density of a dual black hole in anti de Sitter spacetime, η=s/4π(in natural units) [5]. This was conjectured to place a lower bound on the ratio η/s &1/4πfor strongly-coupled phases of matter [6]. Recalling that the entropy density is proportional to the area of the bulk black hole, this relation is universal in the sense that the complicated dependence on the boundary sources is encapsulated as a simple combination of data defined at the black hole horizon [7]. In a relativistic plasma, which is described at long wavelengths by relativistic hydrodynamics (see [8] for a review of relativistic hydrodynamics), the shear viscosity controls the diffusion of transverse momentum D⊥=η/(+p), where and pare the energy density and pressure. – 1 – JHEP10(2019)068 At nonzero density, relativistic hydrodynamics predicts the existence of another, incoherent diffusion constant σ0, associated to processes without any momentum drag. This diffusivity can also be computed in terms of horizon data [9–11], as well as the thermal diffusivity that governs energy diffusion [12]. Taken together, these results point at an interesting link between universal low energy transport properties (diffusivities) and the dynamics close to black hole horizons. These results have been extended to less symmetric cases describing inhomogeneous states, such as the so-called holographic lattices that break translations explicitly with a periodic potential [13–15]. At long distances, charge and energy diffuse in these inhomogeneous systems [16,17]. The diffusivities are proportional to the dc thermoelectric conductivities of the system via Einstein relations, which relates them to horizon data under very general assumptions [18]. More recent developments have considered the effect of breaking translations spontaneously in one or more spatial directions, which can be implemented either in a homogeneous1[19–22] or an inhomogeneous way [23–28]. In either case, the incoherent conductivity and associated diffusivity of the boundary theory can be expressed in terms of horizon data for thermodynamically stable phases [21,22,29,30]. For thermodynamically unstable phases, the incoherent conductivity also features an integral over the whole spacetime of some combination of background fields. The effective theory [31,32] capturing the low energy dynamics of such states with spontaneous translation symmetry breaking contains more than the two characteristic diffusivities just mentioned, due to the presence of additional gapless degrees of freedom (the Goldstones of spontaneous translation symmetry breaking or in other words, the phonons) and their mixing with charge and energy fluctuations. The theory of Wigner crystal (WC) hydrodynamics is succinctly reviewed in section 2and the diffusivities we are interested in are presented there. Then, in section 3, we give some details of our holographic setup. A first objective of this work is to extend previous analyses and show that all of these diffusivities can be expressed in terms of the background black hole solution, by a combination of data on the horizon and on the rest of the spacetime. This is done in section 4, and our main new results are equations (4.18) and (4.19). Positivity of entropy production in WC hydrodynamics places a bound on a combination of the diffusivities, see equation (2.9). The holographic diffusivities obey this bound, as we discuss in section 5. Interestingly, this bound can be saturated at low temperatures. This obtains when the phonons relax into the heat current and leads to simple relations between the diffusivities. We expect this universal relaxation channel to be at play in generic states of matter at finite temperature. We give a criterion for the saturation of this bound as a function of the values of the scaling exponents characterizing the infra-red fixed point of the system and the irrelevant deformations away from it. Universal relaxation of phonons into a hydrodynamic operator has also been recently discussed in the context of the melting of the field-induced Wigner solid in [33]. 1The broken translation generator then combines with an internal symmetry such that a diagonal combination is preserved. – 2 – JHEP10(2019)068 In section 6, we conclude with an analysis of the spectrum of the system at nonzero wavevector by computing the longest-lived pair of transverse quasinormal modes (QNMs) of the dual black hole. We show that they obey the expected dispersion relation from WC hydrodynamics. At low wavevector and intermediate temperatures, there is a pair of shear sound modes, characteristic of the long wavelength dynamics of solids. However, either at very low or very high temperatures, these modes collide on the imaginary frequency axis and become pseudo-diffusive, with a purely imaginary gap controlled by a ratio of thermodynamic data and diffusivities. Both collisions occur in the hydrodynamic regime and the dispersion relation of the modes is well-captured by the hydrodynamic dispersion relation, see equation (6.1). These results were originally presented in [34], but have been moved and expanded upon in the present work for clarity. We give a number of technical digressions as well as details of our numeric scheme in some appendices. 2 Review of Wigner crystal hydrodynamics We first recap the main features of two-dimensional isotropic Wigner crystal hydrodynamics. More details can be found in [31,32]. As translations are broken spontaneously along both spatial directions, the usual conserved densities (energy, charge, momentum) need to be coupled to two Goldstone modes, ϕi,i=x, y. The free energy is supplemented by terms capturing the effect of the Goldstones: f=1 2K|q·ϕq|2+1 2Gq2|ϕq|2.(2.1) Kand Gare the bulk and shear moduli, and characterize the stiffness of phase fluctuations around the ordered state. It is convenient to parameterize the Goldstones by their longitudinal and transverse contributions, λk=∇·ϕand λ⊥=∇×ϕ. The corresponding sources sk,⊥are defined by requiring that λk,⊥=δf/δsk,⊥. To leading order in gradients and keeping only linear terms, the Goldstones obey the following ‘Josephson’ relations ∂tλk=∇·v+γ1∇2µ+γ2∇2T+ξk K+G∇2sk+. . . , ∂tλ⊥=∇×v+ξ⊥ G∇2s⊥+. . . , (2.2) where vis the velocity, µthe chemical potential, and γ1,2and ξk,⊥are diffusive transport coefficients. These equations are supplemented by current, heat and momentum conservation equations (energy can be traded for entropy to linear order): ∂tρ+∇·j= 0 , ∂ts+∇·(jq/T)=0, ∂πi+∇jTji = 0 ,(2.3) – 3 – JHEP10(2019)068 together with the constitutive relations j=ρv−σo∇µ−αo∇T−γ1∇sk+... , jq/T =sv−αo∇µ−(¯κo/T)∇T−γ2∇sk+... , Tij =δij (p+(K+G)∇·ϕ)+2Gh∇(iϕj)−δij∇·ϕi−η2∇(ivj)−δij∇·v+... (2.4) The underlying conformal symmetry of the holographic setup implies that the stress-energy tensor is traceless, which sets the bulk viscosity to zero. The hydrodynamic retarded Green’s functions at nonzero frequency and wavevector are derived by following the Kadanoff-Martin procedure [8,35]: GR AB(ω, q) = MAC h(iω −M)−1iCD χDB (2.5) where the vevs A, B = (δρ, δs, πk, λk, π⊥, λ⊥) and the corresponding sources are (δµ,δT, vk,sk,v⊥,s⊥). Mis the matrix MAB =           σoq2αoq2iρq γ1q20 0 αoq2¯κo Tq2isq γ2q20 0 iρq isq ηq2iq 0 0 γ1q2γ2q2iq ξk K+Gq20 0 0 0 0 0 ηq2iq 0 0 0 0 iq ξ⊥ Gq2           .(2.6) Relativistic symmetry of the holographic setup enforces that the momentum and energy current densities must be equal π=je, which places constraints on the transport coefficients: αo=−µ Tσo,¯κo=µ2 Tσo, γ2=−µ Tγ1.(2.7) Observe that this also means that the heat current jq≡je−µj =π−µj. Finally, the susceptibility matrix is2 χAB =          χρρ χρs 0 0 0 0 χρs χss 0 0 0 0 0 0 χππ 0 0 0 0 0 0 1 K+G0 0 0 0 0 0 χππ 0 0 0 0 0 0 1 G          .(2.8) Positivity of entropy production can be ensured by requiring all the eigenvalues of the matrix Mto be positive [32], which leads to the following constraints: η≥0, σo≥0, γ2 1≤σo ξk K+G.(2.9) 2We have set to zero off-diagonal terms χρλkand χsλk. As pointed out in appendix A of [32], they affect the dispersion relation of modes in the longitudinal sector, but are not important here. – 4 – JHEP10(2019)068 Using (2.5) and identities between the Green’s functions stemming from (2.3), we obtain in the q→0 limit GR TxyTxy =G−iωη, (2.10) GR jj =ρ2 χππ −iωσo, GR jπk=ρ, GR jϕk=−γ1−iρ χππ ω,(2.11) GR πkϕk=GR ϕkπk=i ω, GR ϕkϕk=1 χππ ω2−ξk K+G i ω, GR ϕ⊥ϕ⊥=1 χππ ω2−ξ⊥ G i ω.(2.12) We defined e.g. GR ϕiϕj=qiqj q4GR λkλk+δij −qiqj q2GR λ⊥λ⊥ q2,(2.13) and q2GR ϕkϕk=GR λkλk, q2GR ϕ⊥ϕk=GR λ⊥λ⊥.(2.14) In this q= 0 limit, for an isotropic crystal, GR λkλk=GR λ⊥λ⊥, since there should be no distinction between the longitudinal and tranverse phonons. This leads to the constraint ξk K+G=ξ⊥ G≡X . (2.15) 3 Holographic model 3.1 Setup We consider the holographic model S=Zd4x√−g"R−1 2∂φ2−V(φ)−1 4Z(φ)F2−1 2 2 X I=1 Y(φ)∂ψ2 I#,(3.1) with the scalar couplings behaving near the AdS boundary, i.e. in the small φlimit, as Vuv(φ) = −6−φ2+O(φ3), Zuv(φ) = 1 + O(φ), Yuv(φ) = φ2+O(φ3).(3.2) The model (3.1) enjoys a global shift symmetry ψI7→ ψI+cI. In this work, we will be interested in states that break translations homogeneously [36,37],3with ψI=kxI, xI={x, y}.(3.3) This Ansatz breaks the shift symmetry as well as spacetime translations to a diagonal U(1). As a consequence, the background metric, scalar φand gauge field only depend on the holographic radial coordinate, ds2=−D(r)dt2+B(r)dr2+C(r)(dx2+dy2), φ =φ(r), A =A(r)dt . (3.4) t, x, y are the time and space coordinates of the dual field theory. Given (3.2) and (3.3), the dual boundary theory is deformed by two complex scalar operators ΦI≃φeiψI. These deformations break translations through the spatial dependence of the phases ψI. 3See [38] for a recent field theoretic investigation of such states. – 5 – JHEP10(2019)068 As explained in [19,21,22], whether translations are broken explicitly or spontaneously in the boundary theory depends on the asymptotic behaviour of the scalar φnear the AdS boundary. For our choice of scalar potential, φdecays towards the boundary as φ(r→0) = λr +φ(v)r2+O(r3).(3.5) In the usual quantization scheme, λis the source of the operator dual to the bulk field φ and φ(v)is related to its vev. If λ= 0, the breaking is spontaneous. If λ6= 0, it is explicit. In this work we will only consider the spontaneous case. The condensation of the order parameter itself (i.e. the phase transition between the normal and ordered phase) is not captured by the model. This is typically mediated by an instability towards a spatially modulated, inhomogeneous phase which minimizes the free energy, [25,26,28,39].4Instead, the holographic model (3.1) with zero scalar source λ= 0 directly describes the low energy dynamics of the phonons in the ordered phase, as we shall demonstrate in the remainder of this work.5As we show below, the fluctuations of the bulk fields ψIare dual to the phonons. The bulk global symmetry should then be understood as encoding the shift symmetry of the NG bosons, not as a symmetry of the fundamental UV theory.6 Finite temperature, finite density states are modeled by charged black holes in the bulk, which implies the existence of a regular black hole horizon at r=rh. In the rest of this work, we use a subscript hto denote quantities evaluated at r=rh. The temperature Tand the entropy density sare given by: s= 4πC(rh), T =1 4πs−B0(r)D0(r) B(r)2r=rh ,(3.6) with the following near-horizon expansion ds2=−4πT(rh−r)dt2+dr2 4πT(rh−r)+s 4π(dx2+dy2) + . . . , At=Ah(rh−r) + . . . , φ =φh+. . . . (3.7) We will also be interested in the low temperature behaviour Tµ(where µis the chemical potential) of these translation-breaking black holes. More precisely, we would like to study the interplay between spontaneous translation symmetry breaking and quantum criticality. To do so, we will assume that the scalar φhas a runaway behaviour as T→0 and r→+∞(or equivalently that the horizon value of the scalar diverges at low temperature), and that the scalar couplings behave for large φas V(φ→ ∞) = V0e−δφ , Z(φ→ ∞) = Z0eγφ , YIR =Y(φ→ ∞)eνφ .(3.8) 4This can also be realized in models breaking translations homogeneously [21,22,40]. 5Minimizing the free energy would lead to k= 0, i.e. no breaking of translations at all, [21,22,29]. This can be remedied by turning on higher-derivative terms in the action with small couplings [21,22]. 6A very similar model was considered in [41] to study holographically spin density waves. – 6 – JHEP10(2019)068 The near-extremal, near-horizon geometry of the black hole is then described by a Lifshitz, hyperscaling-violating solution ds2=ξθ−f(ξ)dt2 ξ2z+L2dξ2 ξ2f(ξ)+d~x2 ξ2, f(ξ) = 1 −ξ ξh2+z−θ , A=Aoξθ−2−zdt , ψI=kδIixi, φ =κlog ξ . (3.9) ξis a radial coordinate valid close to rh, simultaneously taking the small temperature limit Tµ.ξhis the location of a Killing event horizon, with the associated Hawking temperature T∼ξ−1/z hand Hawking-Bekenstein entropy s∼ξθ−2 h. Combining both formulæ the entropy density scales as s∼T2−θ z. More details of these solutions can be found in [61] or more recently [11]. Here, we simply recall that in general the couplings appearing in (3.8) are related to the scaling exponents characterizing the solution as κδ =θ , κγ = 4 −θ−2∆Ao, κν = 2∆k−2.(3.10) The exponents ∆Ao,k ≤0 are the scaling dimension of sources of irrelevant operators breaking particle/hole symmetry or translations of the IR T= 0 critical solution. For concreteness, in our numerical calculations we work with V(φ) = −6 cosh(φ/√3) , Z(φ) = exp(−φ/√3) , Y (φ) = (1 −exp φ)2.(3.11) This implies that the critical solution (3.9) is characterized by θ→ −∞,z→+∞,−θ/z = 1 and ∆Ao,k = 0. As a consequence, it has a vanishing entropy density s∼T. Our results can easily be generalized to other scalings. The numerical results presented in this paper are for k/µ = 0.1 and λ= 0. 3.2 Holographic renormalization Holographic renormalization for the spontaneous case was explained in [21], to which we refer for details. The salient features are the following. The UV expansion of the fields ψIare modified compared to that of ordinary massless scalars due to their coupling to the scalar φ. If φis not sourced λ= 0, then close to the AdS boundary r→0 ψI=ψI,(−1) r+ψI,(0) +O(r) = 1 r2ψI,(−1)r+ψI,(0)r2+O(r3),(3.12) where we have used that the operator dual to φhas dimension ∆ = 2 due to (3.11). This makes clear that the operators dual to the ψIhave the same scaling dimension as the operator dual to φand that the whole scalar sector of deformations of the UV CFT is better understood as complex scalar operators dual to bulk complex scalar fields ΦI≃φeiψIfor small φ(ie close to the AdS boundary). φand the ψIare respectively the modulus and phase of the complex scalar, and inherit its scaling dimension. See [21] for details. Going back to our background Ansatz (3.3), we see that indeed this corresponds to spontaneous translation breaking, since the source term ψ(−1) is not turned on. These powers can be generalized easily to the case ∆ 6= 2 and spatial dimension d6= 2. – 7 – JHEP10(2019)068 Near the boundary, the metric and gauge field behave in Fefferman-Graham gauge as D(r) = 1 r21 + d(3)r3+O(r4), C(r) = 1 r21−d(3) 2r3+O(r4), B(r) = 1 r2, A(r) = µ−ρr +O(r3), (3.13) where subleading coefficients are fixed in terms of, ρ,φ(v)and d(3). This allows to read off the one-point correlation functions of the dual field theory, after computing the on-shell action renormalized by suitable counterterms [42]. We find [21] hTtti==−3d(3) ,hTxxi=hTyyi=−3 2d(3) hJti=ρ , hOφi=φ(v). (3.14) Another useful relation is sT =−µρ −9d(3) 2−k2Zrh 0 dr √BDY , (3.15) which results from evaluating the radially conserved bulk expression "ρA(r) + C2(r) pB(r)D(r)D(r) C(r)0 −k2Zrh r d˜r√BDY #0 = 0 ,(3.16) both at the horizon and at the boundary. This is the Noether charge associated to the bulk time-like Killing vector [30,43]. By evaluating the on-shell action for the background solution, we also find that p=sT +µρ −. From the analysis above, χππ was obtained exactly in k[21,22]: χππ =sT +µρ +k2IY, IY=Zrh 0 dr √BDY . (3.17) In this paper, we will only be interested in the transverse sector of fluctuations δgx t(r, t, x) = δhx t(r)e−iωt+iqy,δgy x(r, t, x) = δhy x(r)e−iωt+iqy,δgx r(r, t, x) = δhx r(r)e−iωt+iqy, δax(r, t, x) = δax(r)e−iωt+iqy and δψx(r, t, x) = δψx(r)e−iωt+iqy. We have used the homogeneity of our Ansatz to expand the fluctuations in plane waves. In the radial gauge δgx r= 0, and setting q= 0 for now, their boundary expansions are δax(r) = δa(0) +δa(1)r+. . . , δhx t(r) = δh1,(0) +δh1,(3)r3+. . . , δhy x(r) = δh2,(0) +δh2,(3)r3+. . . , δψx(r) = 1 rδψ(−1) +δψ(0)r+. . . , (3.18) where (δa(0), δh1,(0), δh2,(0), δψ(−1)) are sources and (δa(1), δh1,(3), δh2,(3), δψ(0)) are vevs. Then, the renormalized action at quadratic order in the fluctuations reads: S(2) ren =Zdωδa(0)δa(1)−ρδa(0)δh1,(0)−3 2δh1,(0)δh1,(3) +3 2d(3)(δh1,(0))2+(φ(v))2δψ(−1)δψ(0) −3 2δh2,(0)δh2,(3)+3 2d(3)(δh2,(0))2. (3.19) – 8 – JHEP10(2019)068 10-40.001 0.010 0.100 1 T μ 1 2 3 4 5 6 |γ1|χqπ/(μσo) 1.×10-42.×10-45.×10-40.001 0.002 1.00 1.05 1.10 1.15 1.20 1.25 10-40.001 0.010 0.100 1 T μ 100 105 108 X(χqπ)2/(μ2σo) 1.×10-42.×10-45.×10-40.001 0.002 1.0 1.2 1.4 1.6 Figure 3. At low temperatures, the entropy bound is saturated and the diffusivities γ1,Xare controlled by the thermal diffusivity through (5.8). where we have also used (2.7). For the θ=−∞,z= +∞,θ/z =−1 state we have considered in detail in this work, these expressions capture the correct low temperature behaviour of the diffusivities (see figure 3), and lead to the saturation of the entropy bound (5.1). At higher temperatures, other, non-universal relaxation channels will open. For the θ=−∞,z= +∞,θ/z =−1 holographic state, these non-hydrodynamic operators also contribute to the Hamiltonian (5.6) but give subleading corrections as T→0.8 8See [60] for another holographic example where momentum relaxation due to explicit translation symmetry breaking is controlled by the heat current. – 15 – JHEP10(2019)068 It is interesting to assess how generic the universal relaxation mechanism we have just described is. To do so, we consider the more general family of IR end points (3.9). It is clear that if there is no running scalar φand the IR is AdS2×R2, the entropy bound is saturated since all factors in (5.2) scale like T0at low temperatures, except of course for the explicit T2factor. When there is a running scalar, there are four classes of IR end points, characterized by the presence or absence of certain dangerously irrelevant deformations. These are discussed in some detail in [21,22,61] and more recently in [11], so we will be brief here. There are two potentially dangerously irrelevant deformations, depending on whether the bulk fields dual to the conserved current and dual to the phonons support the deep IR solution or whether they decay faster than other fields in the bulk. These deformations can be usefully characterized by the scaling dimension of the coupling, which we denote by ∆Ao and ∆k, respectively. In order for the coupling to source an irrelevant/marginal deformation, ∆Ao,k ≤0. The four classes differ by whether both deformations are marginal, or only one of them is, or both of them are irrelevant. In the case we have studied numerically and for which the bound is saturated, both deformations are marginal ∆Ao= ∆k= 0. Their expressions in terms of the action couplings have been given in (3.10) Evaluating the right-hand side of (5.2) on the solutions (3.9) together with the expressions (3.8), (3.10) for the IR couplings and recalling that s∼T(d−θ)/z, the entropy bound is saturated provided 2 z(z−2+∆Ao+ ∆k)≥0.(5.9) We find that this condition is met provided z > 2 and θ < 2 and 2 −z < ∆Ao,k ≤0.(5.10) In particular, this implies the bound is never saturated for IR end points with relativistic symmetry z= 1, which only happens when both deformations are irrelevant. If only one deformation is irrelevant, then the dimension of the coupling is bounded from below. If both are marginal ∆Ao,k = 0, then the bound is saturated if the dynamical exponent zis large enough, z > 2. In the parameter space spanned by (5.10), we observe that (2 + z−θ)>−∆Ao,k. Recalling that sT ∼T(2+z−θ)/z at low temperature, this suggests that sT terms can be neglected at low temperatures in (4.17)–(4.19) against other terms that have a non-vanishing zero temperature limit, such as µρ or k2IY.9Doing so leads to σ0=k2IY2 (µρ +k2IY)2Zh+4πρ2 sk2Yh,(5.11) γ1=−µk2IY (µρ +k2IY)2Zh+4πρ2 sk2Yh,(5.12) X=µ2 (µρ +k2IY)2Zh+4πρ2 k2sYh.(5.13) 9The irrelevant deformation ∆Ao,k also source deviations from zero temperature. The inequality (2 + z−θ)>−∆Ao,k implies that those sourced by sT type terms are less important at low temperatures. – 16 – JHEP10(2019)068 We have verified that these expressions are consistent with our numerics at low temperature. It is straightforward to check that the expressions (5.11)–(5.13) saturate the entropy bound (5.1). The prefactors are all temperature independent at low T, so all the low temperature dependence comes from the common factor in parenthesis. Since this common factor is the same for all three quantities, the saturation of the bound comes from the cancelation between the prefactors in the limit of low temperature, rather than from the vanishing of each individual term in (5.1). On the other hand, requiring (2 + z−θ)>−∆Ao,k does not imply that the entropy bound is saturated. It would be interesting to further investigate these cases and to understand why the universal mechanism (5.6) is not at play there. 6 Transverse collective excitations at nonzero wavevector WC hydrodynamics also predicts the existence of a pair of low energy modes in the transverse channel. They appear as zeroes of the denominator of the hydrodynamic retarded Green’s functions and their dispersion relation is:10 ωshear =1 2 −iq2ξ⊥+η χππ ±qs4G χππ −q2η χππ −ξ⊥2 .(6.1) At sufficiently low wavector q, these modes are gapless and propagate transverse sound waves: ωshear =±sG χππ q−i 2q2ξ⊥+η χππ +O(q3).(6.2) On the other hand, for larger values of q > 2√Gχππ/(η−ξ⊥χππ), the modes collide on the imaginary axis and are no longer propagating. Instead, they obey a pseudo-diffusive dispersion relation: ω−=−iξ⊥q2−iG η−ξ⊥χππ +O(q−1), ω+=−iη χππ q2+iG η−ξ⊥χππ +O(q−1).(6.3) Observe that while the sign of the q0damping term is positive for one of the modes, this does not lead to an instability since qis large. This collision is captured by the hydrodynamic dispersion relation, since the inequality on qis true if Gis small enough. We expect Gto be small also close to a phase transition between a translationally ordered and a translationally disordered phase. We compare the exact location of the quasinormal modes determined numerically to the hydrodynamic approximation in figure 4as a function of q/µ, and find they agree very well, including at very low temperatures. We typically observe deviations when q&T, which suggests that temperature is the cut-off of the effective description in terms of WC hydrodynamics, whether Tis large or small compared to the chemical potential µ. This gives strong evidence that the low energy excitations of our system are well described 10This dispersion relation is strictly speaking valid to order q2(6.2), but keeping the expression (6.1) exact in qderived from first order in gradients WC hydrodynamics leads to a better match to the numerics. – 17 – JHEP10(2019)068 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 -1. -0.75 -0.5 -0.25 0. 0.25 0.5 0.75 1. -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0. q/μ Re(ω)/μ Im(ω)/μ T/μ=1 0.02 0.03 0.04 0.05 0.06 -0.0004 -0.0003 -0.0002 -0.0001 0. -0.00175 -0.0015 -0.00125 -0.001 -0.00075 -0.0005 -0.00025 0. q/μ Re(ω)/μ Im(ω)/μ T/μ=0.1 0.005 0.010 0.015 0.020 -0.0005 -0.0004 -0.0003 -0.0002 -0.0001 0. -0.001 -0.0008 -0.0006 -0.0004 -0.0002 0. q/μ Re(ω)/μ Im(ω)/μ T/μ=0.0065 Figure 4. Comparison of the exact location (dots, numerics) and hydrodynamic approximation (6.1) (solid lines) of the real (blue, left axis) and imaginary (magenta, right axis) part of the transverse hydrodynamic QNM at T/µ = 1 (top), T/µ = 0.1 (center) and T/µ = 0.0065 (bottom) vs q/µ. In the top panel, the modes are purely imaginary, but would acquire a real part at smaller q(not displayed). by WC hydrodynamics at all temperatures, and further confirms that our holographic model should be interpreted as describing the low energy dynamics of (pseudo)phonons coupled to conserved currents. We emphasize that the same dispersion relation is valid at both low and high temperature. Similar results on the low-temperature hydrodynamics of translation-invariant black holes have been reported in past literature [62–64]. – 18 – JHEP10(2019)068 0.005 0.010 0.050 0.100 0.500 1 T μ 5.×10-5 1.×10-4 5.×10-4 0.001 0.005 -Im(ω)/μ q/μ=0.02 0.01 0.05 0.10 0.50 T μ 5. ×10-6 1. ×10-5 5. ×10-5 1. ×10-4 5. ×10-4 10-3 Re(ω)/μ Figure 5. Comparison of the exact location (numerics, solid blue) and hydrodynamic approximation (6.1) (dashed, red) of the real (inset) and imaginary part of the transverse hydrodynamic QNM at q/µ = 0.02 vs T/µ. At intermediate temperatures, they follow very well the dispersion relation of shear sound waves predicted by WC hydrodynamics. At both very high and very low T, the modes collide and become purely imaginary and pseudo-diffusive. Keeping T/µ fixed, they modes acquire a nonzero real part as qis lowered, see figure 4. In figure 5, we display the QNMs location as a function of T/µ and compare it to (6.1). Increasing temperature, the shear sound modes collide on the imaginary axis. At these temperatures, the shear modulus of the system becomes very small (see figure 1), and this is the underlying reason behind the high temperature pole collision (see figure 5). The same loss of shear rigidity would occur in a real crystal right before the phase transition to a liquid phase: the solid would no longer support gapless, transverse sound modes. The two high temperature branches are well-approximated by (6.3), with ω+giving the longestlived mode. In other words, as the spontaneous component of the system is becoming very small, the system resembles more and more a fluid (consistently with the gap of (6.3) being proportional to G, which is going to zero as T→+∞). At low temperatures, the QNMs also undergo a collision, which is now driven by the fact that ξ⊥∼1/T is becoming very large. Once again, the longest-lived mode is wellapproximated by ω+in (6.3). The gap is inversely proportional to ξ⊥and decreases with T, see figure 6. The system resembles more and more a fluid since the phonons relax faster (they have a larger diffusivity). Similar collisions between high temperature, gapped, pseudo-diffusive modes leading to low temperature propagating modes have been reported in previous holographic literature [65–68]. There are important differences. In our case, translations are broken spontaneously, not explicitly. As a consequence, at the longest distances q→0, the modes are always gapless, sound modes. The collision we report is ‘coherent’ or in other words, – 19 – JHEP10(2019)068 |G/(η-ξ⊥χππ )| G/(ξ⊥χππ ) 10-40.001 0.010 0.100 1 T μ 10-6 10-4 10-2 1. ×10-42. ×10-45. ×10-40.001 0.002 0.0868 0.0870 0.0872 0.0874 0.0876 0.0878 Figure 6. At low temperatures, the gap at large wavevector is dominated by the phonon diffusivity ξ⊥rather than by the shear viscosity η. can be described by a hydrodynamic effective theory (WC hydrodynamics). In contrast, in [65,67,68] the breaking is always explicit and the effective theory is momentum-relaxed fluid hydrodynamics, and breaks down below a certain temperature where momentum is no longer approximately long-lived. In [66], translations are broken spontaneously, but the transverse propagating modes are gapped. Gapless propagating modes are found for a nonzero density of defects, and it would be interesting if they became gapped at shorter distances, while remaining within the regime of validity of the hydrodynamic description. Recently, [69] have also reported similar collisions in a holographic massive gravity model. 7 Outlook The main results we have obtained are as follows. Firstly, we have shown that all of the diffusivities that characterize the low energy dynamics of a holographic state breaking translations spontaneously can be computed in terms of a combination of horizon and UV-sensitive data. We have focussed on a specific model breaking translations homogeneously, and it would be natural to extend this computation to inhomogeneous, more realistic setups. Moreover, in [21,22,29,30], the incoherent conductivity was computed for thermodynamically stable phases and solely depends on horizon data.11 We have also shown how the low temperature behaviour of these diffusivities and the saturation of an entropy bound stemming from Wigner crystal hydrodynamics is explained 11This statement is strictly true for the conductivity associated to the incoherent current defined as jinc =χπqj−χπj jq. Then, σinc =χ2 ππσo. For other choices of normalization, some overall factors of χππ appear in the incoherent conductivity. – 20 – JHEP10(2019)068 by the universal relaxation of the phonons into the heat current. We have studied whether this depends on the infra-red end point controlling the system at zero temperature. We found that the entropy bound is not saturated in the presence of sufficiently irrelevant deformations, or for values of the dynamical exponent 1 ≤z≤2. Finally, we have studied the spectrum of transverse collective excitations and discovered that for sufficiently low wavectors the pair of longest-lived modes are well captured by the dispersion relation following from WC hydrodynamics, including at low temperatures where the system is governed by the IR end point consistent with previous low temperature holographic studies [62–64]. We observed deviations as q&T. In particular, this means that the hydrodynamic description extends outside of the regime qTto q≃Twhere gradients are of the same order as the thermal scale. This ‘unreasonable effectiveness’ of hydrodynamics could perhaps be understood along the lines of the recent work [70]. We have only carried out explicit numerical calculations for the special case z=−θ= +∞. As we have observed, the saturation of the entropy bound will not occur for phases with z≤2, in particular for z= 1 in the presence of irrelevant deformations. It was shown in [10,11] that in these cases new long-lived modes emerge at low temperature and lead to a breakdown of the hydrodynamic description at earlier times than that set by temperature. A natural future direction would be to study these cases with spontaneous translation symmetry breaking. If translations are explicitly broken, the phonons obtain a mass mand are damped, with a phase relaxation rate Ω (this phase relaxation rate is of a different microscopic origin than the phase relaxation rate sourced by the proliferation of topological defects [71] but appears in the effective theory in exactly the same way). WC hydrodynamics in the presence of relaxation also predicts a bound on the relaxation parameters [32] γ2 1≤min σoX, σoΩ Gm2.(7.1) In [34], we showed that turning on a small source for the scalar φin our holographic model led to a phase whose low energy effective theory was relaxed WC hydrodynamics, until very low temperatures. The bound (7.1) is also saturating as temperature decreases, suggesting that the pseudo-phonons also relax universally into the heat current. It would be interesting to understand how this depends on the infra-red end point, the relaxation mechanism and the details of how translations are broken in the holographic model. By now, many other models of pinned translational order are available [40,47,55,69,72–74]. In the presence of a magnetic field, the longitudinal and transverse phonons hybridize into a magnetophonon and a magnetoplasmon [75]. For strong magnetic fields and in the presence of disorder, the magnetophonon is weakly gapped and a hydrodynamic theory can be formulated [33], which appears to fit well some of the data on two-dimensional electron systems. Universal relaxation of the magnetophonon into the electric current may dominate (compared to dislocation-mediated melting) in more disordered samples, and it would be interesting to study this holographically (see [76] for a recent study of plasmonic response at zero field in the holographic model studied here). – 21 – JHEP10(2019)068 Acknowledgments We would like to thank Riccardo Argurio, Marco Fazzi, Saso Grozdanov, Javier Mas, Alfonso Ramallo and Javier Tarr´ıo for stimulating and insightful discussions. BG would especially like to thank Luca Delacr´etaz, Sean Hartnoll and Anna Karlsson for numerous insightful discussions on Wigner crystal hydrodynamics. BG has been partially supported during this work by the Marie Curie International Outgoing Fellowship nr 624054 within the 7th European Community Framework Programme FP7/2007-2013 and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreements No 341222 and No 758759). DM has been funded by the Spanish grants FPA2014-52218-P and FPA2017-84436-P by Xunta de Galicia GRC2013- 024, by FEDER and by the Mar´ıa de Maeztu Unit of Excellence MDM-2016-0692. D.A. is supported by the ‘Atracci´on del Talento’ programme (Comunidad de Madrid) under grant 2017-T1/TIC-5258 and by Severo Ochoa Programme grant SEV-2016-0597 and FPA2015- 65480-P (MINECO/FEDER). D.A. and D.M. thank the FRont Of pro-Galician Scientists for unconditional support. A AC boundary correlators at zero density First, we need to figure out how to extract the retarded Green’s functions (2.12) from the asymptotic data. The relevant part of the renormalized on-shell action at quadratic order in the fluctuations (3.19) (keeping only h1,(0) and δψ(−1) nonzero) is S(2) ren =Zdω −3 2δh1,(0)δh1,(3) +3 2d(3)(δh1,(0))2+ (φ(v))2δψ(−1)δψ(0)δa(0),δh2,(0)=0 (A.1) so that the retarded Green’s function are GR TtxTtx (ω, q = 0) = 3δh1,(3) δh1,(0) , GR Ttxδψx(ω, q = 0) = 3δh1,(3) δψ(−1) , GR δψxTtx (ω, q = 0) = (φ(v))2δψ(0) δh1,(0) , GR δψxδψx(ω, q = 0) = (φ(v))2δψ(0) δψ(−1) . (A.2) We start by explicitly identifying the source of the boundary phonon. This is done by plugging the UV expansions (3.18) for the bulk fluctuations and solving the equations of motion, which give the following relation δh1,(3) =i ω k(φ(v))2δψ(−1) 3(A.3) from which we immediately deduce GR Ttxδψx(ω, q = 0) = 3δh1,(3) δψ(−1) =k(φ(v))2i ω.(A.4) In order for this expression to match the off-diagonal Green’s function GR πϕ in (2.12), we see that we should identify the phonon source as δs =k(φ(v))2δψ(−1). – 22 – JHEP10(2019)068 To continue, we now need to solve the equations of motion away from the boundary. It is helpful to define the variable [11] Πx≡ −hx t0+iωhx r (D/C)0,(A.5) which obeys the decoupled equation of motion d dr "rD B 1 Y C C2 √BD D C02 Π0 x#+ω2rB D 1 Y C2 Cd/2+1 √BD D C0!2 Πx= 0.(A.6) Then we can expand Πxin ω Πx(r) = sT +k2Zrh r √BDY 1−r rh−iω/(4πT)πo+iω 4πT π1(r) + . . . (A.7) and solve order by order, imposing ingoing boundary conditions. At leading order, πois just a constant. The solution for π1which is regular at the horizon is π1(r) = iπoZr rh dr1   1 r1−rh + (4πT)3Ch Yh √BCY √DsT +k2Rrh r1√BDY 2  .(A.8) To go through these manipulations, it is helpful to keep in mind the relation (3.16): C2 √BD D C0 =−sT −k2Zrh r √BDY. (A.9) The boundary expansion of the Πxvariable is Πx=i ωk(φ(v))2δψ(−1)+iω 2k(φ(v))2δψ(−1)r2+1 3k(φ(v))2kδh1,(0) +iωδψ(0)r3+... (A.10) To establish this, it is necessary to push the boundary expansions to higher order in r than written in (3.18). Also, we note that the combination that appears at O(r3) is gauge invariant. So from (A.2) the retarded Green’s function for δψxis GR δψxδψx(ω, q = 0) = (φ(v))2 2ω2lim r→0 Π(3) x(r) Πx(r).(A.11) Putting together (A.7) and (A.8), we obtain after expanding at low frequency GR δψxδψx(ω, q = 0) = k(φ(v))2"1 9 2d(3)ω2+4πsT2 k2Yh9 2d(3)2 i ω#(A.12) The momentum static susceptibility was computed in [21] χππ =−9 2d(3) =sT +k2Z0 rh √BDY . (A.13) – 23 – JHEP10(2019)068 Observe that our boundary expansions imply that the integral over the whole bulk does not diverge at the boundary. Plugging into (A.12) and comparing with the phonon retarded Green’s function GR ϕϕ in (2.12) leads to identifying the phonon as ϕx=δψ(0) k(φ(v))2(A.14) and the ratio of the transverse crystal diffusivity to the shear modulus as ξ⊥ G=4πsT2 k2YhsT +k2Rrh 0√BDY 2.(A.15) This result matches the zero density limit of (4.19). B Numerics In this appendix we describe how we construct the numerical solutions of the holographic model (3.1) relevant for the analysis presented in the main text. B.1 Black hole geometries The action (3.1) admits black hole solutions asymptotic to AdS which realize holographically the (pseudo-)spontaneous breaking of translations. In order to find those geometries we take the following Ansatz for the metric and matter fields ds2=1 r2−u(r)dt2+1 u(r)dr2+c(r)(dx2+dy2),(B.1) A=At(r)dt , φ =φ(r), ψI=kxI, xI={x, y}.(B.2) The resulting equations of motion can be reduced to a system of four ordinary differential equations (three are second order and one is first order). For the potentials in eq. (3.11) it is easy to find the following UV asymptotic solution: φ(r) = λ r +v r2+λk2+7 36λ2r3+O(r4),(B.3a) At(r) = µ−ρ r −λ ρ 2√3r2−ρ 36 5λ2+ 4√3vr3+O(r4),(B.3b) u(r) = 1 −λ2 4r2+u3r3+O(r4),(B.3c) c(r) = 1 −λ2 4r2−1 3λ v r3+O(r4),(B.3d) where higher order coefficients are functions of λ,v,ρ, and u3. This is not the most general asymptotic solution, but that with AdS asymptotics. In the IR one can find the following near-horizon solution φ(r) = φh+O(rh−r), At(r) = Ah,1(rh−r) + O((rh−r)2),(B.4a) u(r) = uh,1(rh−r) + O((rh−r)3), c(r) = ch+ch,1(rh−r) + O((rh−r)2),(B.4b) – 24 – JHEP10(2019)068 [55] A. Donos and C. 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