On the hydrodynamic description of holographic viscoelastic models

Citation for this paper:

Ammon, M., Baggioli, M., Gray, S., Grieninger, S., & Jain, A. (2020). On the hydrodynamic

description of holographic viscoelastic models. Physics Letters B, 808, 1-8.

https://doi.org/10.1016/j.physletb.2020.135691.

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On the hydrodynamic description of holographic viscoelastic models

Martin Ammon, Matteo Baggioli, Seán Gray, Sebastian Grieninger, Akash Jain

July 2020

© 2020 Martin Ammon et al. This is an open access article distributed under the terms of

the Creative Commons Attribution License. https://creativecommons.org/licenses/by/4.0/

This article was originally published at:

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

On

the

hydrodynamic

description

of

holographic

viscoelastic

models

Martin Ammon

a

,

Matteo Baggioli

b

,

Seán Gray

a

,

∗

,

Sebastian Grieninger

a

,

c

,

Akash Jain

d

a_{Theoretisch-PhysikalischesInstitut,Friedrich-Schiller-UniversitätJena,Max-Wien-Platz1,D-07743Jena,Germany}

b_{InstitutodeFisicaTeoricaUAM/CSIC,c/NicolasCabrera13-15,UniversidadAutonomadeMadrid,Cantoblanco,28049Madrid,Spain} c_{DepartmentofPhysics,UniversityofWashington,Seattle,WA98195-1560,USA}

d_{DepartmentofPhysics&Astronomy,UniversityofVictoria,POBox1700STNCSC,Victoria,BC,V8W 2Y2,Canada}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received11May2020

Receivedinrevisedform30July2020 Accepted6August2020

Availableonline11August2020 Editor:N.Lambert

We show that the correct dualhydrodynamic description of homogeneousholographic models with spontaneously broken translations must include the so-called “strain pressure” – a novel transport coefficient proposed recently. Taking this new ingredient into account, we investigate the near-equilibriumdynamicsofalargeclassofholographicmodelsandfaithfullyreproduceallthehydrodynamic modespresent inthe quasinormalmodespectrum.Moreover,whilestrainpressure ischaracteristicof equilibriumconfigurationswhichdonotminimisethefreeenergy,weargueandshowthatitalsoaffects modelswithnobackgroundstrain,throughitstemperaturederivatives.Insummary,weprovideafirst completematchingbetweentheholographicmodels withspontaneouslybrokentranslationsand their effectivehydrodynamicdescription.

©2020TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

Contents

1. Introduction . . . 1

2. Viscoelastichydrodynamics . . . 2

2.1. Constitutiverelations . . . 2

2.2. Linearmodes . . . 2

2.3. Unstrainedequilibriumconfigurations . . . 3

3. Holographicframework . . . 3

3.1. Holographicmassivegravity . . . 3

3.2. Strainedholographicmodels . . . 4

3.3. Unstrainedholographicmodels . . . 5

4. Conclusions . . . 6

Declarationofcompetinginterest . . . 6

Acknowledgements . . . 6

Appendix A. Holographicrenormalisation . . . 6

References . . . 7

1. Introduction

Models with broken translational invariance have attracted a great deal of interest in the holographic community in recent years,especiallyinrelationtotheirhydrodynamic description[1–

*

Correspondingauthor.

E-mailaddresses:martin.ammon@uni-jena.de(M. Ammon),

matteo.baggioli@uam.es(M. Baggioli),sean.gray@uni-jena.de(S. Gray),

sebastian.grieninger@gmail.com(S. Grieninger),ajain@uvic.ca(A. Jain).

9] and their possiblerelevance forstrange metal phenomenology [10–13].Particularemphasishasbeengiventotheso-called homo-geneous models, e.g. massive gravity [14–17]; Q-lattices [18,19]; andhelicallattices[20,21],duetotheirappealingsimplicity.

Despite the sustained activity in the field, there still remain a number of open questions. For instance, it has been unclear what hydrodynamic framework appropriately describesthe near-equilibriumdynamics offield theories dual tothese models.The authorsof[3] wrotedownagenerictheoryoflinearised hydrody-namicswithbrokentranslations(see also[22,2]),whichhasbeen https://doi.org/10.1016/j.physletb.2020.135691

0370-2693/©2020TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

2 M. Ammon et al. / Physics Letters B 808 (2020) 135691 widelyusedinholography [23,24,8,9,19,11,10,25,26].However,the

first indication that something was amiss came from [9], in the formof a disagreementbetween theholographic resultsandthe hydrodynamic predictions of [3] regardingthe longitudinal diffu-sion mode. Similarly, [6] found inconsistencies between the hy-drodynamictheory of [3] and thequasinormalperturbations ofa bulk modelwithexplicitly brokentranslations.Considering these results,itbecameclearthat the understandingofhydrodynamics waslacking somefundamental detailsneededinordertocapture theholographicresults.

Recently, a newfullynon-linear hydrodynamic theory for vis-coelasticity was proposed in [5]. At the linear level, this formu-lationdiffers from previous formulationsof viscoelastic hydrody-namicsduetothe presenceofan additionaltransportcoefficient,

P

, called the lattice- or strainpressure. Physically,

P

is the dif-ference between the thermodynamic and mechanical pressures; intuitively,

P

can be understoodasan additional contributionto themechanicalpressureasaresultofworkingaroundauniformly strainedequilibriumstate.Inthissensethestrainpressureis anal-ogous to the magnetisationpressure which appears in the pres-enceof an external magnetic field [27,28].

P

is non-zero inthe holographicmodels mentionedabove and,asweillustrate inthis paper,isfundamentalinordertomatchtheholographicresultsto hydrodynamics.

Itismisleading,however,todismissthisnewcoefficientpurely asanartifactofbackgroundstrain.

P

certainlyvanishesinan un-strainedequilibriumstatethat minimisesthe freeenergy(as dis-cussedin[29]),butaswewillillustrateinthispaper,its tempera-turedependencestillcarriesvitalphysicalinformationandaffects various modes through

P



= ∂

T

P

. For instance, in scale invari-anttheories thisleads toa non-zero bulkmodulus B

= −

T

P



/

2. Hence, the preceding hydrodynamic frameworks would still fall shortin capturingthe near-equilibrium behaviour of holographic modelswithoutbackgroundstrain.

In thispaper, we consider the mostgeneral isotropic Lorentz violating massivegravity theoriesintwo spatial dimensions[17]. The dual field theories correspond to isotropic, conformal, and generically strained viscoelastic systems withspontaneously bro-ken translations.Bycarefully studyingthe quasinormalmodesin these systems,we illustrate that they are perfectly described by thehydrodynamicframeworkof [5].Wealsobuildanew thermo-dynamicallystableholographicmodelwithzerobackgroundstrain. Using thisunstrained model, we show that the effectsof

P

 are stillpresentwhen

P

vanishesinequilibrium.

2. Viscoelastichydrodynamics

Letus briefly review the formulation of viscoelastic hydrody-namicsfrom [5];wewill startwiththegeneric constitutive rela-tions for an isotropic viscoelastic fluid, including strain pressure, andwrite downthelinearmodespredictedby thehydrodynamic framework.Wefurtherextendtheworkof [5] bydiscussing ther-modynamically stableconfigurations with zerostrain pressure in equilibrium,butwithnonzerotemperaturederivatives,anddrawa comparisonwiththepreviouslyknownresultsof [3].Wework in

d

=

2 spatialdimensionsforsimplicity.

2.1. Constitutiverelations

Thefundamental ingredientsinthetheory arethefluid veloc-ityuμ, temperature T , andtranslationGoldstone bosons



I.We define eI

μ

= ∂

μ



I, which is used to further define hI J

=

eIμeJμ, eIμ

=

h−I J1e

J

μ,hμν

=

h−I J1eIμe

J

ν ,andthestraintensoruμν

=

12

(

h− 1

I J

−

δ

I J

/

α

2

)

eI_μeν ,J forsomeconstant

α

.Theconstitutiverelationsofan

isotropic neutralviscoelasticsystem,written inasmallstrain ex-pansion,aregivenas [5]

Tμν

=





+

p

+

T

P

uλλ



uμuν

+



p

+

P

uλλ



η

μν

+

P

hμν

−

η σ

μν

− ζ

Pμν

∂

ρuρ

−

2G uμν

− (

B

−

G

)

uλλhμν

,

(1a) with the thermodynamic identities dp

=

sdT ,



=

T s

−

p and P μν

=

η

μν

₌

_{uμuν . Here p and}

_P

_{are the thermodynamic and}

strain pressures respectively;



and s are energy and entropy densities; and G and B are the shear and bulk moduli.

σ

μν

₌

2P ρ(_{μ P ν})σ

_∂

ρ uσ

−

P μν

∂

ρ uρ is thefluidsheartensor,while

η

and

ζ

areshearandbulkviscosities.Allthecoefficientsappearinghere are functions of T ; prime denotes derivative with respect to T

forfixed

α

.Dynamicalevolutionof uμ andT is governedby the energy-momentum conservation equation

∂

μ Tμν

=

0; these are

accompanied by the configuration (Josephson) equations for the Goldstones uμeI_μ

=

h I J

σ

∂

μ



P

eμ_J

− (

B

−

G

)

uλλeμJ

−

2GuμνeJν



,

(1b)

where

σ

isadissipativecoefficientcharacteristicofspontaneously brokentranslations.

2.2. Linearmodes

The

P

dependent terms in (1) have important consequences forthelowenergydispersionrelationofthehydrodynamicmodes. In summary, around an equilibrium state with uμ

= δ

_tμ, T

=

T0,

and



I

=

α

xI, we find two pairs of sound modes, one each in longitudinal and transverse sectors, anda diffusion mode in the longitudinalsector

ω

= ±

v_,⊥k

−

i 2



,⊥k

2

_{+ . . . ,}

_ω

_{= −}

_{i D}

k2

+ . . . .

(2) The sound velocities v_,⊥,attenuation constants



_,⊥, and diffu-sionconstantD_aregivenas

v2_⊥

=

G

χ

π π

,

v2_

=

(

s

+

P



₎

2 s

χ

π π

+

B

+

G

−

P

χ

π π

,



_⊥

=

η

χ

π π

+

G

σ

s2T2

χ

2 π π

,

D_

=

s 2

σ

s B

+

G

−

P

χ

π πv2_

,



_

=

η

+ ζ

χ

π π

+

T2s2v2_

σ χ

π π



1

−

s

+

P

 T sv2_



2

.

(3)

Here

χ

π π

=



+

p

+

P

isthemomentumsusceptibility1;all

func-tions are evaluated at T

=

T0. Note that the pair of transverse

sound modesare not presentwhen G

=

0; instead,they are re-placed by asingle sheardiffusionmode

ω

= −

i D_⊥k2 with D_⊥

=

η

/

χ

π π .2 We can obtain formulasfor various coefficients

appear-ing in (1) in terms of the free-energy density



, stress-tensor one-point function, and(uptocontact-terms) retardedtwo point functions



= 

Ttt

 ,

p

= − ,

P

= 

Txx

 +  ,

χ

π πv2_

=

lim ω→0klim→0Re G R Txx_Txx

,

1 _{Theobservationthatχ}

π π =+p ingenericholographicmodelsof

viscoelastic-ity(i.e.thatthethermodynamicandmechanicalpressuresarenotnecessarilyequal) wasfirstmadein [24].

2 _{ThelimitG→0 issubtleandmustbeperformedatthelevelofthetransverse}

sectordispersionrelations,ω2_sT+₁₊ωT s iσ

 

G

=

χ

π πv2⊥

=

_ωlim_→0klim_→0Re GRTxy_Txy

,

η

= −

lim ω→0klim→0 1

ω

Im G R Txy_Txy

,

(



+

p

)

2

σ χ

2 π π

=

lim ω→0klim→0

ω

Im G R x_x

.

(4)

Thebulkmodulus B canbeobtainedindirectlyusingthev2_ Kubo formula. Inthe equationsin thefirst lineabove, therelation be-tweenthestrain pressure

P

,thermodynamic pressure p,andthe mechanicalpressure



Txx

_

_,ismanifest.

Forourapplicationtoholographyweshall,inthefollowing,be interestedin scale-invariant viscoelastic fluids, wherein T μμ

=

0. Thisleadstoasetofidentities



=

2

(

p

+

P

) ,

T

P



=

3

P

−

2 B

,

ζ

=

0

.

(5)

Takingderivativeofthefirstrelation,wealsofindthespecificheat

cv

=

T s

=

2

(

s

+

P



)

. Using the above equations, we can derive a relation betweensound velocities, i.e. v2



=

1

/

2

+

v2⊥ [30]. For

scale-invarianttheories v_⊥ and



_⊥ stay thesameasin(3), how-evertheexpressionsforthelongitudinalsectorsimplifyto



_

=

η

χ

π π

+

T2s2G2

σ χ

3 π πv2_

,

D_

=

T s 2

_/

_σ

s

+

P

 B

+

G

−

P

χ

π π

+

2G

.

(6)

Interestingly, apart from the implicit dependence in

χ

π π , in a

scale-invariant viscoelasticfluid only D_ dependsexplicitly on

P

and

P

, whichexplainsthe discrepancyreportedin thediffusion modein [9].Notethatusing(5),thebulkmoduluscanbe rewrit-tenasB

= (

3

P −

T

P



)/

2.Consequently,ascale-invariant viscoelas-ticsystemonlyrespondstobulkstressif

P,

P



=

0.3

2.3.Unstrainedequilibriumconfigurations

Let us now extend the analysis of [5] by considering equi-librium states without background strain, i.e. states where the equilibrium strain pressure is zero,

P(

T0

)

=

0. In such a setup

the temperature derivative of the strain pressure need not van-ish,hence

P



(

T0

)

=

0.4Nevertheless,themomentumsusceptibility

reducesto a familiar expression

χ

π π

=



+

p. For generic

scale-non-invarianttheories,wearriveatthemodes

v2_⊥

=

G T s

,

v 2 

=

(

s

+

P



₎

2 T ss

+

B

+

G T s

,



⊥

=

η

T s

+

G

σ

,

D_

=

s

σ

T s B

+

G v2_

,





=

η

+ ζ

T s

+

T sv2_

σ



1

−

s

+

P

 T sv2_



2

.

(7) Inthescale-invariantlimit,thelongitudinalmodesfurthersimplify tov2 

=

1

/

2

+

v2⊥ alongwith



_

=

η

T s

+

2G2

/

σ

T s

+

2G

,

D

=

T s2

/

σ

s

+

P

 B

+

G T s

+

2G

.

(8)

The appearance of

P

 in the denominator of D_ suggests that thetemperature dependenceof strain pressure still plays an im-portant role in an unstrained equilibrium configuration. Indeed,

P

 _{iscrucialforthermodynamicallystableholographicmodels,as}

3 _{Nevertheless,thecompressibilityβ≡ (−1/V)∂T}

xx/∂V isfiniteeveninthe

ab-senceofthestrainpressure,andinthescale-invariantcaseitisgivenbyβ−1= (3/4)[31].Itispossibletoshowthatintermsofthecompressibilitythe longitu-dinalspeedcanbewrittenasv2

= (β−1+G)/χπ π[9,23].

4 _{Wewillreturntothispointinfurtherdetailbelow.}

we illustrate below. In the absence of scale invariance, the ef-fects of

P

 willalso contaminate the expression forthe longitu-dinalsoundmode. Other signaturesofstrain pressure ina scale-invariant viscoelastic system include non-canonical specific heat,

cv

=

2

(

s

+

P



)

=

2s,andnonzerobulkmodulus B

= −

T

P



/

2

=

0.5 Comparingourresultsto [3],wefindthat(7) matchesthe ex-pressions derived using the hydrodynamic framework of [3] for neutralrelativisticviscoelasticfluidsonlyifwefurtherset

P



=

0. As a consequence,the resultsof [3] do not apply to general un-strained viscoelasticsystems withnonzero

P

.Notably, the anal-ysis of [3] can be extended to include certain couplings in the free-energydensitythathavebeenswitchedofftherein(see(A.7) of [3]). We find that such couplings are indeed important and precisely capturethe effects ofnonzero

P

 via themapping b

=

−

P



_/

_s_.

3. Holographicframework 3.1. Holographicmassivegravity

We will consider a simple holographic model with

(

d

+

2

)

-dimensional Einstein-AdS gravity coupled to d copies of Stückel-bergscalars

φ

I Sbulk

=



dd+2x

√

−

g

R 2

+

d

(

d

+

1

)

2



2

−

m 2_V

₍

_I

I J

₎

,

(9) where

I

I J

₌

_gab

_∂

a

φ

I

∂

b

φ

Jisthekineticmatrix;



istheAdS-radius, whichwesettooneinthefollowing;andm isaparameterrelated tothegravitonmass.Wehaveset8πGN

/

c4

=

1.Fortheisotropic case in d

=

2, we can generically take V

(I

I J

₎

₌

_V

₍

_X

_,

_Z

₎

_where

X

=

1₂tr

I

andZ

=

det

I

[16,17,23].Thescalars

φ

I aredual tothe boundary operators



I andbreak the translational invariance of thedualfieldtheory(see[32] and[17] forthespecificsofthe sym-metry breaking pattern). Depending on the boundary conditions imposed on

φ

I,thisbreakingcan eitherbe explicit,spontaneous, orpseudo-spontaneous[15,33,8,23,5].Presently,weshallbe inter-estedinmodelswithspontaneouslybrokentranslationsleadingto phonondynamicsinthedualfieldtheory [24,9,23,34,31].

We consider a black brane solution of (9) in Eddington-Finkelstein(EF)coordinateswiththemetric

ds2

=

1 u2



−

f

(

u

)

dt2

−

2 dt du

+

dx2

+

d y2



,

(10) and a radially constant profile for the scalars,

φ

I

=

α

xI_{, for} someconstant

α

.The radialcoordinateu

∈ [

0

,

uh

]

spansfromthe boundary u

=

0 to the horizon u

=

uh. The emblackening factor

f

(

u

)

takesasimpleform

f

(

u

)

=

1

−

u 3 u3_h

−

u 3 uh



u m2

ℵ

4 V

(

α

2

_ℵ

2

_,

_α

4

_ℵ

4

₎

_d

_{ℵ .}

₍₁₁₎

Linear perturbations around the black brane geometry capture near-equilibrium finite temperature fluctuationsin the boundary fieldtheory [35,36,23,31,37].

Temperatureandentropydensityinthe boundaryfieldtheory areidentifiedwiththeHawkingtemperatureandareaoftheblack brane,respectively T

= −

f 

₍

_u h

)

4

π

=

3

−

m2Vh 4

π

uh

,

s

=

2

π

u_h2

,

(12)

5 _{Notethat [5] assumesP} _{toalsovanishintheorieswithzerostrainpressure,}

4 M. Ammon et al. / Physics Letters B 808 (2020) 135691

Fig. 1.andDforV(X,Z)=XN_{modelsforN=3,4,5 (fromtoptobottom)asfunctionsofthedimensionlessparameterm/T ,alongsidetheirhydrodynamicpredictions}

from(3) (solidlines).

with Vh

=

V

(

u_h2

α

2

,

u4_h

α

4

)

. The free energy density is defined as the renormalised euclidean on-shell action [38]. The expectation value



T μν



canberead offusingtheleadingfall-off ofthe met-ric at the boundary. Using the first row of (4), this leads to the thermodynamicquantities p

=

1 2u3_h

−

m2 u3_h

1 2Vh

−

Uh

,



=

1 u3_h

−

m2 u3_hUh

,

P

=

m2 u_h3

1 2Vh

−

3 2Uh

.

(13) We have defined Uh

= −

u3h



uh 0

ℵ

− 4_V

₍

_α

2

_ℵ

2

_,

_α

4

_ℵ

4

₎

_d

_ℵ

_{, assuming}

V

(

X

,

Z

)

to fall off faster than

∼

u3 at the boundary.6 Details of holographic renormalisationfor thesemodelshave beengivenin Appendix A.Using the expressions in(13) togetherwith (5), we canfindthebulkmodulus

B

=

m 2 4u3_h



3Vh

−

9Uh

+

uh

∂

uhVh

(

m 2_V h

−

3

)

m2



_V h

−

uh

∂

uhVh



−

3



,

(14)

Finally,usingtheresults of[11,8,4], we canderive ahorizon for-mulafor

σ

,whichreads

σ

=

m2 2

α

2_u3 h

∂

Vh

∂

uh

,

(15)

andagreeswellwiththenumericalresultsobtainedwiththeKubo formulain(4).The remaining coefficients, G and

η

,must be ob-tainednumerically.

Thenon-trivialexpressionfor

P

in(13) indicatesthepresence ofbackground strain inthese holographic models.This is associ-ated with the equilibriumstate

φ

I

=

α

xI _{not being a minimum} of free energy [39,19,24]. To wit, using (13) one can check that d

/

dα

|

T

= −

dp

/

dα

|

T

=

0 leadsto

P =

0.However,asisevident from(3), the presenceof

P

by itself doesnot lead toany linear instability orsuperluminality [24,9,23].Setting

P =

0 in (13), we canfindathermodynamicallyfavoured state

α

=

α

0 asanon-zero

solutionof Vh

=

3Uh.Noticethat

P



|

α=α0

=

0, whichmeansthat

strain pressure still plays a crucialrole in thedual hydrodynam-icsthroughitstemperaturederivatives,asdiscussedaround(8).In particular,thesemodels canhavenon-zero bulkmodulus despite beingscaleinvariant.

Simple monomial models considered previously in the litera-ture [23,17,24,36,31,37],such as V

(

X

,

Z

)

=

XN

,

ZM,do notadmit

6 _{For potentials that fall of slower than ∼u}3 _{near the boundary, such as}

V(X)=XN _{with N}

<3/2, this integral is divergent. Nevertheless, perform-ingholographic renormalisation carefully (seeAppendix A),the thermodynamic quantities above can be computed explicitly and amounts to defining Uh= u3

h

∞

uhℵ

−4_V(α2_ℵ2_,α4_ℵ4_{)dℵinstead.}

P =

0 states with non-zero

α

.7 _{The simplest models admitting} states with

P =

0 have polynomial potentials such as V

(

X

,

Z

)

=

X

+ λ

X2_{.Unfortunately, thisnaive modelisplaguedby linear}

in-stabilities.Nevertheless,itcanbeusedasatoymodeltoillustrate the importanceof

P



=

0;we returnto thedetails ofthismodel below.

3.2. Strainedholographicmodels

Let us first specialize to the strained models with V

(

X

,

Z

)

=

XN

_,

_ZM _{and N}

_>

₅

_/

_{2, M}

_>

₅

_/

_{4 to numerically obtain G and}

_η

_, andtest theagreement betweenquasinormalmodesandthe hy-drodynamic predictions. We can compute the full spectrum of quasinormal modes, in both the transverse and longitudinal sec-tors, using pseudo-spectral methods following [9,23,24,40,41]. As we discussedaround (6), thestrain pressure doesnot appear ex-plicitly in the transversesound modes, leading to the same pre-dictions by [3] and[5], modulo the definitionof

χ

π π . Since the

discrepancyin

χ

π π hasalreadybeenidentifiedandtestedagainst

holographicresults [24,23],hereweonlyfocusonthelongitudinal sector.

We start with V

(

X

,

Z

)

=

XN _{models.Note that V}

h

=

α

2Nu2Nh andUh

=

α

2Nu2Nh

/(

3

−

2N

)

.Using(12)-(15),wecanexplicitlyfind

T

=

3

−

m 2_V h 4

π

uh

,

s

=

2

π

u2_h

.

p

=

1 2u3_h

1

−

2N

−

1 2N

−

3m 2_V h

,



=

1 u_h3

1

+

m 2_V h 2N

−

3

,

P

=

N 2N

−

3 m2_V h u3_h

,

P



= −

4

π

u2_h Nm2Vh 3

+ (

2N

−

1

)

m2_V h

,

B

=

Nm 2_V h 2u3_h

3 2N

−

3

+

3

−

m2Vh 3

+ (

2N

−

1

)

m2_V h

,

σ

=

Nm2Vh

α

2_u4 h

,

cv

=

4

π

u2_h 3

−

m2_V h 3

+ (

2N

−

1

)

m2_V h

.

(16)

Computing G and

η

numerically using (4), we can compare the hydrodynamic predictionforthelongitudinalattenuationconstant



anddiffusionconstant D_in(3) withthenumericalresults ob-tained forthe quasinormalmodesin theholographic model. The resultsareshowninFig.1.Theagreementisextremelygoodand isvalidindependentof N.Weno longerseeadiscrepancyinthe diffusionmode.

7 _{However,thewould-bepreferredstateα=0 isnotagoodvacuumofthe}

the-ory,sincethemodelisstronglycoupledaroundthatbackground[24].Therefore,in thesetheories,itisincorrecttocomparefreeenergiesofstateswithα =0 against thestateα=0.

Fig. 2.and Dfor the model V(X,Z)=Z2_{, as a function of the dimensionless parameter m/T , and the hydrodynamic prediction from (3).}

Fig. 3. Left: v2

⊥forV(X)=X+X2/2 modelwithP=0 alongsidethehydrodynamicpredictions(solidlines).Wehavechosenuh=1 settingα=1. Right: DforV(X)= X+X2_{/2 modelwithP=0 alongsidethehydrodynamicpredictions(solidlines).Wehavechosenu}

h=1 settingα=1. Letusnow considermodels V

(

X

,

Z

)

=

ZM.In thiscase, Vh

=

α

4M_u4M

h andUh

=

α

4M_u4M

h

/(

3

−

4M

)

.Theexpressionsfor thermo-dynamicquantitiesremainthesameasin(16) butwithN

→

2M. Generically, X -independent potentialsV

(

X

,

Z

)

=

V

(

Z

)

enjoy a largersymmetry group –the dualfield theory isinvariant under volumepreservingdiffeomorphisms,modelling afluid.These mod-elshaveG

=

0,leading totheabsenceoftransversephonons [17], and

η

saturating the Kovtun-Son-Starinets bound [35]. In Fig. 2

weshowacomparisonbetweenthehydrodynamicpredictionand numericalresultsforquasinormalmodesforV

(

X

,

Z

)

=

Z2_.The

ex-cellent agreementconfirms that the hydrodynamic framework of [5] isvalidforageneralclassofviscoelasticmodelswithnon-zero strainpressure.

3.3.Unstrainedholographicmodels

Inthissection,weconsiderholographicmodelswithzerostrain pressure in equilibrium. These are thermodynamically favourable modelswhich admit translationally broken phasesthat minimise freeenergy.Wewillillustratethatevenforsuchmodels,thestrain pressureplaysacrucialroleinthedualhydrodynamicsthroughits temperaturederivatives and hence the hydrodynamic modes are governedbytheexpressionsineq.(8).

LetusconsiderthesimplestmodelV

(

X

,

Z

)

=

X

+λ

X2.As men-tioned above, this model is unstable: (I) the shear modulus is negative,(II)the speedoftransversesoundisimaginary,and(III) thelongitudinaldiffusionconstantbecomesnegativeatlargem

/

T .

Itcanbeverifiedthatall themodels V

(

X

,

Z

)

=

XN1

+ λ

_XN2 _with

spontaneousbreakingof translationsand

P =

0 suffer fromsuch linearinstabilities,orhaveghostlyexcitationsinthebulk.8 _Clearly, the model V

(

X

,

Z

)

=

X

+ λ

X2 _{cannot describe a stable physical}

system,butit canbe usedasa toy exampletoillustrate the im-portanceofstrain pressure.Wefindthat Vh

=

α

2u_h2

+ λ

α

4u4_h and

8 _{Moreprecisely,formodelswithN}

1<3/2 theshearmodulusisnegative;see

appendixof[24] forformulae.Hence,alsothemodelconsideredin[5] is dynami-callyunstable.

Uh

=

α

2u2_h

− λ

α

4u_h4. Setting

P

in(13) to zero, we find the pre-ferredvalueof

α

=

0 tobe

α

2

=

1

2

λ

u2_h

,

(17)

whichmatches theresultof[19] in thezerocharge densitylimit

ρ

=

0.9

Weobtainthehydrodynamicparameters

T

=

3 4

π

uh

1

−

m 2 4

λ

,

s

=

2

π

u2_h

,

p

=

1 2u3_h

1

−

m 2 4

λ

,



=

1 u3_h

1

−

m 2 4

λ

,

P



=

−

4

π

3u2_h m2

λ

+

5m2

_/

₁₂

,

B

=

m 2 2

λ

u3_h

λ

−

m2

_/

₄

λ

+

5m2

_/

₁₂

,

σ

=

2m 2 u2_h

,

cv

=

4

π

u2_h

λ

−

m2

/

4

λ

+

5m2

_/

₁₂

.

(18)

Notice that the potential behaves as

∼

u2 nearthe boundary, so the alternatedefinition ofUh givenin footnote6 hasto be used informulas(13)-(14).G and

η

havetobefoundnumericallyusing (4). We see that

P



=

0 leading to B

=

0 and cv

=

2s in these models,asdiscussedabove.

We can also compute the quasinormalmodes forthis system numericallyandcomparethemagainst thehydrodynamic predic-tions presentedin eq.(8),andthat of[3] without

P

. Wesee in Fig.3that thetransverse speedofsound v_⊥ is imaginarydueto negative shear modulus G; nevertheless the predictionfrom hy-drodynamicsmatchesperfectly.Weagainfindadiscrepancyin D_

similar to[9] comparedto [3],which isresolved byincluding

P

 contributions,asineq.(8);seeFig.3.

9 _{Thenotationalrelationshipsareα≡k and}

λ≡ λ2,wheretheright-handsidesof

theidentificationsarethenotationof[19].Noticealsothateq.(45)in[19] contains typos;itshouldreadk2_I

Y1(0)+2λ2k

4_I

Y2(0)− λ1ρ

2_k2_I

6 M. Ammon et al. / Physics Letters B 808 (2020) 135691 Despite the simplicity and linear instability of this model, it

sharesvariousfeaturesofinterestwithsimilarholographicmodels withoutbackgroundstrain,suchastheonediscussedin[19]. Simi-larmodelscanalsobeconstructedintheframeworksof [42,43,19,

20,44]. The requirement of thermodynamic stability forisotropic modelscanbeimplementedas



= −

Txx

_

_{[29],whichaccording} to (4) is precisely

P =

0. Irrespective of the particular model at play, while we mightbe able toset

P =

0 by judiciously choos-ing

α

in theequilibriumstate, we will genericallybe left witha non-zero

P

,whichmustbetakenintoaccountinthedual hydro-dynamictheory.10

At this stage, we are not aware ofany massive gravity or Q-lattices models which are both thermodynamically and dynami-callystable.11

4. Conclusions

Inthispaperwe illustrated thatthe theory ofviscoelastic hy-drodynamics formulated in [5] is the appropriate hydrodynamic description for the (strained) homogeneous holographic models of [17] with spontaneously brokentranslations. We showed that the theory faithfully predicts all the transport coefficients and thebehaviour ofthelow-energy quasinormalmodesin the holo-graphicsetup.Moreover,itresolvesthetensionsbetweenthe pre-vioushydrodynamic frameworkof[3] andtheholographic results reportedin [9].

Moreover,weextendedtheanalysisbeyond[5] andarguedthat theeffectsofthetemperaturederivativeofthestrainpressureare present even in unstrained equilibrium configurations. We con-structed athermodynamically stableholographic model,analysed itslow-lyingQNMs,andfoundagreementwiththeexpressionsin equation (8). We have also noted issues (dynamical instabilities) with the physicality of this thermodynamically favoured model (andothersimilarsetups[5,19]).

Generally,weexpectthatthehydrodynamic formulationof [5], withtheadditionoftheresultsanddiscussionspresentedinthis paper, will continue to work for all homogeneous holographic modelswithspontaneouslybrokentranslations[19,7,6,20],dueto thesamesymmetry-breakingpattern.

Theanalysisinthispaperopensupthestageforvarious inter-estingfutureexplorations.Animmediategoalwouldbetoinspect variousholographicmodelsofviscoelasticityintheliterature,with zero background strain, and identify the role ofnon-zero

P

 on thequasinormalspectrum.Inparticular,therelationbetween dy-namic instability and the absence of strain pressure, which has been presented in thiswork, isworthy of further investigations. Furthermore,anotherinterestingdirectionistobetter understand theroleofstrainpressure,anditstemperaturederivative,in phys-icalsystems(seee.g. [46]).

Theadditionofa smallexplicitbreakingoftranslationstothe hydrodynamicframeworkof[5] couldalsoprovidean understand-ingofthe universalphase relaxationrelation



∼

M2

_/

σ

_(with

_

the Goldstonephase relaxation rate;M themass of the pseudo-Goldstonemode).Thisrelationwasproposedin[11] andwaslater verified for the models presented in this paper in [8]. It could also provide an explanation for the complex dynamics found in the pseudo-spontaneous limit in [23]. Furthermore,



seems to be tightly connected to the presence ofglobal bulk symmetries, whicharenot expectedtoappearinproperinhomogeneous peri-odic latticestructures. The physical interpretationof theseglobal

10

Seealso[6] forabulkanalysis.

11 _{PreliminaryresultssuggestthatthemodelinsectionII-Bof[19] isdynamically}

unstableaswell[45].Thisissomewhatexpectedgiventhesimilaritieswithour

V(X)=X+ λX2_model.

structures hasrecentlybeendiscussedin[47],andstillrepresents animportantpuzzleinthefield.

Onemayalsoconsidertheviscoelastichydrodynamictheoryof [5] beyond linear response in order to explore the full rheology oftheholographic modelsconsideredinthiswork,asinitiated in [37].

In conclusion, this work marks an important development in understanding thenature ofthefield theoriesdual tothe widely used holographicmodels withspontaneouslybrokentranslational invariance, andprovides anotherrobust bridge between hologra-phy, hydrodynamics (in its generalised viscoelastic form) and ef-fectivefieldtheory.

Declarationofcompetinginterest

Theauthorsdeclarethattheyhavenoknowncompeting finan-cialinterestsorpersonalrelationshipsthatcouldhaveappearedto influencetheworkreportedinthispaper.

Acknowledgements

We thankAristomenis Donos,Blaise Goutéraux,SeanHartnoll, Christiana PantelidouandVaiosZiogas forseveralhelpful discus-sions andcomments. S. Graywouldlike to thankIFT Madrid for hospitality during the initial stages of this work. S. Grieninger thanks the University ofVictoriafor hospitalityduring the initial stages of this work. MA is funded by the Deutsche Forschungs-gemeinschaft (DFG, German Research Foundation) – 406235073. MB acknowledges the support of the Spanish MINECO’s “Centro de Excelencia Severo Ochoa” Programme under grant

SEV-2012-0249. The work of S. Gray has been funded by the Deutsche

Forschungsgemeinschaft(DFG)underGrantNo.406116891within the Research Training Group RTG 2522/1. S. Grieningergratefully acknowledges financial support by the DAAD (GermanAcademic Exchange Service) for a Jahresstipendium fürDoktorandinnen und Doktoranden in 2019. AJ is supported by the NSERC Discovery GrantprogramofCanada.

Appendix A. Holographicrenormalisation

Inthisappendixwegivesomedetailsregardingtheholographic renormalisationunderlyingthemodelsdiscussedinthemaintext. The bulk action (9) has to be supplemented with appropriate boundary counter terms to have a well-defined variational prin-ciple Scounter

=



u= dd+1x



−

γ

K

−

d



+

m 2_V

¯

_{( ¯}

_I

I J

₎

,

(A.1)

where

γ

μν

=

limu→ gμν istheinducedmetricattheboundary, K

is theextrinsic curvature,and

¯I

I J

₌

_γ

μν

_∂

μ

φ

I

∂

ν

φ

J. V

¯

( ¯

I

I J

)

is an

appropriateboundarypotentialfixedbyrequiringthattheon-shell action of theblack brane solution (10) to be finite. Forinstance, ind

=

2,forV

(

X

)

=

XN _modelswithN

_>

₃

_/

_{2 wehave V}

¯

_{( ¯}

_X

₎

₌

_0, while for N

<

3

/

2 weget V

¯

( ¯

X

)

= ¯

X

/(

3

−

2N

)

, where X

¯

=

1₂tr

¯I

. ForV

(

X

)

=

X

+ λ

X2_{,weinsteadfindV}

¯

_{( ¯}

_X

₎

_{= ¯}

_{X .}

Due to its novelty, we will in the remainder of this section mainlyfocusonholographicrenormalisationforV

(

X

)

=

X

+ λ

X2_.

To implement spontaneous symmetry breaking for models

whose boundary behaviour goes as V

(

X

,

Z

)

∼

XN

_,

_ZM _{with N}

_<

5

/

2

,

M

<

5

/

4, one needs to apply alternativequantisation for the

scalars.12_{Moreprecisely, one needstodeform theboundary} the-orywithaterm

Salt

=



u= dd+1x



−

γ



I

φ

I

,

(A.2) where



I

=

1

√

₋

γ

δ(

S

+

Scounter

)

δφ

I

= δ

I J



V

(

X

)

na

∂

a

φ

J

+ ∇

₍μ_γ)



¯

V

( ¯

X

)∂

μ

φ

J



.

(A.3)

∇

(γ)

μ isthecovariantderivative associatedwith

γ

μν andnaisthe outwardpointingnormalvectorattheboundary.The(A.2) termin theactionturns



I attheboundaryintothedynamical operator, whiletheassociatedsourceisnowgivenbytheboundaryvalueof



I.Weareinterestedindualhydrodynamicmodelsintheabsence of sources for the scalars. Hence, in alternative quantisation we imposetheboundaryconditions

lim

→0

1



d+1



I

=

0

.

(A.4)

Finally,for the metric we always impose the standard boundary conditions

lim

→0



2

_γ

μν

=

η

μν

.

(A.5)

Notethat inthe alternative quantisation schemethe background profile forthe scalars,

φ

I

=

α

xI_{, is no longer an external source} providingthe explicit breakingof translations.This is the funda-mentalreasonwhymodelslike V

(

X

)

=

X

+ . . .

,usingalternative quantisation [5], realize the spontaneous (and not explicit [15]) breakingoftranslations.

References

[1] B.I.Halperin,D.R.Nelson,Theoryoftwo-dimensionalmelting,Phys.Rev.Lett. 41(1978)121–124,https://doi.org/10.1103/PhysRevLett.41.121.

[2] P.C.Martin,O.Parodi,P.S.Pershan,Unifiedhydrodynamictheoryforcrystals, liquidcrystals,andnormalfluids,Phys.Rev.A6(1972)2401–2420,https:// doi.org/10.1103/PhysRevA.6.2401.

[3] L.V. Delacrétaz,B.Goutéraux, S.A.Hartnoll,A.Karlsson,Theoryof hydrody-namictransportinfluctuatingelectronicchargedensitywavestates,Phys.Rev. B96 (19)(2017)195128, https://doi.org/10.1103/PhysRevB.96.195128, arXiv: 1702.05104.

[4] A.Donos,D.Martin,C.Pantelidou,V.Ziogas,Hydrodynamicsofbrokenglobal symmetriesinthebulk,J.HighEnergyPhys.10(2019)218,https://doi.org/10. 1007/JHEP10(2019)218,arXiv:1905.00398.

[5]J. Armas, A. Jain, Viscoelastic hydrodynamics and holography, arXiv:1908. 01175,2019.

[6]A.Donos,D.Martin,C.Pantelidou,V.Ziogas,Incoherenthydrodynamicsand densitywaves,arXiv:1906.03132,2019.

[7] S. Grozdanov, N.Poovuttikul, Generalizedglobalsymmetries instates with dynamicaldefects:thecaseofthetransversesoundinfieldtheoryand holog-raphy,Phys.Rev.D97 (10)(2018)106005,https://doi.org/10.1103/PhysRevD. 97.106005,arXiv:1801.03199.

[8] M.Ammon,M.Baggioli,A.Jiménez-Alba,Aunifieddescriptionoftranslational symmetrybreakinginholography,J.HighEnergyPhys.09(2019)124,https:// doi.org/10.1007/JHEP09(2019)124,arXiv:1904.05785.

[9] M.Ammon,M.Baggioli,S.Gray,S.Grieninger,Longitudinalsoundanddiffusion inholographicmassivegravity,J.HighEnergy Phys.10(2019) 064,https:// doi.org/10.1007/JHEP10(2019)064,arXiv:1905.09164.

[10] A. Amoretti, D. Areán, B. Goutéraux, D. Musso, DCresistivity ofquantum critical, charge density wave states from gauge-gravity duality, Phys. Rev. Lett.120 (17)(2018)171603,https://doi.org/10.1103/PhysRevLett.120.171603, arXiv:1712.07994.

12 _{ForpotentialsV}

(X,Z)=XN

,ZM _withN

>5/2,M>5/4,oneinsteadneedsto followstandardquantisation inordertohavespontaneoussymmetrybreaking,as shownin[24].

[11] A.Amoretti,D.Areán,B.Goutéraux,D.Musso,Universalrelaxationina holo-graphicmetallicdensitywavephase,Phys.Rev.Lett.123 (21)(2019)211602,

https://doi.org/10.1103/PhysRevLett.123.211602,arXiv:1812.08118.

[12] R.A.Davison,K.Schalm,J.Zaanen,Holographicdualityandtheresistivityof strange metals,Phys. Rev.B89 (24) (2014)245116,https://doi.org/10.1103/ PhysRevB.89.245116,arXiv:1311.2451.

[13] L.V.Delacrétaz,B.Goutéraux,S.A.Hartnoll,A.Karlsson,Badmetalsfrom fluc-tuatingdensitywaves,SciPostPhys.3 (3)(2017)025,https://doi.org/10.21468/ SciPostPhys.3.3.025,arXiv:1612.04381.

[14]D.Vegh,Holographywithouttranslationalsymmetry,arXiv:1301.0537,2013.

[15] T.Andrade,B.Withers,Asimpleholographicmodelofmomentumrelaxation, J.HighEnergyPhys.05(2014)101,https://doi.org/10.1007/JHEP05(2014)101, arXiv:1311.5157.

[16] M.Baggioli, O.Pujolas,Electron-phononinteractions, metal-insulator transi-tions,andholographicmassivegravity,Phys.Rev.Lett.114 (25)(2015)251602,

https://doi.org/10.1103/PhysRevLett.114.251602,arXiv:1411.1003.

[17] L. Alberte, M. Baggioli, A. Khmelnitsky, O. Pujolas, Solid holography and massivegravity,J.HighEnergyPhys. 02(2016)114,https://doi.org/10.1007/ JHEP02(2016)114,arXiv:1510.09089.

[18] A.Donos,J.P.Gauntlett,HolographicQ-lattices,J.HighEnergyPhys.04(2014) 040,https://doi.org/10.1007/JHEP04(2014)040,arXiv:1311.3292.

[19] A.Amoretti,D.Areán,B.Goutéraux,D.Musso,Effectiveholographictheoryof charge densitywaves,Phys.Rev.D97 (8)(2018)086017, https://doi.org/10. 1103/PhysRevD.97.086017,arXiv:1711.06610.

[20] T. Andrade, M. Baggioli, A.Krikun, N. Poovuttikul, Pinning oflongitudinal phononsinholographicspontaneoushelices,J.HighEnergyPhys. 02(2018) 085,https://doi.org/10.1007/JHEP02(2018)085,arXiv:1708.08306.

[21] T. Andrade, A. Krikun, Coherent vs incoherent transport in holographic strangeinsulators,J.HighEnergyPhys.05(2019)119,https://doi.org/10.1007/ JHEP05(2019)119,arXiv:1812.08132.

[22] P.Chaikin,T.Lubensky,PrinciplesofCondensedMatterPhysics,Cambridge Uni-versityPress,2000,https://books.google.es/books?id=P9YjNjzr9OIC.

[23] M.Baggioli,S. Grieninger,Zoologyofsolid&fluidholography—Goldstone modesandphaserelaxation,J.HighEnergyPhys.10(2019)235,https://doi. org/10.1007/JHEP10(2019)235,arXiv:1905.09488.

[24]L.Alberte, M.Ammon,M. Baggioli,A.Jiménez-Alba,O. Pujolàs,Holographic phonons,2017,arXiv:1711.03100.

[25] A.Amoretti,D.Areán,B.Goutéraux,D.Musso,Diffusionanduniversal relax-ationofholographic phonons, J.High Energy Phys.10 (2019)068, https:// doi.org/10.1007/JHEP10(2019)068,arXiv:1904.11445.

[26]A.Amoretti,D.Areán,B.Goutéraux,D.Musso,Gaplessandgappedholographic phonons,arXiv:1910.11330,2019.

[27] S.A.Hartnoll,P.K.Kovtun,M.Muller,S.Sachdev,TheoryoftheNernsteffect near quantum phasetransitions incondensed matter, and in dyonic black holes, Phys. Rev. B 76 (2007) 144502, https://doi.org/10.1103/PhysRevB.76. 144502,arXiv:0706.3215.

[28] M.M.Caldarelli,A.Christodoulou,I.Papadimitriou,K.Skenderis,Phasesof pla-narAdSblackholeswithaxioniccharge,J.HighEnergyPhys.04(2017)001,

https://doi.org/10.1007/JHEP04(2017)001,arXiv:1612.07214.

[29] A.Donos,J.P.Gauntlett,OnthethermodynamicsofperiodicAdSblackbranes, J.HighEnergyPhys.10(2013)038,https://doi.org/10.1007/JHEP10(2013)038, arXiv:1306.4937.

[30] A.Esposito,S.Garcia-Saenz,A.Nicolis,R.Penco,Conformalsolidsand hologra-phy,J.HighEnergyPhys.12(2017)113,https://doi.org/10.1007/JHEP12(2017) 113,arXiv:1708.09391.

[31]M.Baggioli,V.C.Castillo,O.Pujolas,Scaleinvariantsolids,arXiv:1910.05281, 2019.

[32] A. Nicolis, R. Penco, F. Piazza, R. Rattazzi, Zoology of condensed matter: framids,ordinarystuff,extra-ordinarystuff,J.HighEnergyPhys.06(2015)155,

https://doi.org/10.1007/JHEP06(2015)155,arXiv:1501.03845.

[33] L.Alberte,M.Ammon,M.Baggioli,A.Jiménez,O.Pujolàs,Blackholeelasticity andgappedtransversephononsinholography,J.HighEnergyPhys.01(2018) 129,https://doi.org/10.1007/JHEP01(2018)129,arXiv:1708.08477.

[34] M.Baggioli,U.Gran,A.J.Alba,M.Tornsö,T.Zingg,Holographicplasmon re-laxationwithandwithoutbrokentranslations,J.HighEnergyPhys.09(2019) 013,https://doi.org/10.1007/JHEP09(2019)013,arXiv:1905.00804.

[35] L. Alberte, M.Baggioli, O. Pujolas,Viscosity boundviolationinholographic solids and the viscoelastic response, J. High Energy Phys. 07 (2016) 074,

https://doi.org/10.1007/JHEP07(2016)074,arXiv:1601.03384.

[36]T.Andrade,M.Baggioli,O.Pujolàs,Viscoelasticdynamicsinholography,arXiv: 1903.02859,2019.

[37]M.Baggioli,S.Grieninger,H.Soltanpanahi,Nonlinearoscillatorysheartestsin viscoelasticholography,arXiv:1910.06331,2019.

[38] K.Skenderis, Lecturenoteson holographic renormalization,Class.Quantum Gravity 19 (2002) 5849–5876, https://doi.org/10.1088/0264-9381/19/22/306, arXiv:hep-th/0209067.

[39] A.Donos,J.P.Gauntlett,V.Ziogas,Diffusion forholographic lattices,J.High Energy Phys. 03(2018)056,https://doi.org/10.1007/JHEP03(2018)056,arXiv: 1710.04221.

8 M. Ammon et al. / Physics Letters B 808 (2020) 135691

[40] M.Ammon,S.Grieninger,A.Jimenez-Alba,R.P.Macedo,L.Melgar,Holographic quenchesandanomaloustransport,J.HighEnergyPhys.09(2016)131,https:// doi.org/10.1007/JHEP09(2016)131,arXiv:1607.06817.

[41]S. Grieninger, Holographic quenches and anomalous transport, arXiv:1711. 08422,2016.

[42] A.Donos,J.P.Gauntlett,Blackholesdualtohelicalcurrentphases,Phys.Rev. D86(2012)064010,https://doi.org/10.1103/PhysRevD.86.064010,arXiv:1204. 1734.

[43]M. Baggioli, A. Buchel, Holographicviscoelastic hydrodynamics, arXiv:1805. 06756,2018.

[44] A. Donos, C. Pantelidou, Holographictransport and density waves, J. High EnergyPhys. 05(2019)079,https://doi.org/10.1007/JHEP05(2019)079,arXiv: 1903.05114.

[45] PrivatecommunicationwithKeun-YoungKimandHyun-SikJeong.

[46]J.Armas, A.Jain, Hydrodynamicsfor charge density wavesand their holo-graphicduals,arXiv:2001.07357,2020.

[47]M.Baggioli,Arethehomogeneousholographicviscoelasticmodels quasicrys-tals?,arXiv:2001.06228,2020.