Longitudinal Sound and Diffusion
in Holographic Massive Gravity
We consider a simple class of holographic massive gravity models for which the dual field theories break translational invariance spontaneously. We study, in detail, the longitudinal sector of the quasi-normal modes at zero charge density. We identify three hydrodynamic modes in this sector: a pair of sound modes and one diffusion mode. We numerically compute the dispersion relations of the hydrodynamic modes. The obtained speed and the attenuation of the sound modes are in agreement with the hydrodynamic predictions. On the contrary, we surprisingly find disagreement in the case of the diffusive mode; its diffusion constant extracted from the quasi-normal mode data does not agree with the expectations from hydrodynamics. We comment on some possible reasons behind this puzzle. Finally, we extend the analysis of the collective longitudinal modes beyond the hydrodynamic limit by displaying the dynamics of the higher quasi-normal modes at large frequencies and momenta.
Hydrodynamics and effective field theories (EFTs) are powerful tools which can be used to describe the macroscopic and low energy dynamics of physical systems. EFTs have been widely and successfully applied in a broad variety of research areas, from cosmology to particle physics, and finally to condensed matter. Symmetries are the fundamental pillars upon which such effective frameworks are built; furthermore, the symmetries of a system strongly constrain the nature of its collective excitations and their low energy dynamics.
However, in realistic settings, and especially in the context of condensed matter systems, part of the fundamental symmetries appear to be broken. Paradigmatic examples are the spontaneous breaking of the symmetry in superconducting materials, as well as the spontaneous breaking of translational (and rotational) invariance in crystals with long range order, i.e. systems modelled by periodic lattices. Nevertheless, even in cases where some of the symmetries are broken, effective field theory frameworks still provide an important guidance for theoretical descriptions bro (). In particular, the various symmetry breaking patterns can be used to classify the corresponding phases of matter Nicolis:2015sra (). Historically, such methods of classification have been of great importance for the understanding of the dynamics of fluids, liquid crystals, nematic crystals and ordered crystals PhysRevB.22.2514 (); PhysRevA.6.2401 ().
Nevertheless, for systems of the types mentioned above, it is challenging to set up the hydrodynamic theory, i.e. finding a consistent and complete gradient expansion. Over the last decade, holographic dualities (also known as AdS/CFT or gauge/gravity dualities) have provided a powerful and effective tool set which can be used to tackle strongly coupled condensed matter systems Hartnoll:2016apf (); Ammon:2015wua (); zaanen2015holographic (). In particular, holography has provided novel insights into hydrodynamics concerning new transport coefficients; bounds of transport coefficients; and the applicability of hydrodynamics in the presence of large gradients. Moreover, holography goes beyond hydrodynamics in the sense that it allows for direct computations of transport coefficients, which in the EFT description appear only as undetermined parameters.
Historically, holography has been focused on the description of strongly coupled fluids Policastro:2002se (); Policastro:2002tn (). More recently, the attention switched to the description of strongly coupled solids or, in general a more general sense, systems with elastic properties. The long-term scope of this programme is to bring the theoretical framework nearer to realistic systems in order to make concrete predictions which are testable in the lab.
The propagation of sound in compressible materials is a direct macroscopic manifestation of the spontaneous symmetry breaking (SSB) of translational invariance. Sound propagation admits the effective quasiparticle description by acoustic phonons, which are the Goldstone bosons corresponding to the spontaneous breaking of translational symmetry Leutwyler:1996er (). Sound can propagate in two different forms, transverse and longitudinal sound, which correspond to transverse and longitudinal phononic vibrational modes, with dispersion relation
where denotes the sound speed and denotes the diffusion constant. Such modes are tightly correlated to the mechanical properties of the material, in particular to its shear and bulk elastic moduli , respectively Lubensky (); landau7 (). The speed of transverse sound is determined by the simple expression
where is the shear elastic modulus and the momentum susceptibility. The property of supporting propagating shear elastic waves is usually considered the main distinction between solids and fluids.111This statement is imprecise and true only at low frequency and low momentum . It is well known that fluids at large frequency, i.e. beyond the so-called Frenkel frequency, support propagating shear waves doi:10.1021/j150454a025 (). Simultaneously, it is suggested by recent experiments noirez2012identification (); PhysRevLett.118.215502 () and theoretical developments 2016RPPh…79a6502T (); Baggioli:2018vfc (); Baggioli:2018nnp () that the same might happen at large momentum, beyond the usually called -gap. Longitudinal sound appears both in solids and in fluids and, in two spatial directions, propagates with speed
where is the bulk elastic modulus, which is the coefficient determining the response of the system under a compressive strain. The diffusion constants appearing in expression (1) are determined by the dissipative mechanisms of the system; in a conformal system, where the bulk viscosity vanishes, they are both given in terms of the shear viscosity . In particular, for a relativistic CFT with two spatial directions, we have
which can be derived using both relativistic hydrodynamics Kovtun:2012rj () and holographic techniques Policastro:2002se (). The presence of a diffusive term in the dispersion relation of the phonons (1) indicates that the system has also a viscous component, i.e. that it is a viscoelastic material. In an unrealistic perfect crystal, with no dissipation nor effective viscosity, sound would propagate indefinitely, which is not realistic.
In a system with spontaneously broken translational symmetry the longitudinal sector contains an additional diffusive mode with dispersion relation
with diffusion constant . This diffusive mode is intimately connected with the presence of Goldstone degrees of freedom, in fact its hydrodynamic nature is simply related to the conservation of the phase of the Goldstone bosons, i.e. the phonons. The presence of such a mode can be predicted by performing an effective hydrodynamic description for crystals PhysRevA.6.2401 (); PhysRevB.22.2514 (); Delacretaz:2017zxd (). Notably, the underlying physics is very similar to that of superfluids/superconductors Davison:2016hno (), where a Goldstone diffusion mode appears as well.
All the features just mentioned can be formally derived using hydrodynamic techniques PhysRevA.6.2401 (); PhysRevB.22.2514 (); Delacretaz:2017zxd (). The analysis of the transverse modes within the holographic framework and its successful matching with the hydrodynamic predictions have already been presented in Alberte:2017oqx (). The scope of this paper is to extend the analysis to the longitudinal sector, in particular to the study of the diffusion constants of the sound and diffusive modes, in a simple holographic system with spontaneously broken translational invariance. Concretely, we use the holographic massive gravity theory introduced in Baggioli:2014roa (); Alberte:2015isw () to implement the SSB breaking of translations as explained in Alberte:2017oqx ().
The hydrodynamic degrees of freedom present in the longitudinal sector have previously been investigated in a more complicated model in Andrade:2017cnc (), with more emphasis on the electric transport properties of the dual field theory. In particular the longitudinal sound mode and the diffusive mode have been identified numerically but without any systematic study or theoretical background. Related results, in the presence of electromagnetic interactions, have recently appeared in Baggioli:2019aqf (). In this work, thanks to the simplicity of the model at hand, we improve that analysis. In particular:
We study in the dynamics of the longitudinal collective modes dialing the SSB scale.
We compare numerical results obtained from the holographic model with the hydrodynamics expectations.
We numerically describe the higher modes, extending the analysis beyond the hydrodynamic limit towards large frequencies and momenta, .
From the numerical study we obtain the expected hydrodynamic modes, namely the longitudinal sound and the diffusive mode. Both modes display the correct dispersion relation. Moreover, the propagation speed and the attenuation constant of the sound mode are in agreement with the hydrodynamic predictions PhysRevA.6.2401 (); PhysRevB.22.2514 (); Delacretaz:2017zxd () at every value of the SSB scale. Instead, we find a disagreement between the diffusion constant of the diffusive mode and the value predicted by hydrodynamics. We will comment on this puzzle below. Our results indicate that a complete understanding of these holographic models, in terms of an hydrodynamic effective description, may still be lacking.
2 Holographic Set-up
with and is a coupling which relates to the graviton mass Alberte:2015isw (). Here, are Stückelberg scalars which admit a radially constant profile, with . In the dual field theory the Stückelberg fields represent scalar operators breaking the translational invariance of the system. Whether the translational symmetry breaking is explicit or spontaneous depends crucially on the bulk potential of the scalar fields. In particular, depending on the choice of the potential , the bulk solution may either correspond to the non-normalisable mode and hence it acts as a source term for the dual operators ; or it corresponds to the normalisable mode and hence it gives rise to a non-zero expectation value . In the following we will concentrate on potentials of the type
which in a simple way realise the spontaneous breaking of translational invariance Alberte:2017oqx ().
The thermal state is dual to an asymptotically AdS black hole geometry which in Eddington-Finkelstein (EF) coordinates is described by the metric
where is the radial holographic coordinate. The conformal boundary is located at and the horizon is defined by . Hence the emblackening factor takes the simple form
The corresponding temperature of the dual QFT reads
while the entropy density is . For simplicity, and without loss of generality, we will fix in the rest of the manuscript.
The heat capacity is , with being the energy density and being the temperature, and it has been studied in Baggioli:2015gsa (); Baggioli:2018vfc (). The dual field theory has been shown to possess viscoelastic features and was studied in detail in Baggioli:2015zoa (); Baggioli:2015gsa (); Baggioli:2015dwa (); Alberte:2016xja (); Alberte:2017cch (); Alberte:2017oqx (); Ammon:2019wci ().222The viscoelastic nature of these holographic systems and their features have recently inspired important results in the understanding of amorphous solids and glasses Baggioli:2018qwu ().
It is important to stress that our model only has one dimensionless scale controlling the strength of the spontaneous translational symmetry breaking. More specifically, we have
where the limiting case corresponds to the translationally invariant system.
In the following we consider fluctuations around the thermal equilibrium state and compute the quasi-normal modes. The transverse sector of the fluctuations has been studied in detail in Alberte:2017oqx () and it revealed the presence of damped transverse phonons whose speed are in perfect agreement with the theoretical predictions from hydrodynamics. Within this manuscript, we continue this analysis by considering instead the quasi-normal modes within the longitudinal sector.
3 Hydrodynamics and Longitudinal Quasi-Normal Modes
In this section we will study the hydrodynamic modes appearing in the longitudinal sector of the quasi-normal mode spectrum.333By “hydrodynamic modes” we mean the modes present in the system within the range , . Furthermore, we compare the dispersion relations of the quasi-normal modes to the corresponding, analytic, hydrodynamic expectations.
We defer the relevant equations of motion for the fluctuations and the technicality concerning the holographic computation to appendices A and B. Moreover, in appendix C we review and (partially) extend the hydrodynamic effective description of systems with broken translations following the lines of Delacretaz:2017zxd ().444All the physics of this section is already included in Delacretaz:2017zxd (); nevertheless we will display results beyond the specific approximations assumed in Delacretaz:2017zxd ().
3.1 Hydrodynamic Regime of Quasi-Normal Modes
First we analyse the quasi-normal modes in the low-frequency and long-wavelength regime. In particular the spectrum exhibits:
A pair of longitudinal damped sound modes with dispersion relation
with the speed and the diffusion constant .555Sometimes people refer to the sound attenuation constant which corresponds to . The ellipsis stands for higher momenta corrections. The mode (12) is exactly a longitudinal damped phonon mode, which is expected both in fluids and solids.
A diffusive mode with dispersion relation
where is the diffusion constant. The ellipsis stands for higher momenta corrections. The presence of such a diffusive mode is typical of systems which break translational invariance spontaneously. This hydrodynamic mode can be viewed as the diffusive mode of the Goldstone parallel to the momentum.
3.2 Correlators of the Goldstone Modes
Before comparing the numerically obtained dispersion relations of the low-lying quasi-normal modes to the expected dispersion relations of hydrodynamics, let us discuss the retarded Green’s functions which will be relevant for the future analysis. We are particularly interested in correlators appearing due to the SSB of translations, i.e. those containing the Goldstone fields which are dual to the Stückelberg fields in the bulk. To be more precise, we discuss the retarded Green’s functions and , where and (with ) are the two Goldstone modes and the momentum operators, respectively.
Using the hydrodynamic description of phases with spontaneously broken translational symmetry, found in PhysRevB.22.2514 (); Delacretaz:2017zxd () and repeated in our Appendix C, we can derive the form of the previous Green’s functions at low energy/frequency
where is the momentum susceptibility and relates to the Goldstone diffusion.
The holographic correlators can be derived from the equations for the fluctuations of the bulk fields using the standard holographic dictionary Skenderis:2002wp (); see Appendix B for further details, in particular regarding the mapping between the scalar Stückelberg fields and the Goldstone operators .
We have conducted numerical studies of the behaviour of the Green’s functions (14) within the context of our holographic model and for various potentials with . One example of the results is shown in figure 3. We find perfect agreement between the Green’s functions obtained from holography and from hydrodynamics. The transport coefficient can be read off from the imaginary part of the Green’s function as
where, in this limit, it is no longer necessary to to distinguish between or .666At zero momentum, , and for isotropic backgrounds, there is no distinction between the various directions. A plot of the parameter for various is shown in figure 4. Furthermore, the numerical results for are in perfect agreement with the horizon formula
given in Amoretti:2018tzw () and used at zero chemical potential .777Notice the different notations. The map between the conventions is (17)
3.3 Matching to Hydrodynamics
Let us compare the dispersion relations obtained from the quasi-normal mode analysis presented above with the theoretical expectations from hydrodynamics. For details regarding the hydrodynamic analysis see Appendix C.
3.3.1 Sound Mode
where is the energy density; is the thermodynamic pressure; and is the momentum susceptibility. Moreover, and are the bulk and shear elastic modulus, respectively. Both and vanish in the absence of translational symmetry breaking and the formula (18) reduces to the well-known speed formula for sound ; or in the special case of a -dimensional CFT.
Reminiscent of solids, we can re-express the speed of the longitudinal sound modes as
where is the thermodynamic bulk modulus obtained from the inverse of the compressibility
where is the volume of the system and is the mechanical pressure defined by (no sum over implied). Let us notice that, in our model
In other words, the first term in the above formula is a contribution which is finite even in the absence of SSB, and it is indeed what determines the speed of sound in a pure relativistic CFT. The second term captures additional effects due to spontaneous symmetry breaking.
Let us now determine the coefficients , and within our model. The elastic shear modulus is given in terms of the Kubo formula888For our definition of the retarded Green’s function see appendix B.
The bulk modulus can be determined in various ways. Using the underlying conformal invariance of the dual QFT, as well as the structure of the energy-momentum tensor, the bulk modulus reads Baggioli:2018bfa ()
In fact, using as well as999See also oriol () for more discussions on this point.
which is valid for two-dimensional conformal solids, we can check that the consistency relation (22) is indeed satisfied. Finally, we may also compute by a Kubo formula; for more details see appendix C.4.
in a two-dimensional conformal solid, for any value of the SSB parameter .
The results for the sound speed are shown in figure 5. In the left panel we show the value of from equation (18) as a function of the dimensionless symmetry breaking scale , for various potentials of the form . At we recover the CFT result , which acts as a lower bound for the speed of longitudinal sound. Increasing , the speed grows until it reaches a constant value at large . Depending on the choice of the potential, the speed of sound might become superluminal and the system can exhibit causality pathologies (see oriol ()). However, this is not the case for the choice , provided . In the right panel we compare the numerical data extracted from the QNMs and the theoretical formula written down in equation (18). Within the hydrodynamic limit the agreement is excellent.
Let us now move on and discuss the dissipative part of the phonon dispersion relation. Because of finite temperature effects, or in other words due to the presence of an event horizon and its associated shear viscosity , the longitudinal phonons obtain a finite diffusive damping . The same phenomenon occurs also in the transverse sector, the effects of which were studied in Alberte:2017oqx (). The interplay of elasticity (a propagating term) and viscosity (damping) qualify the system at hand to represent the gravity dual of a viscoelastic material. Hydrodynamics provides us with the formula for the diffusion constant, which reads
where we define the specific heat and the parameters , , and as
where is the Goldstone operator (parallel to ) of the dual field theory (not to be confused with the scalar bulk field ).
The equation for the diffusion constant can be considered as a sum of leading and sub-leading terms. The leading contribution, at , is
where the dots signify the sub-leading terms. The above expression reduces to the known result for sound damping, , in the absence of SSB Policastro:2002tn (); Baier:2007ix (). In figure 6 we compare the behaviour of the dimensionless damping constant with the numerical results computed for the potential with , as functions of the symmetry breaking scale . As expected, the leading order result (32) represents a good approximation only for small values of .
Note that at large the damping coefficient goes to zero and the phonons become purely propagating modes. In other words, at zero temperature no dissipation is present. The results are analogous for various , see figure 7. The tendency is a smooth decrease when going to small temperature, which is consistent with the hypothesis that no dissipation can be at work at ; thus, in this limit our system can be considered a “perfect elastic solid.”
We verified numerically, using the Kubo formulas (30) and (31), that both coefficients and are zero in our holographic model, which is in agreement with Kim:2016hzi ().101010We thank Keun-Young Kim for explaining us this point. Thus, the full expression for the diffusion constant of the sound modes reduces to
In figure 8 we plot this improved formula next to the numerical data, for . It is evident that the complete formula, including the sub-leading terms in equation (33), is now in very good agreement with the numerical data.
3.3.2 Diffusive Mode
As mentioned above, spontaneous breaking of translational invariance gives rise to a non-propagating diffusive mode in the longitudinal sector.111111Notice that such a mode does not appear when translations are broken explicitly as in Davison:2014lua (). For an effective hydrodynamic description of this mode we refer the reader to PhysRevA.6.2401 (); PhysRevB.22.2514 (); Delacretaz:2017zxd () or to our appendix C. This additional diffusive mode was first observed numerically in the context of holography Andrade:2017cnc () and later discussed in Baggioli:2019aqf (). In this subsection we will study this mode, in particular its diffusive constant .
The numerical results for the dimensionless form of the diffusion constant are shown in figure 9, for a range of values of the power . We notice that the value of is finite even at , suggesting that its nature can be understood entirely in the limit when the Goldstone modes decouple.121212Similar observations appeared in Amoretti:2018tzw (). Moreover, decreases monotonically when increasing the SSB parameter .
In Appendix C we derive the following analytic expression for the diffusion constant
In the case of , equation (34) displays a leading contribution
where the dots represent corrections. Surprisingly, we find that the complete hydrodynamic formula in expression (34) does not agree with the numerical quasi-normal mode analysis, which is evident in the plot in the right panel of figure 10.
The discrepancy between the numerical data and the analytic expression for points towards a disagreement between the hydrodynamic predictions and the holographic results. Below we discuss some possible reasons behind this issue. Let us first analyse the various parameters entering in the formula (34):
As stated in section 3.3.1, the coefficients and are expected to vanish in absence of a finite charge density. Our numerical checks show that this is consistent in our model.
The bulk modulus has been computed in the literature and is, additionally and successfully, tested in this manuscript using the speed of longitudinal sound.
In summary, we do not believe that the discrepancy between the hydrodynamic result and the holographic QNM data can be caused by a mistake in the computation of the parameters appearing in formula (34).
Let us continue by mentioning two interesting observations. The first requires a simple bulk analysis of the diffusion coefficient done at small . In the limit of very small graviton mass we can assume that the scalars decouple from the gravitational dynamics. Denoting the fluctuations parallel and transverse to the momentum by and , respectively, the equations of motion for the fluctuations take the simple form
The above equations coincide when , since at that value the distinction between transverse and parallel directions is no longer well defined. In the decoupling limit the fluctuations in both sectors display a diffusive mode with dispersion relation
The perpendicular diffusive constant reads
which can be used to immediately compute the diffusion constant of the longitudinal diffusion mode as
The plot shown in the left panel of figure 10 indicates that the formula (41) is in good agreement with the numerical data in the limit of soft SSB, roughly . It is important to note that a missing factor of is not able to fix the disagreement between hydrodynamics and the holographic results at arbitrary values of . This is clear from the left panel of figure 10 where the discrepancy is still present at values of .
The second observation appears when we apply a constant shift to the hydrodynamic formula (34). The predictions of hydrodynamics combined with such a shift reproduces the numerical values of the diffusion constant very well, even until large values of (see right panel of figure 10).
We are currently unable to find the precise reason for the mismatch described above. Assuming the correctness of our computations, we cannot discard the possibility that the hydrodynamic description is missing some effect encoded in a novel transport coefficient. At the same time, it is not guaranteed that the holographic models considered in this work are the exact duals of a system with spontaneously broken translations, as those described by hydrodynamics. Our results are nevertheless surprising given the large amount of verifications in this direction.
4 Quasi-Normal Modes Beyond Hydrodynamics
In the previous section we investigated hydrodynamic modes, i.e. poles of the retarded Green’s function located at . Hence, these modes determine the late time and long distance behaviour of the system.
Using holographic techniques we can study further, so-called non-hydrodynamic, poles of the Green’s functions. These poles correspond to higher quasi-normal modes which we determine numerically. In particular we are interested in the behaviour of the higher quasi-normal modes as functions of the SSB scale , and momentum .
The results for the potential are shown in figure 11. The left panel of the figure displays the tower of quasi-normal modes at zero momentum while dialing the dimensionless parameter . For any given ratio there is a clear gap in the system. Moreover, as the mass is increased all the higher non-hydrodynamic modes move away from the origin of the complex plane and the imaginary part of the frequency grows. Hence the late time and long distance behaviour is determined by the three hydrodynamic modes which we investigated in the previous section. The other poles do not give rise to quasi-particles due to their short lifetime.
In the right panel of figure 11 we show the quasi-normal mode spectrum as a function of the momentum . In addition, the more refined results for the first seven modes are shown in figure 12. Note that the dynamics of the higher quasi-normal modes as a function of the momentum appears to be quite complicated. Especially noticeable is the large imaginary part of the QNM depicted in blue in figure 12 (corresponding to the diffusive mode in the hydrodynamic regime) upon increasing .
Finally, we analyse the large momentum behaviour of the quasi-normal modes corresponding to the two hydrodynamic sound modes. In particular, we consider the case which clearly exceeds the hydrodynamic regime. As discussed above, at small momenta the expected dispersion relation is given by (12). On the contrary at very large momenta we obtain the dispersion relation of a sound mode within the relativistic UV AdS fixed point. In this limit, the real part of the dispersion relation is given by .
In contrast to the sound mode in the transverse sector (see Ammon:2019wci ()), the real part of the frequency of the sound mode never becomes zero; in fact, as shown in 12 the real part of the dispersion relation of the sound mode interpolates smoothly between in the hydrodynamic limit and in the large momentum limit.
In this paper we studied the longitudinal quasi-normal mode spectrum of a simple holographic system with spontaneously broken translational symmetry, at zero charge density. We successfully identify the three hydrodynamic modes: the pair of longitudinal sound modes and a diffusion mode. We compared the numerical results obtained from the holographic model with the predictions of hydrodynamics given in Delacretaz:2017zxd () and in our appendix C.
The numerical results for the propagation speed and the attenuation constant of the sound modes are in agreement with the hydrodynamic formulas at any strength of the SSB, at arbitrary graviton mass. Surprisingly, we find disagreement between the diffusion constant of the diffusion mode obtained numerically and the formula given by hydrodynamics. At this stage we are not able to identify the reason behind this puzzle.
Some comments are in order. Mistakes present in our computations are of course a possibility. A second, more interesting, option is that the hydrodynamic framework considered is missing some effect. A lacking transport coefficient, for example, could potentially explain the discrepancy we observe. A last scenario is that the holographic model discussed in this work is not the gravity dual of the type of systems described by the hydrodynamic picture, i.e. a dissipative and conformal system with spontaneously broken translational invariance. Needless to say, finding the reason for this discrepancy is certainly the most important open question that our manuscript poses. It would be interesting to perform the same check in similar holographic models Andrade:2017cnc (); Grozdanov:2018ewh (); Amoretti:2018tzw (); Baggioli:2018bfa ().
As a final remark, it would be beneficial to gain a better understanding regarding the propagation of sound and the elastic properties of quantum critical materials, both from EFT methods Alberte:2018doe () and holographic methods oriol (). There are several recent hints suggesting that phonons, viscoelasticity and glassy behaviours may play an important role in quantum critical systems ishii2019glass () and High-Tc superconductors setty2019glass (). Moreover, electron-phonon coupling might play an important role in the high-Tc puzzle He62 ().
In conclusion, we have in this paper continued the study of simple holographic massive gravity models with spontaneously broken translational symmetry. We have tested several predictions obtained from the hydrodynamic theory of such systems. Finally, we provided food for thought in the form of a discrepancy between the hydrodynamic and holographic theories concerning the dispersion relation of the diffusive mode. In this case, we would be very happy to be proven wrong, and even more happy to find a novel and interesting physical reason behind the mismatch we observed.
We thank Mike Blake, Alex Buchel, Markus Garbiso, Blaise Gouteraux, Saso Grozdanov, Matthias Kaminski, Keun-Young Kim and Napat Poovuttikul for useful discussions and comments about this work and the topics considered. We are particularly grateful to: Daniel Arean for sharing with us precious insights and help us deciphrate criptic papers on the topic; Amadeo Jimenez Alba for collaboration on closely related topics and for never-ending critical discussions; Oriol Pujolas and Victor Cancer Castillo for collaboration on closely related topics and for sharing with us unpublished results.
MA is funded by the Deutsche Forschungsgemeinschaft (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. SG gratefully acknowledges financial support by the Fulbright Visiting Scholar Program, which is sponsored by the US Department of State and the German-American Fulbright Commission in 2018 and by the DAAD (German Academic Exchange Service) for a Jahresstipendium für Doktorandinnen und Doktoranden in 2019.
MA would like to thank the Erwin Schrödinger International Institute for Mathematics and Physics as well as NORDITA for the hospitality during the completion of this work. MB would like to thank MIT, Perimeter Institute, University of Montreal, University of Alabama and Universidad Autonoma de Mexico for the warm hospitality during the completion of this work. MB would like to thank Marianna Siouti for the unconditional support.
Appendix A Equations of Motion
In this work we determine the spectrum of quasi-normal modes in the longitudinal sector. Choosing the momentum along the -direction, the relevant time-dependent fluctuations are and . We choose to work in radial gauge i.e. , with . Since we work in in-falling Eddington-Finkelstein coordinates, the fluctuations already satisfy in-going horizon conditions. The Fourier transformed equations of motions to first order in the fluctuations read
where we introduced the short-hand notations and for the symmetric combination and the anti-symmetric combination , respectively. In terms of these combinations it is straightforward to solve equation (49) by integrating twice, . In order to determine the integration constants, we compare the solution to the near the boundary expansion at the conformal boundary given by
In the context of the QNM computations we do not allow for sources of fluctuations and have to vanish leading to the trivial solution .131313In order to compute the Kubo formulas we have to source some of the metric functions; in cases where we source or we can not set to zero. In this case, the equations of motion simplify to
Appendix B Numerical Techniques
with and being differential operators. The quasi-normal modes are the complex frequencies to which is a regular solution to equation (58) (and fulfills the constraint equations). For given and , we may solve the eigenvalue problem by means a of pseudo-spectral method on a Gauß-Lobatto grid, as explained in Grieninger:2017jxz (); Ammon:2016fru (). Note, that the sources of the fluctuations have to vanish which we implement by a suitable redefinition of the fields.
We checked that our solutions fulfill the equations of motions and all constraint equations. To check the convergence of the numerical solution, we monitor the change of the solution for finer discretizations. As depicted in the l.h.s. of figure 13, the change of the quasi-normal mode frequency decays exponentially with a growing number of grid points; the same is valid for the corresponding eigenfunctions. Another check for the numerical method are the Chebychev-coefficients of the solution, displayed in the r.h.s. of figure 13; the coefficients decay exponentially, indicating exponential accuracy of our numerical method.
In order to calculate the correlation functions, outlined in equation (14), we have to source the scalar field explicitly. Note, that for it is sufficient to consider the sector since the and decouple. The corresponding asymptotic expansions are of the form141414We assume that . For the term proportional to corresponds to the expectation value and the logarithmic term is proportional to the same power in .
where we set and . Note, that the logarithmic terms are not present for and . In order to ensure convergence in presence of logarithmic term (e.g. in the cases ), we use a suitable mapping for the radial coordinate , for instance (for more details see Grieninger:2017jxz (); Ammon:2016fru ()).
The retarded Green’s function of operators and is given by
The Fourier-transformed retarded Green’s function reads151515From now on, we drop the tilde to simplify notation.
and it describes the linear response to for a given perturbation by161616The coupling of to is described by replacing the QFT Lagrangian by
Using holographic techniques the retarded Green’s functions are given by
where , , and are the boundary values as given in (59). The prefactors of the Green’s function may be explained as follows: the multiplicative factor arising in holographic renormalisation reduces to for the -dimensional QFT considered here. The factors are due to the non-canonical normalisation of the kinetic term for the fluctuations of the scalar field. In particular, the prefactor of the kinetic term reads where denotes evaluated on the background.
Next we determine the correct normalization of the Goldstone operator. Note that the Goldstone operator has to satisfy , while on the gravity side we numerically confirmed
Hence we can establish the following map between the Goldstone operator , on the quantum field theory side, and the Stückelberg field on the gravity side,
Appendix C The Hydrodynamic Theory
In this section we present the hydrodynamic calculations which lead to the dispersion relations used in this paper. We follow the reasoning of Delacretaz:2017zxd () and expand on their results, focusing on the case of zero density. First we present the effective hydrodynamic theory and then compute the dispersion relations in the transverse and longitudinal sector.
We will consider the linearised hydrodynamics of small fluctuations around a dimensional system at equilibrium. Without loss of generality we only allow excitations to depend on and , hence the momentum is in the -direction. We also adopt the convention of denoting the -component of a quantity by , and the -component by , to indicate the orientation with respect to the momentum.
The hydrodynamic variables which are of interest for our analysis are
for, in order, energy density; momentum density parallel and transverse to the momentum; and , are the divergence and curl of the two-component Goldstone field , respectively (not to be confused with the scalar bulk fields ). To be more precise, for the system at hand, the field is a vector of the form
From the above Goldstone field we define
which reduces to and in the case of -independent fluctuations.
The sources of the theory are
which denote the temperature; the parallel and transverse components of ; and linear displacements parallel and transverse to the momentum, respectively. The ordering of the above sources indicates the corresponding variable when compared to the list (69).
The hydrodynamic variables, collectively denoted , are related to the their corresponding sources, , through the thermodynamic susceptibilities via the following equation
In the linear hydrodynamic regime the susceptibilities will be between small fluctuations of the variables and sources. Hence, for the system under consideration, the above relations can be expressed in matrix form as follows