Hydrodynamic diffusion and its breakdown near AdS 2 fixed points
Daniel Arean, Richard A. Davison, Blaise Goutéraux, Kenta Suzuki
CCPHT-089.112020, IFT-UAM/CSIC-20-165
Hydrodynamic diffusion and its breakdown near AdS fixed points Daniel Are´an ∗ Instituto de F´ısica Te´orica UAM/CSIC and Departamentode F´ısica Te´orica, Universidad Aut´onoma de MadridCampus de Cantoblanco, 28049 Madrid, Spain
Richard A. Davison † Department of Mathematics and Maxwell Institute for Mathematical Sciences,Heriot-Watt University, Edinburgh EH14 4AS, U.K.
Blaise Gout´eraux ‡ and Kenta Suzuki § CPHT, CNRS, Ecole polytechnique,IP Paris, F-91128 Palaiseau, France (Dated: November 26, 2020)
Abstract
Hydrodynamics provides a universal description of interacting quantum field theories at suffi-ciently long times and wavelengths, but breaks down at scales dependent on microscopic details ofthe theory. We use gauge-gravity duality to investigate the breakdown of diffusive hydrodynamicsin two low temperature states dual to black holes with AdS horizons. We find that the breakdownis characterized by a collision between the diffusive pole of the retarded Green’s function with apole associated to the AdS region of the geometry, such that the local equilibration time is setby infra-red properties of the theory. The absolute values of the frequency and wavevector at thecollision ( ω eq and k eq ) provide a natural characterization of all the low temperature diffusivities D of the states via D = ω eq /k eq where ω eq = 2 π ∆ T is set by the temperature T and the scalingdimension ∆ of an infra-red operator. We confirm that these relations are also satisfied in an SYKchain model in the limit of strong interactions. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] a r X i v : . [ h e p - t h ] N ov ONTENTS
I. Introduction 3II. Diffusive hydrodynamics 7III. Diffusion in a neutral holographic state 7A. Hydrodynamic mode 8B. Infra-red modes 9C. Breakdown of hydrodynamics 9IV. Diffusion in a charged holographic state 11V. Comparison with SYK chain 13VI. Outlook 15Acknowledgments 16References 17A. Perturbations in the neutral translation-breaking model 231. Gauge-invariant perturbations 232. Near-horizon perturbation equations 253. Low temperature dispersion relation 26a. Leading order solution 27b. First order correction when Im( ω ) > − πT ω ) 304. Analytic description of pole collisions 32a. Leading order ingoing solution 33b. Correction to the leading order solution 34c. Low temperature limit of the Green’s function 365. Comparison with numerical results 39B. Perturbations of AdS -Reissner-Nordstrom 441. Gauge-invariant perturbations 442. Numerical results 45C. SYK chain model 48 I. INTRODUCTION
Interacting quantum field theories are notoriously challenging, especially when there isno quasiparticle-based description of the state. To describe the late time, long wavelengthdynamics of these states, one can instead rely on effective approaches such as hydrodynamics.This approach has been used to gain insight into both the quark-gluon plasma [1–3] andelectronic transport in metals [4–10].At long times and wavelengths, hydrodynamics provides an effective description of a sys-tem in terms of a few conserved quantities dictated by symmetries [11–13]. Their evolutionis governed by local conservation equations for the densities and associated currents, alongwith constitutive relations expressing the currents in terms of the densities in a gradientexpansion. The late time relaxation of the system back to equilibrium is governed by thehydrodynamic modes: poles of the retarded Green’s functions of the densities with gaplessdispersion relations [14].While extremely powerful, hydrodynamics breaks down at sufficiently short scales setby the local equilibration time and length. At such scales, the dynamics of the systemcan no longer be truncated to just the evolution of the conserved densities. Additionaldegrees of freedom play a significant role, and appear as additional poles of the retardedGreen’s function with lifetimes comparable to those of the hydrodynamic modes. In caseswhere the density response exhibits a parametrically slow mode arising due to a weaklybroken symmetry, the breakdown of hydrodynamics manifests itself as a collision in thecomplex frequency plane at nonzero wavevector k eq between the hydrodynamic mode andthe slow mode. In this case, it is often possible to augment the hydrodynamic description toincorporate this slow mode [4, 15–23]. But typically the modes relevant for the breakdownof hydrodynamics are not of this nature, and a more complete knowledge of a system’smicroscopic details is required to understand them.Gauge-gravity duality can be exploited to address this fundamental question, by mappingthe late time dynamics of certain large N c quantum field theories (where N c is the rank3f the gauge group) to theories of gravity with a negative cosmological constant [13, 24,25]. The relaxation of conserved densities back to equilibrium is captured exactly by theevolution of perturbations of asymptotically anti de Sitter (AdS) black holes, which canbe studied to obtain a precise understanding of the breakdown of hydrodynamics and themodes responsible for it [26, 27].Even in the absence of a weakly broken symmetry, the breakdown of hydrodynamics canbe characterized by an energy scale ω eq and wavenumber k eq , which are sensitive to thesystem’s microscopic details. ω eq and k eq are defined as the absolute values of the complexfrequency ω and complex wavenumber k at which the hydrodynamic pole of the retardedGreen’s function first collides with a non-hydrodynamic pole [28–30]. The convergenceproperties of the real-space hydrodynamic gradient expansion are governed by k eq [32, 33],which also coincides with the radius of convergence of the small- k expansion of the hydro-dynamic dispersion relation ω hydro ( k ). See [34–37] for recent applications of this.In this work, we study the breakdown of hydrodynamics in certain low temperature( T ) states dual to black holes with nearly-extremal AdS × R near-horizon metrics. Suchstates are closely related to the Sachdev-Ye-Kitaev (SYK)-like models of electrons in strangemetals, which are governed by the same type of infra-red fixed point in the limit of largenumber of fermions and strong interactions [38–49]. Specifically, we study the AdS neutral,translation-breaking black brane of [50, 51] and the AdS -Reissner-Nordstr¨om (AdS -RN)black brane, and the breakdown of the hydrodynamics governing the diffusive transportof energy, charge and momentum in their dual states. The states we are interested in donot include any slow modes in the sense described above. Instead, local equilibration iscontrolled by the incoherent dynamics that follow from the presence of an AdS × R fixedpoint. We identify simple, general results for the local equilibration scales ω eq and k eq andconfirm that these also apply to the SYK chain model studied in [37] in the limit of stronginteractions.Our first result is that the breakdown is caused by modes associated to the AdS regionof the geometry, and as a consequence ω eq is set by universal (i.e. infra-red) data via ω eq → π ∆ T as T → , (1) Non-analyticities due to interactions between hydrodynamic modes provide an independent mechanismfor the breakdown of hydrodynamics (see e.g. [12] for a review), but these are expected to be suppressedin the large N limit [31]. ω eq (cid:28) T . More precisely, we find that at small k and T the Fourier space locations of thelongest-lived non-hydrodynamic poles are inherited from infra-red Green’s functions, and arelocated at ω n = − i ( n + ∆)2 πT for non-negative integers n . The breakdown is characterizedby a collision, parametrically close to the imaginary ω axis, between the n = 0 mode (whichhas a weak k -dependence) and the hydrodynamic mode.Secondly, we find that at low temperatures the corrections to the quadratic approximation − iDk to the exact hydrodynamic dispersion relation are parametrically small such that thecollision occurs when k is almost real and k eq → ω eq D as T → . (2)In other words, the scales k eq and ω eq governing the regime of validity of hydrodynamicsare set simply by the diffusivity D and the scaling dimension ∆. In some of the exampleswe study (those involving diffusion of energy), the relevant diffusivity is controlled by anirrelevant deformation of the AdS fixed point and in these cases the result (2) indicates that k eq is controlled by the same irrelevant deformation. A priori, the result (2) is quite surpris-ing: it relates the radius of convergence to just the leading order term in the hydrodynamicexpansion. This is a consequence of the AdS fixed point.By rearranging equation (2) we obtain an answer to the question raised in [5] of whatthe underlying velocity and time scales are that govern the diffusivity in non-quasiparticlesystems. In all our examples they are set by the local equilibration scales D → v eq τ eq , as T → , (3)where v eq ≡ ω eq /k eq and τ eq ≡ ω − eq are the velocity and timescale associated to local equili-bration.In the cases where diffusive hydrodynamics breaks down due to a parametrically slowmode protected by a weakly broken symmetry, D is typically set by τ eq and the speed ofthe propagating mode that dominates following the breakdown. We emphasize that thebreakdown of hydrodynamics is qualitatively different in the cases we study: there is not Unlike here, τ eq is often defined by the lifetime of a k = 0 mode. In our conventions, these examples have D → v eq τ eq / modes with parametrically similar lifetimes setby ω n , and the breakdown does not produce a propagating mode with velocity v (cid:39) v eq .More generally, the local equilibration time has been argued to set an upper bound on thediffusivity in [52, 53]. All examples that we study are consistent with a bound of the form D (cid:46) v eq τ eq for the range of parameters we have investigated.In the absence of a slow mode, it was proposed that low temperature diffusivities are setby the butterfly velocity v B and Lyapunov time τ L that characterize the onset of scramblingfollowing thermalization of the system [54]. This was shown to robustly apply to the diffu-sivity of energy density D ε in holographic theories and SYK-like models [47, 48, 55–58]. Forthe examples we study, τ − L = 2 πT and D ε → v B τ L as T → , (4)which is furthermore true in general for states governed by an infra-red AdS with theuniversal deformation [56].As for our result (3), equation (4) can be viewed as a consequence of the excellent appli-cability of the quadratic approximation to the exact hydrodynamic dispersion relation upto the relevant scale. Specifically, pole-skipping analysis suggests that the energy diffusionmode satisfies ω hydro ( k = iv − B τ − L ) = iτ − L [59–61], from which (4) follows assuming correc-tions to the quadratic, diffusive form − iD ε k at k = iv − B τ − L are parametrically small as T → ω eq , k eq ), while only the energy diffusionmode satisfies the pole-skipping constraint above [62]. As a consequence we provide a newperspective on, and generalization of, the relations between equilibration, transport andscrambling and their applications in AdS /SYK-like models of electrons in strange metals.In the remainder of this work, we explain how we arrive at equations (1) and (2) beforeclosing with comments on implications and the more general applicability of our results.6 I. DIFFUSIVE HYDRODYNAMICS
The spectrum of hydrodynamic modes is dependent on the system under consideration,and by our definition each hydrodynamic mode has its own associated local equilibrationscales ω eq and k eq . We will focus on hydrodynamic diffusion modes, which arise when asystem has a current density j with constitutive relation j ( ρ ) = − D ρ ∇ ρ + O ( ∇ ) , (5)where ρ is the corresponding conserved density. ρ will then obey the diffusion equationwith diffusivity D ρ at leading order in the derivative expansion, and has a retarded Green’sfunction [12, 14] G ρρ ( ω, k ) = D ρ χ ρρ k + . . . − iω + D ρ k + . . . , (6)where χ ρρ ≡ lim ω → G ρρ ( ω, k ) is the static susceptibility of ρ , and ellipses denote terms withhigher powers of ω and k . The dispersion relation of the hydrodynamic diffusion mode isthen ω hydro ( k ) = − iD ρ k + O ( k ) . (7)Hydrodynamic diffusion is a very general phenomenon. Even within the restricted class ofsystems that we study, the set of conserved densities that exhibit hydrodynamic diffusionvaries. In this work, we will be interested in the diffusion of energy ε and of transversemomentum Π. III. DIFFUSION IN A NEUTRAL HOLOGRAPHIC STATE
We begin with the AdS neutral translation-breaking model [50, 51] which is a classicalsolution of the action S = (cid:90) d x √− g (cid:32) R + 6 − (cid:88) i =1 ( ∂ϕ i ) (cid:33) , (8)with spacetime metric ds = − r f ( r ) dt + r dx + dr r f ( r ) , (9)supported by two scalar fields ϕ i = mx i ( i = 1 ,
2) that break translational symmetry. Theemblackening factor of the solution is f ( r ) = 1 − m r − (cid:18) − m r (cid:19) r r , (10)7here r denotes the location of the horizon with associated temperature T . The linearperturbations can be written in terms of four decoupled variables and we will focus on theone exhibiting the single hydrodynamic mode of the system. See Appendix A for furtherdetails on this spacetime, and on the calculations leading to the results below. A. Hydrodynamic mode
The hydrodynamic mode corresponds to diffusion of energy, with the small k dispersionrelation (7) and diffusivity D ε → (cid:112) / m − in the low T limit [15].First we quantify corrections to this result that will enable us to understand the break-down of hydrodynamics. In the low temperature limit T ∼ k ∼ (cid:15) (cid:28)
1, the retarded Green’sfunction of energy density G εε exhibits a pole located at ω ( k ) = − i(cid:15) (cid:114) k m (cid:18) (cid:15) k m + (cid:15) (cid:18) πT m + k m (cid:19) + . . . (cid:19) , (11)where we have explicitly written all (cid:15) dependence. For suitably small k , this is an approxima-tion to the hydrodynamic dispersion relation. It becomes invalid near specific wavenumbers k = k n related to the breakdown of hydrodynamics, which will be addressed shortly.It is important to note that (11) is different than the hydrodynamic expansion: correctionsto the quadratic k term are not being neglected as in the usual gradient expansion, butare parametrically small in this limit under consideration. One consequence of this is thatif we define any wavenumber k ∗ with k ∗ ∼ T at low T , and define ω ∗ to be the locationof the hydrodynamic pole at this wavenumber, then (11) implies D ε → iω ∗ /k ∗ as T → k ∗ = iv − B τ − L results in the chaos relation (4) as described in theIntroduction. We will soon show that the breakdown of hydrodynamics at low T is characterized bya pole collision at k eq ∼ T, ω eq ∼ T and thus the diffusivity can alternatively be expressedsimply in terms of these scales by (2). But prior to exploring the pole collision that char-acterizes the breakdown of hydrodynamics, it is instructive to first understand the origin ofthe non-hydrodynamic mode responsible. See Figure 2 of [61] for a visual representation of this. . Infra-red modes At low T the state is governed by an infra-red fixed point manifest in the emergence of anear-horizon AdS × R metric with SL(2,R) symmetry (see Appendix A). Each linear per-turbation of the spacetime can be characterized by ∆( k ), a wavenumber-dependent scalingdimension of the corresponding operator with respect to this infra-red fixed point, and acorresponding infra-red Green’s function [63, 64] G IR ∝ T k ) − Γ (cid:0) − ∆( k ) (cid:1) Γ (cid:0) ∆( k ) − iω πT (cid:1) Γ (cid:0) + ∆( k ) (cid:1) Γ (cid:0) − ∆( k ) − iω πT (cid:1) . (12)For the spacetime perturbation that exhibits a diffusive mode, ∆( k ) = (1+ (cid:112) k /m ) / G εε from G IR is not easy, for our purposes it isenough to observe that G εε exhibits poles whose locations approach those of the poles of G IR as k, T →
0. Specifically, this means that in this limit G εε exhibits poles at ω n = − i πT ( n + ∆(0)) , n = 0 , , , . . . , (13)with ∆(0) = 2.At low T , the infra-red modes (13) have a parametrically longer lifetime than the othernon-hydrodynamic poles of G εε , and are responsible for the breakdown of hydrodynamics.The wavenumbers at which our calculation of the low T dispersion relation (11) of thehydrodynamic mode is invalid are k n = (cid:112) / n ) πmT + O ( T ), for which the modewould be located at precisely ω ( k n ) = ω n in the limit of low T . The natural interpretationwould therefore be that the invalidity of the calculation at these values of k can be tracedto the nearby presence of the infra-red mode (assuming that the location of the infra-redmode has a weak k -dependence). A more refined calculation below confirms this, as well asthe existence of a collision between these modes for complex k that signals the breakdownof hydrodynamics. C. Breakdown of hydrodynamics
In order to extract the existence of the pole collision, a more refined perturbative com-putation of G εε at the points ω = ω n + δω, k = k n + δ ( k ) is required. This yields G − εε ( ω, k ) ∝ (cid:0) D n δ ( k ) − iδω (cid:1) (1 − iτ n δω ) − iλ n δω, (14)9
20 40 60 80 100 1200.00.51.01.52.02.5 k / T - Im ( ω ) π T / T - Im ( ω ) π T FIG. 1. Left: Frequencies of the hydrodynamic and longest lived infra-red modes at
T /m = 10 − .Black circles are numerical results and red lines are the analytic expressions (11) and (13). Right:close-up of region near ω , with red lines showing the dispersion relations extracted analyticallyfrom (14). The pole collision is not visible on this plot as it happens at a complex value of k . where we show only terms relevant for understanding the collision. The low T limit of eachcoefficient is τ n → m √ n ) π T , λ n → (cid:114)
32 ( n ( n + 4) + 3) πTm , (15)while D n → D ε in the same limit. The comparable size of the δω and τ n ( δω ) terms atfrequencies δω ∼ T indicates that for such frequencies G εε is dominated by two poles,whose dispersion relations are given by solving the quadratic equation (14) for δω ( δk ). InFigure 1 we show that indeed (14) correctly describes the locations of the two poles near ω . The poles will collide (coincide in Fourier space) at the complex value of δk where thediscriminant of the quadratic polynomial vanishes.The collision closest to the origin of k -space ( n = 0) signals the breakdown of hydrody-namics. The absolute value of k and ω at this collision are (as T → ω eq ≡ | ω collision | → πT (cid:32) √ πT m + . . . (cid:33) ,k eq ≡ | k collision | → ω eq D ε (cid:32) − √ πT m + . . . (cid:33) , (16)from which our main results (1) and (2) follow.The collision location asymptotically approaches real (imaginary) values of k ( ω ) as T →
10. More precisely, as T → k and ω at the collision point are φ k → / (cid:18) πTm (cid:19) / , φ ω → − π φ k , (17)where k collision = k eq e iφ k and ω collision = ω eq e iφ ω .Note that even after the pole collision formally indicating the breakdown of hydrodynam-ics, Figure 1 illustrates that the system continues to exhibit a diffusion-like mode describedextremely well by the dispersion relation (11). IV. DIFFUSION IN A CHARGED HOLOGRAPHIC STATE
The AdS -RN solution to Einstein-Maxwell gravity S = (cid:90) d x √− g (cid:18) R + 6 − F (cid:19) , (18)has a metric of the form (9) but supported by a radial electric field A t = µ (cid:16) − r r (cid:17) , (19)such that the emblackening factor is f ( r ) = 1 − (cid:18) µ r (cid:19) r r + µ r r . (20)This solution represents a translationally invariant state with U (1) chemical potential µ ,and its linear perturbations can be written in terms of four decoupled variables. Furtherdetails of the solution and the calculations underlying our results are given in Appendix B.The state exhibits two independent diffusive hydrodynamic modes, each associated toa different such variable. The first, corresponding to diffusion of energy and U (1) chargewith diffusivity D ε , is analogous to the diffusive mode of the previous section. The secondcorresponds to the transverse diffusion of momentum with diffusivity D Π .The variables exhibiting each of these modes have k → ε (0) = 2 , ∆ Π (0) = 1 , (21) In the low T limit both such modes have D ε = κ/c ρ with κ the open circuit thermal conductivity and c ρ the heat capacity [23, 56]. - - / μω eq π Δ T - - / μ k eq2 D ω eq FIG. 2. Numerically obtained local equilibration data for diffusive hydrodynamics in G εε (blackcircles) and G ΠΠ (red squares) of the charged state. and numerical calculations confirm that at small k and T each corresponding retardedGreen’s function exhibits non-hydrodynamic poles at the locations (13). As before, a colli-sion close to the imaginary ω axis between the longest-lived such pole and the hydrodynamicpole signifies the independent breakdown of hydrodynamics in each case.At low T (and until the collision occurs) both hydrodynamic modes are described ex-tremely well by the quadratic approximation to the hydrodynamic dispersion relation, whilethe locations of the longest-lived non-hydrodynamic poles depend only very weakly on k .As a consequence, the equilibration scales in both cases are set by the simple formulae (1)and (2) as shown in Figure 2.The phase of the collision wavenumber φ k in each case is shown in Figure 3, illustratingthat the collision point asymptotically approaches real values of k as T →
0. Both caseshere, and the result (17) in the previous example, are consistent with ∆ controlling the low T scaling of the phase via φ k ∼ T ∆ − / .As in the previous example, the system continues to exhibit diffusion-like modes evenafter the collision formally indicating the breakdown of hydrodynamics. The dispersionrelations of these modes are extremely well approximated by the quadratic approximationto diffusive hydrodynamics, as shown in Figure 9 of Appendix B.12 ■ ■ ■ ■ ■ ■ ■ ■ ■ - - - - / μϕ k FIG. 3. Numerically obtained φ k for G εε (black circles) and G ΠΠ (red squares) of the chargedstate. Dashed lines shows the best-fits to a power law at small T : φ k = 19 . T /µ ) . and φ k =3 . T /µ ) . respectively. V. COMPARISON WITH SYK CHAIN
The SYK model is a (0+1)-dimensional theory of N interacting fermions that, in thelimit of large N and strong interactions, is governed by the same effective action as a theoryof gravity in a nearly-AdS spacetime [38–46]. The SYK chain [47] is a higher-dimensionalgeneralisation of this, which has served as a very useful toy model for studying diffusive en-ergy transport in strange metal states of matter. As it exhibits the local quantum criticalitycharacteristic of AdS × R fixed points, it is natural to ask whether local equilibration inthis explicit microscopic model is governed by our general results (1) and (2).In [37], an SYK chain model with N Majorana fermions per site χ i,x and Hamiltonian H = i q/ M − (cid:88) x =0 (cid:88) ≤ i <...
1. In this limit, the longest-lived non-hydrodynamicmodes are a series of infra-red modes located at precisely the frequencies ω n of equation (13)with ∆ = 2 as k →
0. Hydrodynamics breaks down due to a collision between the hydrody-namic mode and the longest-lived of these infra-red modes, and in Figure 4 we confirm thatthe local equilibration scales are given simply by ω eq → π ∆ T and k eq → ω eq D ε as v → , (23)analogously to (1) and (2). Unlike in the holographic examples, the pole collision at stronginteractions here happens for real k (i.e. φ k = 0). It would be interesting to determinewhether finite 1 /q corrections generate a small non-zero phase φ k . Consistently with theother examples we have presented, following the formal breakdown of hydrodynamics thespectrum still contains a mode whose dispersion relation is very well approximated by thequadratic approximation to diffusive hydrodynamics.14 I. OUTLOOK
There is good reason to expect that at least some of our results will generalise beyondthe specific examples studied here to other states governed by AdS infra-red fixed points.While our key observation that the quadratic approximation to the hydrodynamic dispersionrelation works parametrically well even for wavenumbers k ∼ T seems unusual, it is non-trivially consistent with the result (4) that is indeed true for holographic AdS × R fixedpoints with a universal deformation [56] as well as in related SYK chain models [47, 48].There are more general holographic and SYK-like systems governed by AdS fixed pointsthat exhibit additional diffusive modes beyond the two types we have studied. Of partic-ular interest are non-translationally invariant systems with a U (1) symmetry, for which anEinstein relation relates the electrical resistivity to a diffusivity [5, 48, 56]. If our results (1)and (2) extend to such modes, they will therefore also provide a simple relation between thephenomenologically important electrical resistivity and the local equilibration scales of suchstrongly correlated systems.Confirmation of the broader applicability of our result (3) for AdS × R solutions wouldbe an important step for quantifying diffusivities near general infra-red fixed points. Oneway to do this would be to identify a speed u and timescale τ such that in general D ∼ u τ with the coefficient being T -independent. This is difficult even for the relatively simplecase of holographic energy diffusion, primarily because there are two exceptional types offixed point where dangerously irrelevant deformations take over the properties of the mode:AdS × R fixed points (i.e. dynamical critical exponent z = ∞ ) [56] and relativistic fixedpoints (i.e. z = 1) [57]. If our result does generalize to AdS × R solutions (including thosewith non-universal deformations), both of these exceptional cases will be consistent with theidentification u = v eq and τ = τ eq . Provided that naive T -scaling holds for the equilibrationscales in the other cases ( τ eq ∼ /T and v eq ∼ T − /z ), which seems likely, this identificationwill then work for all fixed points.For cases where the breakdown of hydrodynamics is due to a slow mode, it is theseparation of scales between the decay rate of the slow mode Γ and that of typical non-hydrodynamic excitations T that allows one to augment the hydrodynamic description toincorporate the slow mode. Mathematically, the separation allows one to resum the hydro- We expect the result of footnote 2 to apply to the z = 1 cases due to the existence of a parametricallyslow mode [21, 23]. ω ∼ k ∼ Γ (see e.g. [15, 20, 22]).This makes manifest the convergence properties of the hydrodynamic expansion, k eq ∼ Γ.It would be very interesting if one could extract an analogous effective theory for the casesdescribed here, taking advantage of the separation of scales between the decay rate of theinfra-red modes (set by T ) and that of the other non-hydrodynamic excitations (set by thecurvature of AdS ). Such an effective theory would need to resum the effects of the entiretower of infra-red modes and would give a greater understanding of why the quadratic ap-proximation to diffusive hydrodynamics is valid up to (and indeed beyond) the wavenumber k eq .It would also be interesting to study whether our results continue to hold for AdS fixedpoints supported by a different hierarchy of scales (such as large angular momentum ormagnetic field compared to temperature), and whether analogous results hold for othertypes of hydrodynamic modes near these fixed points.The examples we studied are all consistent with bounds on D of the type proposed in[52, 53] but with the velocity in the bound given by v eq (rather than the operator growthvelocity, characteristic velocity of low energy excitations, or butterfly velocity). Indeed, thearguments in [52, 53] assume that v eq is set by one of these velocities and thus imply thebound D (cid:46) v eq τ eq . In this sense our results support the assumption of [52, 53] that the localequilibration is controlled by an underlying effective lightcone, even though the systems arenon-relativistic. It would be very worthwhile to determine D , v eq and τ eq for other states,and over a wider parameter range, in order to establish the robustness of these observationsand to determine whether v eq is set by a speed such as the butterfly velocity in general. ACKNOWLEDGMENTS
We are grateful to Luca Delacr´etaz, Saˇso Grozdanov, Subir Sachdev and Benjamin With-ers for helpful discussions. D. A. is supported by the ‘Atracci´on de Talento’ programme(2017-T1/TIC-5258, Comunidad de Madrid) and through the grants SEV-2016-0597 andPGC2018-095976-B-C21. The work of R. D. is supported by the STFC Ernest RutherfordGrant ST/R004455/1. The work of B. G. is supported by the European Research Council(ERC) under the European Union’s Horizon 2020 research and innovation program (grant While the SYK chain results shown in Figure 4 do not obey a strict bound D ≤ v eq τ eq , there is noparametric violation of such a relation and thus they are consistent with [52, 53]. [1] P. Kovtun, Dan T. Son, and Andrei O. Starinets, “Viscosity in strongly interacting quan-tum field theories from black hole physics,” Phys.Rev.Lett. , 111601 (2005), arXiv:hep-th/0405231 [hep-th].[2] Michal P. Heller, “Holography, Hydrodynamization and Heavy-Ion Collisions,” Proceedings,56th Cracow School of Theoretical Physics : A Panorama of Holography: Zakopane, Poland,May 24-June 1, 2016 , Acta Phys. Polon.
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2) that break translational symmetry. The horizonis located at r and the asymptotically AdS boundary at r → ∞ . The Hawking temperatureof the solution is T = 3 r π (cid:18) − m r (cid:19) , (A3)and further details of its thermodynamic properties can be found in [51].At zero temperature, m = 6 r and an AdS × R metric emerges near the extremalhorizon. To see this explicitly, one should change coordinates from ( t, r ) to ( u, ζ ) where r = r + (cid:15) ζ, t = u(cid:15) , (A4)and then take the (cid:15) → ds → − ζ L du + L dζ ζ + r dx , (A5)where the AdS radius of curvature is L = 1 / ω hydro ( k ) = − iD ε k + O ( k ) , D ε = 1 m (cid:114) m + 4 π T . (A6)
1. Gauge-invariant perturbations
Perturbations of this spacetime can be conveniently studied by defining suitable gauge-invariant combinations of the Fourier space perturbations of the metric components and23atter fields. Specifically, there is a choice of four such variables for which the equationsdecouple. The decoupling is a reflection of the diagonalisation of the matrix of Green’sfunctions of the dual operators. Of these variables, the one relevant for the energy densityGreen’s function is [15]˜ ψ ( r, ω, k ) = r f ( k + r f (cid:48) ) (cid:20) ddr (cid:18) δg xx + δg yy r (cid:19) − ikr δg xr − rf δg rr − k + r f (cid:48) r f δg yy (cid:21) − mr k + m ) (cid:16) mr ( δg xx − δg yy ) − ikδϕ (cid:17) , (A7)where ω and k here are the frequency and wavenumber after a Fourier transform with respectto the t and x coordinates in which the metric has the form (A2). This variable obeys theequation (cid:16) r f ˜ ψ (cid:48) (cid:17) (cid:48) + (cid:18) ω − k fr f + V ( r ) (cid:19) ˜ ψ = 0 , (A8)where primes denote derivatives with respect to r and V ( r ) = − r ( m − r )2 r (2 k r + m (2 r − r ) + 6 r ) (cid:16) k r + m (cid:0) − r + 6 r r − r (cid:1) + 12 m r (cid:0) r − r r + r (cid:1) − r (cid:0) r + r (cid:1) (cid:17) . (A9)For some calculations it will be convenient for us to convert to an ingoing Eddington-Finkelstein like coordinate system with time coordinate v = t + r ∗ , where the tortoisecoordinate r ∗ is r ∗ = 14 πT log r − r (cid:113) r + rr + r − m + ( m − r )2 r (cid:112) m − r log (cid:32) r + r + (cid:112) m − r r + r − (cid:112) m − r (cid:33) . (A10)In this coordinate system the relevant gauge-invariant variable may be written as ψ ( r, ω, k ) = r f (cid:34) ddr (cid:18) δg xx + δg yy r (cid:19) − iωr f ( δg xx + δg yy ) − ikr (cid:18) δg xr + δg vx r f (cid:19) − rf (cid:18) δg rr + 2 r f δg vr + 1 r f δg vv (cid:19) − k + r f (cid:48) r f δg yy (cid:35) − mr ( k + r f (cid:48) )2 ( k + m ) (cid:16) mr ( δg xx − δg yy ) − ikδϕ (cid:17) , (A11)where ω here denotes the frequency after a Fourier transform with respect to the v coordi-nate. This variable obeys the equation of motion ddr (cid:20) r f e − iωr ∗ ( k + r f (cid:48) ) ψ (cid:48) (cid:21) − Ω( ω, k ) e − iωr ∗ r ( k + r f (cid:48) ) ψ = 0 , (A12)24here Ω( ω, k ) = (2 r − m )3 ir ω + k ( k + m ) . (A13)Up to contact terms, the energy density Green’s function can be expressed as [61] G εε ( ω, k ) = k ( k + m ) ψ − r ψ (cid:48) + iωψ (cid:12)(cid:12)(cid:12)(cid:12) r →∞ , (A14)where in this expression ψ denotes the solution of (A12) that is regular at the horizon r = r in ingoing coordinates. We will fix the overall normalisation of the solution as ψ ( r , ω, k ) = 1,without loss of generality.
2. Near-horizon perturbation equations
We begin by analysing the relevant perturbation equation in the AdS × R spacetimethat emerges near the horizon at low temperatures. To do this, we follow [63] in introducingthe location of the extremal horizon r e via 6 r e = m and then defining r = r e + (cid:15) ζ, r = r e + (cid:15) ζ , ω = (cid:15) ζ ω . (A15)This is similar to the Fourier space version of the limit (A4), but now keeping ω/T fixed as ω →
0. The temperature is proportional to ζ . At leading order in the near-horizon, lowtemperature limit (i.e. (cid:15) → ∂ ζ ˜ ψ + 2 ζζ − ζ ∂ ζ ˜ ψ + ζ ω ζ − ζ ) − (cid:16) k m (cid:17) ζ − ζ ˜ ψ = 0 . (A16)This is the equation of a scalar field in non-zero temperature AdS with effective mass L m = 2 (1 + k /m ). As such, we can associate it with an operator of dimension ∆( k ) =(1 + (cid:112) k /m ) / G IR of such operators can be found explicitly by solving the equation (A16) andimposing the usual AdS/CFT rules at the AdS boundary ζ → ∞ [63, 64] G IR ∝ T k ) − Γ (cid:0) − ∆( k ) (cid:1) Γ (cid:0) ∆( k ) − iω πT (cid:1) Γ (cid:0) + ∆( k ) (cid:1) Γ (cid:0) − ∆( k ) − iω πT (cid:1) . (A17)At any non-zero temperature the infra-red Green’s function has a series of poles along thenegative imaginary frequency axis at the locations ω n = − i πT ( n + ∆( k )) , n = 0 , , , . . . . (A18)25o obtain G εε , one must extend the near-horizon solutions through the rest of the spacetime.In practice this is very difficult to do, but schematically one expects a result of the form[63, 64] G εε = A + B G IR + . . .C + D G IR + . . . , (A19)in the low frequency limit. If the expression written in (A19) were exact (ignoring theellipses), then at the locations (A18) corresponding to poles of G IR there would not be apole of G εε : the purported pole of G εε (arising from a pole of G IR in the numerator) wouldcancel exactly with a zero (arising from a pole of G IR in the denominator). However, theresult (A19) is not exact and numerical calculations show that while there are a pole anda zero of G εε at low T near each location (A18), for any non-zero T, k they are not exactlycoincident. As a consequence, G εε does exhibit poles at the locations (A18) at small T and k and we will present numerical results demonstrating this later in this Appendix. Thequalitative observation that the emergence of an AdS × R spacetime at low temperaturescoincides with the emergence of a series of poles of G εε with imaginary frequencies ∼ T hasbeen made previously for other spacetimes (see e.g. [66, 67]). We are going beyond this byproviding the precise locations of these poles as k, T →
3. Low temperature dispersion relation
In this section we will derive the dispersion relation (11) for a pole of G εε in the limit oflow T (with k /T fixed). The key observation that will allow us to make progress is that theequation (A12) simplifies considerably when Ω = 0 i.e. when ω = ik ( k + m ) / (3 r (2 r − m )), allowing us to formally obtain G εε along this line in Fourier space. We will use thisas a starting point for a perturbation theory, allowing us to determine G εε close to thisline in Fourier space. From this we will find that in the low T limit (with k /T fixed), theGreen’s function near this line typically exhibits a single pole with dispersion relation (11).Exceptional cases arise near particular points on this line, for which a more sophisticatedperturbation theory will be presented in the following section.First we choose a point ( ω ∗ , k ∗ ) in Fourier space that lies on the line Ω = 0, aroundwhich we will perturbatively evaluate G εε . Note that although we will often write ω ∗ and k ∗ independently in the equations that follow, they are not really independent but areconstrained by Ω( ω ∗ , k ∗ ) = 0. It will be simplest to think of us choosing a wavenumber k ∗ ω ∗ ( k ∗ ).We now take frequencies and wavenumbers that are close to this point in Fourier space ω = ω ∗ + δω, k = k ∗ + δ ( k ) , (A20)and look perturbatively for a solution to the equation (A12) of the form ψ ( r, ω, k ) = ψ ( r, ω ∗ , k ∗ ) + δψ ( r, ω ∗ , k ∗ ) , (A21)where we will evaluate δψ ( r, ω ∗ , k ∗ ) to first order in δω and δ ( k ). It will sometimes beconvenient for us to repackage the two parameters δω and δ ( k ) as δω and δ Ω where δ Ω = (2 r − m )3 ir δω + (2 k ∗ + m ) δ ( k ) , (A22)follows from (A13).Substituting (A20) and (A21) into the equation (A12) yields ddr (cid:20) r f e − iω ∗ r ∗ ( k ∗ + r f (cid:48) ) ∂ r ψ ( r, ω ∗ , k ∗ ) (cid:21) = 0 , (A23)at leading order. The first correction obeys ddr (cid:20) r f e − iω ∗ r ∗ ( k ∗ + r f (cid:48) ) ∂ r δψ ( r, ω ∗ , k ∗ ) (cid:21) = 2 r f e − iω ∗ r ∗ ∂ r ψ ( r, ω ∗ , k ∗ )( k ∗ + r f (cid:48) ) ddr (cid:20) iδωr ∗ + δ ( k ) k ∗ + r f (cid:48) (cid:21) + δ Ω e − iω ∗ r ∗ r ( k ∗ + r f (cid:48) ) ψ ( r, ω ∗ , k ∗ ) . (A24)The remaining task is to solve these equations with appropriate boundary conditions. a. Leading order solution The leading order equation (A23) can be integrated trivially for any k ∗ , giving ψ ( r, ω ∗ , k ∗ ) = C + C (cid:90) dr e iω ∗ r ∗ ( k ∗ + r f (cid:48) ) r f , (A25)where C and C are constants. Imposing the ingoing boundary conditions means that werequire that ψ ( r, ω ∗ , k ∗ ) has a Taylor series expansion near the horizon.Near the horizon, the exponential term in the integrand e iω ∗ r ∗ ∼ ( r − r ) iω ∗ πT (which forgeneric ω ∗ corresponds to an outgoing mode). Thus for generic ω ∗ , the ingoing solution is ψ ( r, ω ∗ , k ∗ ) = 1 . (A26)27ubstituting this into the expression (A14) we obtain the leading order Green’s function G εε = k ∗ ( k ∗ + m ) iω ∗ . (A27)This leading order result is exact for points lying exactly on the line Ω = 0 and the fact thatit is finite mean that there are no poles lying exactly on this line when ω ∗ (cid:54) = 0.Although the ingoing solution for generic choices of k ∗ is given by (A26), there are im-portant exceptional cases. Assuming that k ∗ + r f (cid:48) ( r ) (cid:54) = 0, the integrand of (A25) is of theform ∼ ( r − r ) iω ∗ πT − near the horizon. Therefore for the exceptional choices of wavenumber k ∗ = k n such that ω ∗ ( k n ) = ω n = − i πT ( n + 2) , n = − , , , , . . . , (A28)the integral term in (A25) will also obey the appropriate ingoing boundary condition. Forthese cases, the leading order ingoing solution is not simply that in (A26), and we willaddress them separately in the next section. There are two additional exceptional cases thatwe will not discuss as they have been studied extensively elsewhere: k ∗ = − r f (cid:48) ( r ) = ⇒ ω ∗ = + i πT [61], and the usual hydrodynamic expansion k ∗ = ω ∗ = 0 [15]. b. First order correction when Im ( ω ) > − πT Returning now to the generic case, we substitute the leading order ingoing solution (A25)into (A24) to obtain the following equation for the first correction ddr (cid:20) r f e − iω ∗ r ∗ ( k ∗ + r f (cid:48) ) ∂ r δψ ( r, ω ∗ , k ∗ ) (cid:21) = δ Ω e − iω ∗ r ∗ r ( k ∗ + r f (cid:48) ) . (A29)This can be trivially integrated to give the general solution δψ ( r, ω ∗ , k ∗ ) = δ Ω (cid:90) dr (cid:18) ( k ∗ + r f (cid:48) ) e iω ∗ r ∗ r f (cid:90) dr e − iω ∗ r ∗ r ( k ∗ + r f (cid:48) ) (cid:19) . (A30)Each integral in this expression gives rise to an integration constant that we must fix tomaintain the appropriate ingoing boundary conditions.Consider first the inner integrand. Recalling that near the horizon e − iω ∗ r ∗ ∼ ( r − r ) − iω ∗ πT ,the inner integral is only finite at the horizon for Im( ω ∗ ) > − πT . We will assume thiscondition for now, and will show in the next section that it can be relaxed without affectingthe results. With this condition, we write the inner integral as (cid:90) dr e − iω ∗ r ∗ r ( k ∗ + r f (cid:48) ) = C + (cid:90) rr dr e − iω ∗ r ∗ r ( k ∗ + r f (cid:48) ) , (A31)28here C is a constant. Substituting this into (A30), we deduce that C = 0 in order for δψ to obey ingoing boundary conditions at the horizon. Therefore the ingoing solution is givenby δψ ( r, ω ∗ , k ∗ ) = δ Ω (cid:90) rr dr (cid:18) ( k ∗ + r f (cid:48) ) e iω ∗ r ∗ r f (cid:90) rr dr e − iω ∗ r ∗ r ( k ∗ + r f (cid:48) ) (cid:19) , (A32)where we have also imposed the second boundary condition ψ ( r , ω, k ) = 1.We can now evaluate the energy density Green’s function using equation (A14). Fromour point of view the important piece is the term ( − r ψ (cid:48) + iωψ ) r →∞ in the denominator,which contains the information about poles of G εε . At ω = ω ∗ + δω and k = k ∗ + δk , thisterm has the form( − r ψ (cid:48) + iωψ ) r →∞ = iω ∗ + iδω − δ Ω( k ∗ + m ) (cid:90) ∞ r dr e − iω ∗ r ∗ r ( k ∗ + r f (cid:48) ) + iω ∗ δ Ω (cid:90) ∞ r dr (cid:18) ( k ∗ + r f (cid:48) ) e iω ∗ r ∗ r f (cid:90) rr dr e − iω ∗ r ∗ r ( k ∗ + r f (cid:48) ) (cid:19) . (A33)While the perturbative corrections in (A33) are formally small, the key point is that inan appropriate low temperature limit one of these corrections becomes anomalously largeand thus is important. This allows us to deduce that at low T the Green’s function nearthe Ω = 0 line in Fourier space is dominated by a single pole whose dispersion relation wewill shortly compute.The relevant low T limit is T →
0, keeping k ∗ /T fixed (and therefore ω ∗ /T fixed), forwhich the integral on the first line of (A33) diverges like 1 /T . The divergence arises fromthe near-horizon region of the integral, and to isolate it we define the AdS radial coordinate ζ as in equation (A4), rescale T → (cid:15) T , k ∗ → (cid:15) k ∗ , ω ∗ → (cid:15) ω , and then expand the integrandin the (cid:15) → (cid:90) rr dr e − iω ∗ r ∗ r ( k ∗ + r f (cid:48) ) = 1 (cid:15) (cid:90) ζ dζ √ − iω ∗ πT ζ − iω ∗ πT (4 πT + 3 ζ ) iω ∗ πT m (cid:0) √ k ∗ + m πT + 6 mζ (cid:1) + . . . = 1 (cid:15) √ πT + 3 ζ ) − iω ∗ πT (cid:16) ζ πT +3 ζ (cid:17) − iω ∗ πT m (16 π T + 4 ω ∗ ) (cid:0) √ k ∗ + m πT + 6 mζ (cid:1) + . . . . (A34)Evaluating this integral up to a UV cutoff Λ of the near-horizon region (with Λ (cid:29) T ), wefind the following divergent, cutoff-independent contribution to the integral on the first lineof (A33) (cid:90) ∞ r dr e − iω ∗ r ∗ r ( k ∗ + r f (cid:48) ) = 1 (cid:15) √ m (2 πT + iω ∗ ) (2 πT − iω ∗ ) + . . . . (A35)29he ellipsis denotes terms that are subleading in the (cid:15) → /(cid:15) in the same limit.In the expression (A33) for the denominator of G εε , we observe the first and third termswill dominate in the low T limit described above. More specifically, if we scale δ Ω → (cid:15) δ Ωand δω → (cid:15) δω along with the scalings of T, ω ∗ and k ∗ above, then in the limit (cid:15) → − r ψ (cid:48) + iωψ ) r →∞ = (cid:15) (cid:32) iω ∗ − δ Ω 3 √ m (2 πT + iω ∗ ) (2 πT − iω ∗ ) (cid:33) + . . . . (A36)The right hand side vanishes for a suitable δ Ω, which means that G εε is dominated by a poleat the corresponding location in the complex ω plane. More specifically, for a given fixedwavenumber k = k ∗ (i.e. δ ( k ) = 0) we can replace δ Ω by δω using equation (A22) and thensolve for δω ( ω ∗ ). Recalling that ω ∗ and k ∗ are related by Ω( ω ∗ , k ∗ ) = 0, we obtain the polelocation ω ( k ) = − i(cid:15) (cid:114) k m (cid:18) (cid:15) k m + (cid:15) (cid:18) πT m + k m (cid:19) + . . . (cid:19) . (A37)quoted in equation (11) in the main text.The result (A37) contains information about the low T limit of the hydrodynamic dis-persion relation. For example, by comparing (A37) to (A6) we would obtain the correctlow temperature expansion of the diffusivity. But the result (A37) is not simply the hydro-dynamic expansion of the dispersion relation: the terms with higher powers of k are notbeing ignored but are formally subleading in the expansion we have described. Furthermore,although we have assumed that Im( ω ) > − πT until now, we will shortly see that the result(A37) continues to hold beyond this. In fact, even for values of k outside the radius ofconvergence of the hydrodynamic expansion of the dispersion relation, we will see that thereis still a pole at the location indicated by equation (A37). c. Extension to general Im ( ω ) In our derivation of the result (A37) we assumed that Im( ω ) > − πT in order that wecould rewrite an integral appearing in the perturbative solution for δψ as (A31). When thisinequality is not obeyed, we must take more care as the integral formally diverges near thehorizon. To deal with this, we will manually separate out the divergent part of the integralbefore imposing ingoing boundary conditions in an analogous manner to the steps above.30he upshot of this is that G εε continues to be dominated by a pole with dispersion relation(A37) independently of the value of Im( ω ) (within the low T limit under consideration).The only exceptions to this are near the points (A28) that we previously highlighted, whichwe address in the following section.To extend the results to lower in the complex ω plane, we write the inner integral ap-pearing in the solution (A30) for δψ as (cid:90) dre − iω ∗ r ∗ r ( k ∗ + r f (cid:48) ) = (cid:90) dr ( r − r ) − iω ∗ / πT G ( r ) , (A38)where G ( r ) = (cid:16) r + rr + r − m (cid:17) iω ∗ / πT r ( k ∗ + r f (cid:48) ) (cid:32) r + r + (cid:112) m − r r − r − (cid:112) m − r (cid:33) − iω ∗ ( m − r πTr √ m − r . (A39)We can formally expand G ( r ) in a Taylor series near the horizon with coefficients G n andthen write the integral as (cid:90) dre − iω ∗ r ∗ r ( k ∗ + r f (cid:48) ) = (cid:90) dr (cid:40) e − iω ∗ r ∗ r ( k ∗ + r f (cid:48) ) − N (cid:88) n =0 G n ( r − r ) n − iω ∗ πT (cid:41) + N (cid:88) n =0 G n n − iω ∗ πT ( r − r ) n − iω ∗ πT , (A40)where the integral on the right hand side is finite provided that Im ( ω ∗ ) > − πT (2 + N ).With this expression we now evaluate the contribution of the δψ (cid:48) term in the perturbativeexpansion of the Greens function denominator (A14), anticipating that it will again beanomalously large in the relevant low T limit, and findlim r →∞ (cid:0) − r δψ (cid:48) (cid:1) = − (cid:0) m + k ∗ (cid:1) δ Ω lim r →∞ (cid:34) N (cid:88) n =0 G n (1 + n − iω ∗ πT ) ( r − r ) n − iω ∗ πT + (cid:90) rr dr (cid:40) e − iω ∗ r ∗ r ( k ∗ + r f (cid:48) ) − N (cid:88) n =0 G n ( r − r ) n − iω ∗ πT (cid:41)(cid:35) . (A41)The terms on the first line will vanish provided that Im ( ω ∗ ) < − πT (1 + N ). Assumingthis, along with the previous condition, we find a contribution to the Green’s functiondenominator given bylim r →∞ (cid:0) − r δψ (cid:48) (cid:1) = − (cid:0) m + k ∗ (cid:1) δ Ω (cid:90) ∞ r dr (cid:40) e − iω ∗ r ∗ r ( k ∗ + r f (cid:48) ) − N (cid:88) n =0 G n ( r − r ) n − iω ∗ πT (cid:41) , (A42)31or − πT (2 + N ) < Im( ω ∗ ) < − πT (1 + N ). There are an extra set of terms in comparisonto the corresponding term in the Green’s function denominator (A33) for Im( ω ∗ ) > − πT .By setting N = 0 , , , . . . and evaluating this integral, we can extend our previous resultsto successively lower regions of the complex ω plane. To isolate the divergent contributionto the integral arising from the near-horizon region we follow the same procedure as before:take the low T , near-horizon scaling limit of the integrand and integrate the leading termup to a UV cutoff Λ (cid:29) T of the near-horizon region. After doing this explicitly for N =0 , , , N . As a consequence, the results for the Green’s function derivedpreviously (including the dispersion relation (A37) for the pole) are also valid in the regionIm( ω ) < − πT .
4. Analytic description of pole collisions
We now turn to the exceptional points (A28) for which the analysis in the previous sectionmust be modified. Recall that for these points the leading order ingoing solution is not givenby (A26), despite the fact that they lie on the line Ω = 0. In fact, at these points the ingoingsolution is no longer uniquely defined and thus the perturbative expansion is more subtle.These ‘pole-skipping points’ correspond to points where a line of poles of the field theoryGreen’s function intersects with a line of zeroes [59–62]. Here, we will take a direct approachand just describe in practice how to obtain the appropriate perturbative solutions.The main result of this calculation is that in the low T limit, there are in fact two polesof G εε in the vicinity of each exceptional point with n ≥
0. One passes directly throughthe point while the other is separated from it by a distance ∼ T in ω space. These twopoles collide for specific complex values of k , that become real as T →
0. Assuming thatthe collision occurring at the smallest value of | k | characterizes the breakdown of diffusivehydrodynamics (which we will confirm numerically in the next section), this allows us toquantitatively extract ω eq and k eq (or equivalently τ eq and v eq ) in the low T limit.32 . Leading order ingoing solution Our starting point is the result (A25) for the general solution for ψ , at a point ( ω ∗ , k ∗ )lying exactly on the line Ω( ω ∗ , k ∗ ) = 0. At the frequencies ω n given in (A28), this solutionobeys ingoing boundary conditions at the horizon for any values of the constants. Thecorresponding values of k n are found by solving Ω( ω n , k n ) = 0. This equation has twodistinct solutions for k n and ultimately we will be interested in the one with k n ∼ T at low T . Using equation (A3) to trade T and r , explicitly this is k n = 12 (cid:18) − m + (cid:113) − m (5 + 3 n ) + 24 m (2 + n ) r − n ) r (cid:19) = (cid:114)
83 (2 + n ) mπT −
83 (2 + n ) π T + O ( T ) , (A43)where on the second line we have written the small T expansion.To fix the boundary conditions at these points we must consider points in Fourier spaceinfinitesimally away from ( ω n , k n ), where the ingoing solution is unique. Specifically, we takethe equation of motion (A12) at frequencies ω = ω n + δω and wavenumbers k = k n + δ ( k )and construct a Taylor series solution for ψ near the horizon. This solution is ingoing bydefinition, and is unique (up to an overall normalisation). We subsequently expand theTaylor series solution at small δω ∼ δ ( k ), finding the leading order result ψ ( r, ω n + δω, k n + δk ) → c n δ Ω δω ( r − r ) n + . . . , (A44)where the coefficients c n are independent of both δω and δ ( k ). The important aspect ofthis solution is that, although it is unique, it depends on the ratio δ Ω /δω that parameterizesexactly how we have moved away from the point ( ω n , k n ) in Fourier space. By expandingour general leading order solution (A25) near the horizon and matching it to the ingoingsolution (A44), we obtain the values of the constants C and C that correspond to aningoing solution at the point ( ω n + δω, k n + δk n ). Explicitly, the ingoing solution at leadingorder is then ψ ( r, ω n , k n ) = 1 + i πT α (1) n δ Ω δω I (1) n ( r ) , (A45)where we have defined the constants α ( j ) n = 1(2 + n − j )! (cid:18) d n − j dr n − j (cid:18) ( r − r ) n e − πT (2+ n ) r ∗ r ( k n + r f (cid:48) ) (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) r = r , (A46)33nd the integrals I ( j ) n ( r ) ≡ (cid:90) rr dr e iω n r ∗ ( k n + r f (cid:48) ) r f ( r − r ) j − . (A47) b. Correction to the leading order solution Having established the leading order solution (A45) near these special points in Fourierspace, we will now repeat the same procedure at one higher order in the expansion at small δω ∼ δ ( k ) to obtain the first correction to this result. After substituting the leading orderresult (A45) into the equation (A24), we can trivially integrate the resulting equation toobtain the expression δψ ( r, ω n , k n ) = (cid:90) dr (cid:32) e iω n r ∗ ( k n + r f (cid:48) ) r f δ Ω (cid:90) dr (cid:32) − πT α (1) n r f + e − iω n r ∗ r ( k n + r f (cid:48) ) (cid:33)(cid:33) + i πT α (1) n δ Ω δω (cid:90) rr dr (cid:32) e iω n r ∗ ( k n + r f (cid:48) ) r f (cid:90) rr dr e − iω n r ∗ I (1) n ( r ) r ( k n + r f (cid:48) ) (cid:33) + i πT α (1) n δ Ω δ ( k ) δω (cid:90) rr dr e iω n r ∗ ( k n + r f (cid:48) ) r f , (A48)for the correction. There are two arbitrary integration constants arising from the two indef-inite integrals in the first term, and these need to be fixed in accordance with the ingoingboundary conditions.Before doing this, note that for all n ≥ Specifically, we re-write it as (cid:90) dr (cid:32) − πT α (1) n r f + e − iω n r ∗ r ( k n + r f (cid:48) ) (cid:33) = (cid:90) rr dr (cid:32) e − iω n r ∗ r ( k n + r f (cid:48) ) − πT α (1) n r f − n (cid:88) j =2 α ( j ) n ( r − r ) j (cid:33) + β n + n (cid:88) j =2 α ( j ) n (1 − j )( r − r ) j − , (A49)where β n is the integration constant. Substituting this into (A48), we obtain the solution For the special case n = −
1, the summation terms in (A49) and subsequent equations can be set to zero.
34n the form δψ ( r, ω n , k n ) = δ Ω (cid:40)(cid:90) rr dr (cid:32) e iω n r ∗ ( k n + r f (cid:48) ) r f (cid:90) rr dr (cid:32) e − iω n r ∗ r ( k n + r f (cid:48) ) − πT α (1) n r f − n (cid:88) j =2 α ( j ) n ( r − r ) j (cid:33) + β (0) n I (1) n ( r ) + n (cid:88) j =2 α ( j ) n (1 − j ) I ( j ) n ( r ) (cid:33)(cid:41) + δ Ω δω (cid:40) i πT α (1) n (cid:90) rr dr (cid:32) e iω n r ∗ ( k n + r f (cid:48) ) r f (cid:90) rr dr e − iω n r ∗ I (1) n ( r ) r ( k n + r f (cid:48) ) (cid:33) + β (1) n I (1) n ( r ) (cid:41) + δ Ω δ ( k ) δω (cid:40) i πT α (1) n (cid:90) rr dr e iω n r ∗ ( k n + r f (cid:48) ) r f + β (2) n I (1) n ( r ) (cid:41) . (A50)In this expression we fixed one integration constant such that ψ ( r , ω, k ) = 1. The remain-ing integration constant β n has been split up into parts proportional to δ Ω , δ Ω /δω and δ Ω δ ( k ) /δω for later convenience.The integration constants β ( i ) n are fixed by imposing ingoing boundary conditions. Todetermine these conditions explicitly we again use the unique Taylor series solution for ψ ( r, ω n + δω, k n + δk ) near the horizon, and find the correction to equation (A44) at thenext order in the small δω ∼ δ ( k ) expansion. At this order, the series has the form ψ ( r, ω n + δω, k n + δk ) = 0 + . . . + ( r − r ) n (cid:20) γ (0) n δ Ω + γ (1) n δ Ω δω + γ (2) n δ Ω δ ( k ) δω (cid:21) + . . . . (A51)The coefficients γ ( i ) n become increasingly complicated for higher values of n and so we willonly explicitly present the n = 0 expressions, which are those relevant for the breakdown ofhydrodynamics γ (0)0 = 4 m r (6 r − m ) (2 k n + 6 r − m ) − m − r ) r (6 r − m )(2 k n + 6 r − m ) ,γ (1)0 = − ir (6 r − m )(2 k n + 6 r − m ) ,γ (2)0 = 18 i ( m − r ) r (2 k n + 6 r − m ) − i ( m − r ) r (6 r − m )(2 k n + 6 r − m ) . (A52)To impose ingoing boundary conditions on the solution (A50), we expand it near the horizonand match it to the ingoing solution (A51) to fix the values of the three integration constants β ( i ) n . Explicitly, these integration constants are then β (0) n = n + 2 p n γ (0) n − n (cid:88) j =2 σ ( j − n α ( j ) n p n (1 − j ) , β (1) n = n + 2 p n γ (1) n , β (2) n = n + 2 p n γ (2) n − i πT α (1) n k n + r f (cid:48) ( r ) , (A53)35here p n = ( k n + r f (cid:48) ( r )) (4 πT ) n/ r n/ (cid:32) r + (cid:112) m − r r − (cid:112) m − r (cid:33) (2+ n )( m − r r √ m − r , (A54)and σ ( j ) n = 1 j ! (cid:18) d j dr j (cid:18) e πT (2+ n ) r ∗ ( k n + r f (cid:48) ) r f ( r − r ) n (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) r = r . (A55)In summary, the first correction to the ingoing solution in the small δω ∼ δ ( k ) expansionis given by (A50) and (A53). c. Low temperature limit of the Green’s function The ingoing solution near ( ω n , k n ), including the first correction in the small δω ∼ δ ( k )expansion, is given by the sum of (A44) and (A50). It is convenient to rewrite this as ψ ( r, ω n + δω, k n + δk ) = 1 + Φ (0) n ( r ) δ Ω δω + Φ (1) n ( r ) δ Ω + Φ (2) n ( r ) δ Ω δω + . . . , (A56)where the ellipsis represents higher order terms in the small δω ∼ δ Ω expansion, andΦ (0) n ( r ) = i πT α (1) n I (1) n ( r ) , Φ (1) n ( r ) = (cid:18) β (0) n − ir (2 r − m )2 k n + m β (2) n (cid:19) I (1) n ( r ) + n (cid:88) j =2 α ( j ) n (1 − j ) I ( j ) n ( r )+ (cid:90) rr dr (cid:32) e iω n r ∗ ( k n + r f (cid:48) ) r f (cid:90) rr dr (cid:32) e − iω n r ∗ r ( k n + r f (cid:48) ) − πT α (1) n r f − n (cid:88) j =2 α ( j ) n ( r − r ) j (cid:33)(cid:33) + 12 πT r (2 r − m ) α (1) n k n + m (cid:90) rr dr e iω n r ∗ ( k n + r f (cid:48) ) r f , Φ (2) n ( r ) = (cid:32) β (1) n + β (2) n k n + m (cid:33) I (1) n ( r ) + i πT α (1) n k n + m (cid:90) rr dr e iω n r ∗ ( k n + r f (cid:48) ) r f + i πT α (1) n (cid:90) rr dr (cid:32) e iω n r ∗ ( k n + r f (cid:48) ) r f (cid:90) rr dr e − iω n r ∗ I (1) n ( r ) r ( k n + r f (cid:48) ) (cid:33) . (A57)Substituting into (A14), the denominator of the energy density Green’s function near ( ω n , k n )is ( − r ψ (cid:48) + iωψ ) r →∞ ∝ lim r →∞ (cid:34) δ Ω (cid:16) − r Φ (0) n (cid:48) ( r ) + iω n Φ (0) n ( r ) (cid:17) + 2 πT (2 + n ) δω + iδω + δ Ω (cid:16) − r Φ (2) n (cid:48) ( r ) + iω n Φ (2) n ( r ) (cid:17) + δωδ Ω (cid:16) − r Φ (1) n (cid:48) ( r ) + i Φ (0) n ( r ) + iω n Φ (1) n ( r ) (cid:17)(cid:35) . (A58)36he first important result can be seen from this formal expression: it vanishes as δω, δ Ω → ω n , k n ). The dispersion relation of the pole in the immediate vicinity of this point can befound by taking the O ( δω ) and O ( δ Ω) terms in (A58) and solving for δω as a function of δ ( k ).As the O ( δ ) terms are formally small, one might naively conclude that the Green’sfunction is dominated by this single pole in the vicinity of ( ω n , k n ). However the secondimportant result is that, in the low T limit, the coefficient of the δ Ω δω term becomesanomalously large for all n ≥
0. This indicates that there is an additional pole of theGreen’s function near ( ω n , k n ) in these cases. By carefully evaluating the expression (A58)for the Green’s function at low T we will further be able to show that for each n ≥ ω n , k n ). The collisionclosest to the origin of k space ( n = 0) involves the hydrodynamic diffusion mode and thuscharacterizes the breakdown of hydrodynamics.To take the low temperature limit, we formally rescale T ∼ (cid:15) and then evaluate thescaling of each term in (A58) in the limit (cid:15) → For the n = − δ Ω term in (A58) is ∼ T − in this limit while the coefficients of the δωδ Ω and δ Ω termsare both ∼ T − . This is what one might have naively expected: the quadratic correctionsare small provided δω, δ Ω (cid:46) T and thus the Green’s function is dominated by a single poleover a region of Fourier space of size ω ∼ T near ω = − i πT . This is consistent with whatwe found for a generic point near Ω = 0.However, for the cases n ≥
0, the scaling of the coefficients in the low T limit is different:the coefficient of the δ Ω term is ∼ T while the coefficients of the δωδ Ω and δ Ω termsare ∼ T − and ∼ T − respectively. Therefore the quadratic corrections are important atfrequencies δω ∼ T , indicating that the single-pole approximation to the Green’s functionbreaks down here due to the presence of a second pole. In the limit T →
0, these two polesare parametrically closer to one another in Fourier space than the typical separation ∼ T between poles.To make this more quantitative, we can formally rescale δ Ω ∼ δω ∼ (cid:15) and then compute(A58) in the low (cid:15) → n ≥
0. If we keep only the leading order terms in this limit To accurately extract divergent near-horizon contributions in this limit we used the procedure describedaround equation (A34). O ( (cid:15) )), the Green’s function denominator near ( ω n , k n ) has the simple factorised form δ Ω (1 − iτ n δω ) ∝ (cid:0) D n δ ( k ) − iδω (cid:1) (1 − iτ n δω ) , (A59)where we replaced δ Ω using (A22), D n = (cid:112) / m − , and τ n is given in equation (15) inthe main text. The expression (A59) clearly shows that the Green’s function near thepoints ( ω n , k n ) with n ≥ δω = − iτ − n and real wavenumber δ ( k ) = τ − n D − n . But this is not yet really a pole collision: the poles pass through each otherundisturbed, and the dispersion relations δω ( δ ( k )) extracted from (A59) have an infiniteradius of convergence.To observe the finite radius of convergence characteristic of a pole collision, we must gobeyond the result (A59) and include subleading corrections in the (cid:15) expansion. One sourceof these are finite T corrections to τ n and D n . Although these corrections are the largest inmagnitude (e.g. the leading correction to τ n is suppressed by T log T ), we will ignore themas they will not alter the functional form of (A59) and so will not produce a pole collision.The other source of corrections are the δω , δ Ω and δω terms present in the full expression(A58). The first two corrections enter at O ( (cid:15) ) while the latter is O ( (cid:15) ). Of the first two,we will only need to consider that proportional to δω as it is clear from inspecting the firstline of (A59) that adding a term ∝ δ Ω will still result in two nicely factorised poles whosedispersion relations δω ( δ ( k )) each have an infinite radius of convergence. Upon adding theleading δω term correction to (A59), we obtain the expression G − εε ( ω, k ) ∝ (cid:0) D n δ ( k ) − iδω (cid:1) (1 − iτ n δω ) − iλ n δω, (A60)for the energy density Green’s function denominator presented in the main text.The expression (A60) no longer factorizes in a simple manner and solving explicitlyfor the dispersion relations δω ( δ ( k )) of the two poles results in expressions with a non-zero discriminant term. The poles will collide at the wavenumber where the discriminantvanishes, which is when δ ( k c ) = D − n τ − n (cid:0) ± iλ / n (cid:1) = ⇒ δω c = − iτ − n (cid:0) ± iλ / n (cid:1) . (A61) We computed D n , τ n , λ n explicitly for n = 0 , , , n . δω c and δ ( k c ) at low T , and recalling that ω = ω n + δω and k = k n + δ ( k ), gives the collision location ω c = − i πT (2 + n ) (cid:32) √ πT m + . . . ± i (cid:32) / / (cid:112) (2 + n ) − (cid:18) πTm (cid:19) / + . . . (cid:33)(cid:33) k c = k n (cid:32) √ πT m + . . . ± i (cid:32) / / (cid:112) (2 + n ) − (cid:18) πTm (cid:19) / + . . . (cid:33)(cid:33) , (A62)for every n ≥
0. We have shown the leading order real and imaginary terms in eachexpression, and the ellipses denote small T corrections to these. Note however, that thereare corrections to the leading real (imaginary) term in k c ( ω c ) that are larger than the leadingorder imaginary (real) term (for example, coming from T log T suppressed corrections to τ n ).There are two collision points in the complex ω plane with opposite signs of Re( ω ) andthese collision points asymptotically approach the imaginary (real) axis as T →
0. From(A62), the phase of the collision wavenumber and frequency are φ k = 2 / (cid:112) (2 + n ) − / (cid:18) πTm (cid:19) / + . . . , φ ω = − π φ k , (A63)where we define these quantities using the collision in the upper right quadrant of thecomplex k c plane.In summary, we have derived an analytic description of a series of pole collisions hap-pening in the vicinity of the points ( ω n , k n ) for all n ≥
0. The n = 0 collision characterizesthe breakdown of diffusive hydrodynamics (as can be seen from the numerical results in thefollowing section) and from (A62) the local equilibration scales are therefore given by ω eq = 4 πT (cid:32) √ πT m + . . . (cid:33) , k eq = ω eq D ε (cid:32) − √ πT m + . . . (cid:33) , (A64)as quoted in the main text.
5. Comparison with numerical results
In this section we present exact results for the locations of poles of G εε obtained nu-merically. In addition to verifying our analytic results in the appropriate regions of Fourierspace, these numerical results provide a global picture of the pole structure and thereforeshow how the different analytic results we have presented connect together.39o obtain the exact locations of poles of G εε one needs to solve for the relevant linearfluctuations around the black hole geometry (A2). As explained above one can determine G εε by solving for a suitably defined variable ˜ ψ (see (A7)) that satisfies a decoupled equationof motion. For our numerical calculations we followed [61] and solved instead forΨ ( r, ω, k ) = r f ( k + r f (cid:48) ) ddr (cid:104) ( k + r f (cid:48) ) ˜ ψ ( r, ω, k ) (cid:105) , (A65)which satisfies the equation of motion ddr (cid:20) r f ( k + r f (cid:48) ) ω ( k + r f (cid:48) ) − k ( k + m f ) Ψ (cid:48) (cid:21) + ( k + r f (cid:48) ) r f Ψ = 0 . (A66)We solved this equation numerically using a dimensionless radial variable z = r /r . Weintegrated from the horizon at z = 1, where we imposed ingoing boundary conditions Ψ = ψ h, (1 − z ) − iω/ (4 π T ) [1 + O ((1 − z ))], to the boundary at z = 0 where we read off the leadingand subleading contributions of Ψ = ψ + ψ z + O ( z ). We used 8 terms in the near-horizonexpansion, an IR cutoff of 1 − z = 10 − and working precision 50 in Mathematica. Finally,the poles of G εε were identified as zeroes of the ratio ψ /ψ [61].For small wavenumbers k , it is the hydrodynamic pole of G εε with dispersion relation(7) that lies closest to the origin of the complex ω plane. In order to characterize thebreakdown of hydrodynamics we must first identify the non-hydrodynamic poles lying nearto the origin with which this pole could collide. At small enough T /m (and for small real k ), a family of poles with imaginary frequencies are the non-hydrodynamic poles closestto the origin. In Figure 5 we show the locations of the longest-lived such pole in ω spaceat small k , demonstrating agreement with (13) as T /m → G εε of course exhibits othernon-hydrodynamic poles besides these, but at low T /m these are far from the origin and sowill not play any role in the rest of our discussion.We will now examine more carefully how these infra-red poles move in the complex ω plane as real k is varied, and how they fit together with the diffusive hydrodynamic mode.In the low T limit, we have established that there is generically a pole with the dispersionrelation (A37) but that near each of the points ( ω n ≥ , k n ≥ ) there is an additional pole.Given that the infra-red modes lie at ω n ≥ as k →
0, the simplest explanation would bethat the infra-red modes have a very weak k -dependence and are themselves the additional At the lowest temperatures and largest wavenumber we decreased the IR cutoff to 1 − z = 10 − andincreased the working precision to 60. .000 0.002 0.004 0.006 0.008 0.0101.982.002.022.042.062.082.10 T / m - Im ( ω ) π T FIG. 5. Numerical results for the frequency of the longest lived non-hydrodynamic modes of theneutral, translation-symmetry breaking model at k = 10 − T . The dashed black line indicates theinfra-red frequency ω defined in equation (13). / T - Im ( ω ) π T FIG. 6. Numerical results for the frequencies of the hydrodynamic and longest lived infra-redmodes of the neutral, translation-symmetry breaking model at
T /m = 10 − (circles). Away from ω n ≥ , the analytic dispersion relation (A37) (solid red line) provides an excellent approximationto the exact location of a pole. .0005 0.0010 0.0015 0.00202.01202.01252.01302.01352.01402.01452.0150 Re ( ω ) π T - Im ( ω ) π T FIG. 7. Motion of the hydrodynamic and longest-lived infra red mode in the complex ω plane as | k | /T is increased (from approximately 101 .
09 to approximately 101 .
15) at fixed
T /m = 10 − andfixed phase of the wavenumber φ k = 7 . × − . There is a collision for | k | ≈ . T . poles. This explanation is correct, as is shown in Figure 6. Note that while this Figureshows multiple instances of poles coming very close to one another, the collisions themselvescannot be seen as they occur at the complex values of k (A62).The final part of our numerical analysis is to examine more closely the dynamics of thepoles near ( ω , k ) where we expect a collision between the hydrodynamic diffusion mode andthe longest-lived infra-red mode to occur, characterising the breakdown of hydrodynamics.Based on our calculations in the previous section, we expect the collision to occur for thecomplex value of the wavenumber (A62). In Figure 7 we show pole trajectories in thecomplex ω plane as | k | is varied at fixed T and phase of k , illustrating that two poles collidefor an appropriate | k | . In Figure 8 we quantitatively compare features of this pole collision tothe analytic results obtained in the previous section, showing excellent agreement as T → ω eq and k eq , whilethe lower panels show the ratio Dk eq /ω eq and the phase of the collision wavenumber φ k .In summary, the low T numerical results show that as real k is increased, the hydro-dynamic diffusion mode moves down the imaginary ω axis before coming into the vicinityof the longest-lived infra-red mode near ω = − i πT . The breakdown of hydrodynamics ischaracterized by a collision between these two modes near this point when k has a smallimaginary part. The low T collision is captured quantitatively by the analytic expressions42 .0000 0.0005 0.0010 0.0015 0.00202.0002.0052.0102.0152.0202.0252.030 T / m ω eq π T / m k eq T / m k eq2 D ε ω eq / m ϕ k FIG. 8. Temperature dependence of the local equilibration data for the neutral, translational-symmetry breaking model obtained numerically (circles) and analytically in equations (A6), (A63)and (A64) (solid lines) from the n = 0 pole collision. derived in the previous section.We finish this section with some observations on what happens after hydrodynamicsbreaks down. After the collision signaling the breakdown, one may expect the character ofthe collective modes of the system to change. But this does not happen. After the collisionoccurs, there continues to be a diffusive-type mode with dispersion relation (A37) as isapparent from Figure 6 earlier. In fact, this mode then undergoes a second collision near ω = − i πT before quickly re-appearing and so on through a whole series of collisions nearthe frequencies ω n ≥ . So while hydrodynamics formally breaks down, diffusion-like transportcontinues to occur. 43 ppendix B: Perturbations of AdS -Reissner-Nordstrom We study the AdS -RN spacetime as a solution to Einstein-Maxwell gravity with action S = (cid:90) d x √− g (cid:18) R + 6 − F (cid:19) . (B1)The metric is given by ds = − r f ( r ) dt + r dx + dr r f ( r ) , f ( r ) = 1 − (cid:18) µ r (cid:19) r r + µ r r , (B2)and is supported by the gauge field profile A t = µ (cid:16) − r r (cid:17) . (B3)The Hawking temperature T is related to the horizon radius and chemical potential by T = 12 r − µ πr . (B4)
1. Gauge-invariant perturbations
As for the neutral translation-breaking solution, the field theory Green’s functions can beconveniently studied by defining four gauge-invariant combinations of the linearised pertur-bations that obey decoupled equations of motion [68]. With appropriate conditions imposedat the AdS boundary, the quasinormal modes of two of these variables (‘the longitudinalvariables’ ψ L ± ) correspond to the poles of the retarded Green’s function of energy density G εε ( ω, k ), while the quasinormal modes of the other two variables (‘the transverse variables’ ψ T ± ) correspond to poles of the retarded Green’s function of transverse momentum density G ΠΠ ( ω, k ) [66, 67].One can assign an infra-red dimension ∆( k ) to each decoupled variable by examiningits perturbation equation in the near-horizon AdS region. For either pair of variables, thedimensions are [66, 67] ∆ ± ( k ) = 12 (cid:118)(cid:117)(cid:117)(cid:116) k µ ± (cid:115) k µ . (B5)This charged state supports three independent types of hydrodynamic mode: propagatingsound waves, diffusion of energy and charge with diffusivity D ε , and transverse diffusion of44omentum with diffusivity D Π . We focus on the latter two modes, whose diffusivities canbe computed from Einstein relations [12] as D ε = 13 r + 8 r µ + 4 r ) , D Π = 4 r µ + 4 r ) . (B6)Each hydrodynamic mode corresponds to a quasinormal mode of one of the four decoupledbulk variables. The sound modes arise as quasinormal modes of the longitudinal variable ψ L − , diffusion of energy and charge arises as a quasinormal mode of the longitudinal variable ψ L + , and transverse diffusion of momentum arises as a quasinormal mode of the transversevariable ψ T − . From these observations, we obtain ∆ ε (0) = 2 and ∆ Π (0) = 1 for the k →
2. Numerical results
The linearised perturbation equations for the AdS -RN black brane solution are signif-icantly more complicated than for the neutral translation-breaking model of Appendix A.We will therefore establish the results by solving these equations numerically to extract therelevant information about the poles of G εε and G ΠΠ . The poles of the Green’s functionshave previously been studied in [66, 67, 69–73] and more recently the breakdown of hydro-dynamics was looked at in [28, 35]. The breakdown of hydrodynamics in the AdS case wasstudied in [34].To obtain our numerical results for the poles of G εε we used the coupled variables andequations described in [71]. Although in these variables it is not manifest that the polescan be separated into two decoupled sectors, on occasion it will be useful for us to takeadvantage of this fact to simplify the results that we present. As for the perturbations ofthe neutral translation-breaking model we work with a dimensionless radial variable z = r /r and integrated from the horizon at z = 1 (where we imposed ingoing boundary conditions) tothe boundary at z = 0. To determine the poles of G εε we followed the determinant methodof [74]. We used 12 terms in the near-horizon expansion, an IR cutoff of 1 − z = 10 − andworking precision 60 in Mathematica. For the lowest temperatures and largest wavenumberswe worked with 1 − z = 10 − and working precision 80. Our definition of the variables ψ L ± is the opposite of that in [67], and instead agrees with that in [64]. Denoting the chemical potential in [71] by ˜ µ , µ = 2˜ µ due to a different normalisation of the action. G ΠΠ were obtained by solving the perturbation equationfor the decoupled variable ψ T − , as in [66]. The poles arising from ψ T + will not be relevantto the results we present. In this case, we used 9 terms in the near-horizon expansion, anIR cutoff of 1 − z = 10 − , and working precision 50 in Mathematica. As before, for thelowest temperatures and largest wavenumbers we decreased the cutoff to 1 − z = 10 − andincreased the working precision to 70.For this state it is well-established that at low temperatures the longest-lived non-hydrodynamic poles of each Green’s functions are a family of poles with imaginary fre-quencies ∼ T [66, 67]. In Figure 9 we show the motion of the diffusive pole of each Green’sfunction, along with that of the longest-lived non-hydrodynamic poles, as function of real k for a fixed low temperature. For both Green’s functions the diffusive mode moves downthe imaginary ω axis and is extremely well approximated by the quadratic approximation todiffusive hydrodynamics, while the infra-red modes are approximately k -independent and lievery close to the locations (13) of the poles of the infra-red Green’s functions. Note that G εε can be separated exactly into the sum of two independent pieces, each associated to one ofthe variables ψ L ± . In Figure 9, we show only poles of G εε that are associated to the variable ψ L + . The reason is that generically a pole of ψ L − cannot have any effect on the hydrodynamicdiffusion pole (or its radius of convergence) even if these poles coincide, as they are solutionsto independent differential equations. In the vicinity of the frequencies ω n ≥ , the diffusive mode and an infra-red mode becomevery close, but a detailed inspection (Figure 10) shows that there are no collisions nearthese points for any real k . To observe the pole collisions that characterize the breakdownof hydrodynamic diffusion, we must consider the motion of the poles for complex values of k . For G ΠΠ , the first pole collision happens close to ω = − i πT and plots of ω eq , Dk eq /ω eq and φ k for this collision are shown in Figures 2 and 3 in the main text. In [35] it wasshown that k eq ( T → →
0, which is consistent with the more quantitative results we havepresented. At sufficiently high T , the nature of the collision characterising the breakdownof hydrodynamics undergoes a qualitative change [28, 35].For G εε the breakdown of hydrodynamics is characterized by a collision near ω = − i πT An exception to this arises at particular values of k where the equations for the two decoupled variablesbecome equivalent. Such collisions can be relevant to the breakdown of hydrodynamics [28, 34, 35], butnot in the low T regime which we study. / T - Im ( ω ) π T / T - Im ( ω ) π T FIG. 9. Numerical results (black circles) for the locations of the relevant diffusive and infra-redmodes of RN-AdS . The upper plot shows poles of G ΠΠ associated to the bulk variable ψ T − at T /µ = 5 × − and the lower plot shows the poles of G εε associated to the bulk variable ψ L + at T /µ = 5 × − . Solid red lines show the quadratic approximation ω = − iD ε, Π k to thehydrodynamic dispersion relation, with D ε, Π as in (B6). between the diffusive mode and the first infra-red mode of ψ L + . In Figures 2 and 3 in the maintext we show corresponding plots of ω eq , Dk eq /ω eq and φ k . Our results for k eq are consistentwith those in [35], which observed that k eq → T →
0. As for G ΠΠ , the collisionthat characterizes the breakdown of diffusive hydrodynamics in G εε changes qualitatively atsufficiently high T [34, 35].Finally, we mention that it is manifest from Figure 9 that even far outside the formalregime of applicability of hydrodynamics (i.e. far past the first collision of the hydrody-47
040 2060 2080 2100 2120 2140 21600.981.001.021.04 k / T - Im ( ω ) π T / T - Im ( ω ) π T FIG. 10. Closeup of the numerical results for pole locations of G ΠΠ in the vicinity of ω = − i πT at T /µ = 5 × − (left panel) and of G εε in the vicinity of ω = − i πT at T /µ = 5 × − (rightpanel). There is no collision for real k in either case. namic pole), the system still supports modes whose dispersion relations are extremely wellapproximated by the quadratic approximation to diffusive hydrodynamics, with diffusivitiesgiven by (B6). We observed the same phenomenon in the neutral, translational symmetry-breaking model above, and it would be interesting to relate these observations to the resultsof [72, 73] on the hydrodynamic-like properties of zero temperature fluids. Appendix C: SYK chain model
In this Appendix, we summarise the key features of the large- q SYK chain model necessaryfor our purposes. We refer the reader to [37] for more details. The Hamiltonian is H = i q/ M − (cid:88) x =0 (cid:88) ≤ i <...