Incoherent hydrodynamics of density waves in magnetic fields
DDCPT-21/01, CPHT-RR001.012021
Incoherent hydrodynamics of density waves inmagnetic fields
Aristomenis Donos , Christiana Pantelidou and Vaios Ziogas Centre for Particle Theory and Department of Mathematical Sciences,Durham University, Durham, DH1 3LE, U.K. School of Mathematics, Trinity College Dublin, Dublin 2, Ireland CPHT, CNRS, ´Ecole Polytechnique, IP Paris, F-91128 Palaiseau, France
Abstract
We use holography to derive effective theories of fluctuations in sponta-neously broken phases of systems with finite temperature, chemical po-tential, magnetic field and momentum relaxation in which the order pa-rameters break translations. We analytically construct the hydrodynamicmodes corresponding to the coupled thermoelectric and density wave fluc-tuations and all of them turn out to be purely diffusive for our system.Upon introducing pinning for the density waves, some of these modes ac-quire not only a gap, but also a finite resonance due to the magnetic field.Finally, we study the optical properties and perform numerical checks ofour analytical results. A crucial byproduct of our analysis is the identifi-cation of the correct current which describes the transport of heat in oursystem. a r X i v : . [ h e p - t h ] F e b ontents A.1 Perturbations for χ I . . . . . . . . . . . . . . . . . . . . . . . . . . . 40A.2 Constitutive relations for the thermoelectric currents . . . . . . . . . 41A.3 Horizon vector constraint . . . . . . . . . . . . . . . . . . . . . . . . . 44 B Details on dispersion relations 46References 47
The holographic conjecture predicts that in a certain large- N limit, large classes ofconformal field theories possess a dual classical gravitational description. Apart fromits fundamental implications about quantum gravity, it provides a powerful tool tostudy strongly interacting regimes of quantum field theories which are inaccessibleby standard perturbative techniques. 1ver the last decade, the duality has been used to study aspects of stronglycoupled systems. One of its exciting applications concerns condensed matter systemsat finite temperature, chemical potential and magnetic field [1,2]. In that context, thediscussion was sparkled by the discovery of electrically charged black hole instabilitieswhich lead to superfluids/superconductors [3–5] from the field theory point of view.In this case, the order parameter is given by the expectation value of a complexoperator which breaks an internal U (1) symmetry.Soon after the discovery of holographic superfluid phases, black hole instabilitieswhich spontaneously break translations were found in [6]. These phases are expectedto play a crucial role in understanding particular physical aspects of various con-densed matter systems which exhibit instabilities such as charge and spin densitywaves, including the cuprate superconductors. In this paper we wish to constructthe effective theory of long wavelength excitations in holographic phases with spon-taneously broken translations.In order to make contact with realistic condensed matter systems, one needsto tackle the extra complication of the ionic lattice which relaxes the momentumof charge and energy carriers in the system. Momentum relaxation is an essentialingredient in discussing the low frequency transport properties of real materials. Inorder to accomplish this holographically, we need to deform our UV conformal fieldtheory by relevant operators with source parameters which break translations. Inother words, apart from the spontaneous, we also need to implement explicit breakingof translations.The construction of these inhomogeneous black hole backgrounds and the study ofthe corresponding thermodynamics is technically challenging mainly due to the factthat unstable modes naturally lead to inhomogeneous backgrounds where the only ex-pected symmetry left is time translations. However, for certain classes of holographictheories with a bulk action which is invariant under global U (1) symmetries one canfollow a Q-lattice construction [7] in order to implement both the holographic latticeas well as the order parameter that spontaneously breaks translations by simply solv-ing ODEs. This system was introduced in [8, 9] where the focus was on the transportproperties and the derivation of analytic formulae for the DC transport coefficients.In [10] it was shown that the spontaneous breaking of the global U (1) in thebulk introduced additional diffusive hydrodynamic degrees of freedom to the system,which are separate from the universal ones associated to the conservation of heat andelectric charge in the system. From that point of view, the system we are studyingis different from the modulated phases of holography where apart from translations,no additional symmetry breaking occurs. In this work, our aim is to generalise the2esults of [11] in order to include an arbitrary number of internal broken symmetriesas well as a finite magnetic field. Similar systems with spontaneous breaking of translations via an internal symme-try have been studied before in [15]. In the absence of an explicit lattice and at zeromagnetic field, the longitudinal hydrodynamic modes included one pair of sound andtwo diffusive modes. One of the diffusive modes can be accounted to the incoherentthermoelectric mode while the second one was associated to the diffusive mode of theinternal symmetry breaking described in [10]. Similarly, the transverse sector of thesystem contains a single pair of sound modes. It is known that a finite magnetic fieldhas the effect of combining the transverse and longitudinal sound modes to producea gapped mode and a mode whose frequency was growing quadratically with thewavenumber [16, 17]. Based on the hydrodynamic models of [18, 19], reference [20]argued that the corresponding constant of proportionality is complex, using numeri-cal techniques in a holographic massive gravity model; see also [21] for related workin effective theories for weak explicit background lattices.In contrast, in our work, we consider phases with strong explicit backgroundlattices, and we analytically show that all hydrodynamic modes remain diffusive evenin the presence of a magnetic field. Another aspect of our effective theory is theinclusion of explicit deformation parameters which perturbatively pin the densitywaves in the system. As one might expect, such deformations introduce a collectionof gaps for some of the diffusive modes in our theory. Interestingly, we find that atfinite magnetic field pinning also introduces resonance frequencies. As we show, fromthe retarded Green’s functions point of view these show up as poles in the lower halfplane.In section 2 we discuss the class of holographic models we are considering alongwith some important aspects of their thermodynamics. In section 3 we start by in-troducing the model of hydrodynamics that provides an effective description of thelong wavelength excitations, meanwhile identifying the correct current that describesthe transport of heat in our system. We then move on to include the effects of pin-ning in order to compute the resulting gap and resonance of the density waves, aswell as compute the retarded Green’s functions and extract their optical properties.We conclude the section by discussing how to decouple the Goldstone modes fromthe U (1) and heat currents, and by deriving the dispersion relations of our hydrody-namic modes. Section 4 contains a number of non-trivial numerical validity checks For numerical computations of quasinormal modes in 3+1 boundary dimensions, in the presenceof magnetic fields (for systems preserving translations and without spontaneous symmetry breaking),as well as the effects of chiral anomaly, see [12–14].
3f the effective theory of section 3. We summarise our most important observationsand conclude in section 5. Finally, the appendices contain technical details of theanalytical calculations of section 3.
In this section we introduce the holographic model that captures all the necessaryingredients that we would like to include in our theory. For this reason, we considerholographic theories which in addition to the metric, they contain a gauge field A µ and N Y + N Z complex scalar fields Y J and Z I with a global U (1) N Y + N Z symmetry.Essentially, this is a generalisation of the model that we considered in [11] to includean arbitrary number of complex scalars in the bulk.The gauge field will be used to introduce the chemical potential µ and the magneticfield B in the dual field theory. The first N Y complex scalars are going to implementthe explicit lattice and should therefore be relevant operators with respect to the UVtheory. The remaining N Z will provide the density wave order parameters in our sys-tem. For simplicity, we consider only four bulk spacetime dimensions, correspondingto a 2 + 1 dimensional conformal field theory on the boundary, but all our results caneasily be generalized to higher dimensional theories as well.The class of theories we are considering is described by the two-derivative bulkaction S bulk = (cid:90) d x √− g (cid:16) R − V − (cid:32) N Z (cid:88) I =1 G I ∂Z I ∂ ¯ Z I + N Y (cid:88) J =1 W J ∂Y J ∂ ¯ Y J (cid:33) − τ F (cid:17) , (2.1)with G I , W J , τ and V being only functions of the moduli b I = | Z I | and n J = | Y J | .In this case, the global internal symmetries are represented by the invariance underthe global transformations Z I → e iθ I Z I and Y J → e iω J Y J .The equations of motion are L µν ≡ R µν − τ F µρ F ν ρ − g µν F ) − g µν V − (cid:32)(cid:88) I G I ∂ ( µ Z I ∂ ν ) ¯ Z I + (cid:88) J W J ∂ ( µ Y J ∂ ν ) ¯ Y J (cid:33) = 0 , ∇ µ (cid:16) G L ∇ µ Z L (cid:17) − ∂ b L V Z L − ∂ b L τ Z L F Note that, throughout this paper, the scalar indices
I, J will not be summed over unless explicitlystated. (cid:32)(cid:88) I ∂ b L G I ∂Z I ∂ ¯ Z I + (cid:88) J ∂ b L W J ∂Y J ∂ ¯ Y J (cid:33) Z L = 0 , ∇ µ (cid:16) W K ∇ µ Y K (cid:17) − ∂ n K V Y K − ∂ n K τ Y K F − (cid:32)(cid:88) I ∂ n K G I ∂Z I ∂ ¯ Z I + (cid:88) J ∂ n K W J ∂Y J ∂ ¯ Y J (cid:33) Y K = 0 ,C ν ≡ ∇ µ ( τ F µν ) = 0 . (2.2)In order for our bulk theory to admit an AdS solution of unit radius, we demandthe small Z I and Y J expansions V = − − N Z (cid:88) I =1 m Z I | Z I | − N Y (cid:88) J =1 m Y J | Y J | + · · · G I = 1 + · · · , W J = 1 + · · · , τ = 1 + · · · . (2.3)In this case, the conformal dimensions ∆ Z I and ∆ Y J of the dual complex operatorssatisfy ∆ Z I (∆ Z I −
3) = m Z I and ∆ Y J (∆ Y J −
3) = m Y J . For the AdS vacuum we usea coordinate system in which the metric reads ds = r ( − dt + dx + dx ) + dr r . (2.4)In these coordinates, the near conformal boundary expansion for the scalars takesthe form Z I ( t, x i , r ) = z Is ( t, x i ) 1 r − ∆ ZI + · · · + z Iv ( t, x i ) 1 r ∆ ZI + · · · ,Y J ( t, x i , r ) = y Js ( t, x i ) 1 r − ∆ Y J + · · · + y Jv ( t, x i ) 1 r ∆ Y J + · · · , (2.5)where ¯ z Is and ¯ y Js are the source parameters for the dual operators O Z I and O Y J . The constants of integration z Iv and y Jv are related to the VEVs (cid:104)O Z I (cid:105) and (cid:104)O Y J (cid:105) asexplained below.Provided that the operators O Y J are relevant with ∆ Y J <
3, the Q-lattice con-struction [7] picks the deformation parameters y Js ( t, x i ) = ψ Js e ik Jsi x i . In our theory,the operators O Z I take a VEV spontaneously suggesting that for the bulk fields Z I we should have z Is = 0. In this case, the bulk field Z I are going to be zero abovea certain critical temperature. As we lower the temperature of the system, theystart developing instabilities which will yield new branches of black holes breaking In this paper we are using the canonical definition of the one-point function of any real orcomplex scalar operator O . N Z U (1)’s spontaneously. However, in section 3.3 we will turn on z Is perturbatively in order to study the pinning of the density waves.Within the Q-lattice ansatz for the backgrounds, we have that (cid:104)O Z I (cid:105) = (cid:18) ∆ Z I − (cid:19) z Iv ( t, x i ) = (cid:18) ∆ Z I − (cid:19) φ Iv e i ( k Ii x i + c I ) , (2.6)up to potential contact terms. At this point we see that we have a set of N Y wavevec-tors k Jsi which are determined externally as part of the sources related to the explicitbreaking of the lattice. In addition to those, we also have N Z wavevectors k Ii associ-ated to the order parameters (cid:104)O Z I (cid:105) . These are dynamically chosen by the system ina way that the free energy is minimised. For the particular holographic systems weare studying, this happens at k Ii = 0. However, following the logic of [11, 22], we willstill consider background solutions with k Ii (cid:54) = 0. In this way the order parameters ofthe spontaneous breaking also break translations apart from the internal U (1)’s.It is useful to define the real operatorsΩ I = 12 (cid:16) e − i ( k Ii x i + c I ) O Z I + e i ( k Ii x i + c I ) ¯ O Z I (cid:17) , (2.7)which have a constant expectation value (cid:104) Ω I (cid:105) = (cid:0) ∆ Z I − (cid:1) φ Iv = |(cid:104)O Z I (cid:105)| in thebroken phase. We now perform an internal infinitesimal rotation δε I to the bulkscalar Z I . The VEVs of the operators O Z I and ¯ O Z I transform according to δ (cid:104)O Z I (cid:105) = − i (cid:104)O Z I (cid:105) δε I and δ (cid:104) ¯ O Z I (cid:105) = i (cid:104) ¯ O Z I (cid:105) δε I and therefore δ (cid:104) Ω I (cid:105) = (cid:104) S I (cid:105) δε I ≡ i (cid:16) e − i ( k Ii x i + c I ) (cid:104)O Z I (cid:105) − e i ( k Ii x i + c I ) (cid:104) ¯ O Z I (cid:105) (cid:17) δε I . (2.8)The above suggests that from a microscopic point of view, the operator S I = 12 i (cid:16) e − i ( k Ii x i + c I ) O Z I − e i ( k Ii x i + c I ) ¯ O Z I (cid:17) , (2.9)is the right object to focus on in order to study the gapless fluctuations of the system.In order to make this point clearer, we parametrise the spacetime fluctuations of theVEVs (cid:104)O Z I (cid:105) according to δ (cid:104)O Z I (cid:105) ( t, x i ) = (cid:18) ∆ Z I − (cid:19) e i ( k Ii x i + c I ) (cid:16) δφ Iv ( t, x i ) + iφ Iv δc I ( t, x i ) (cid:17) , (2.10)where δc I ( t, x i ) parametrises fluctuations of the phase around its value in the thermalstate. Correspondingly, we see that the fluctuations of the VEV of S I are δ (cid:104) S I (cid:105) ( t, x i ) = (cid:104) Ω I (cid:105) δc I ( t, x i ) . (2.11)6herefore, the operators S I capture the gapless mode we wish to study.Note that the bulk expansions (2.5) imply that the source for Ω I is 2 Re[ e i ( k Ii x i + c I ) ¯ z Is ]and the source for S I is − e i ( k Ii x i + c I ) ¯ z Is ].In order to solve the bulk equations of motion more efficiently, we find convenientto make the field transformations Y J = ψ J e iσ J , Z I = φ I e iχ I . (2.12)bringing the bulk action to the form S bulk = (cid:90) d x √− g (cid:16) R − V − (cid:32)(cid:88) J W J ( ∂ψ J ) + (cid:88) I G I ( ∂φ I ) (cid:33) − (cid:32)(cid:88) J Ψ J ( ∂σ J ) + (cid:88) I Φ I ( ∂χ I ) (cid:33) − τ F (cid:17) , Ψ J ≡ W J ( ψ J ) , Φ I ≡ G I ( φ I ) . (2.13)A consistent ansatz that captures all the necessary ingredients we discussed so far forthe thermal state is ds = − U ( r ) dt + 1 U ( r ) dr + g ij ( r ) dx i dx j ,A = a t ( r ) dt − Bx dx ,φ I = φ I ( r ) , χ I = k Ii x i + c I ,ψ J = ψ J ( r ) , σ J = k Jsi x i . (2.14)According to (2.12), the constants c I in our ansatz (2.14) for the background blackholes translate to an overall phase for the complex scalars Z I . The absence of explicitsources in our asymptotic expansion for the corresponding field does not fix it andwe have to leave it arbitrary. These are essentially the gapless modes associated withthe symmetry breaking in the bulk that we wish to promote to hydrodynamic onesin section 3.We choose our coordinate system so that the conformal boundary of AdS isapproached as we take r → ∞ . In this case, the asymptotic expansions of thefunctions in our ansatz (2.14) take the form U → ( r + R ) + · · · + W ( r + R ) − + · · · ,g ij → δ ij ( r + R ) + · · · + g (3) ij ( r + R ) − + · · · , a → µ + Q ( r + R ) − + · · · ,ψ J → ψ Js ( r + R ) − Y J + · · · + ψ Jv ( r + R ) − ∆ Y J + · · · ,φ I → φ Is ( r + R ) − ZI + · · · + φ Iv ( r + R ) − ∆ ZI + · · · , (2.15)7here we chose to only show the terms where constants of integration of the relevantODEs appear. The constants of integration g (3) ij that appear in the expansion ofthe metric have to satisfy δ ij g (3) ij = − (cid:80) J (cid:0) ∆ Y J − (cid:1) (cid:0) ∆ Y J − (cid:1) ψ Js ψ Jv which is thegravitational constraint and yields the conformal anomaly for the stress tensor.The constant of integration R in (2.15) represents a global shift in the radialcoordinate r . This is fixed by demanding that the finite temperature horizon is at r = 0. Demanding our background solutions to be regular imposes the near horizonexpansion U ( r ) = 4 π T r + · · · , g ij = g (0) ij + · · · , a = a (0) r + · · · ,φ I = φ I (0) + · · · , ψ J = ψ J (0) + · · · . (2.16)The equations of motion (2.2) lead to the following equations for the phases ofthe complex scalars associated to spontaneous breaking ∇ µ (cid:16) Φ I ∇ µ χ I (cid:17) = 0 . (2.17)At this point, it is useful to note that the fields σ J and χ I are not well defined wheneither ψ J or φ I are equal to zero. This certainly happens close to the conformalboundary and in order to avoid misinterpretations with the holographic dictionary,we discuss asymptotic expansions in terms of the complex fields Y J and Z I through(2.12). This is well defined in the regime of perturbation theory that we are interestedin. The asymptotic expansions for the perturbations of φ I and χ I are δφ I ( t, x i , r ) = δφ Is ( t, x i ) 1( r + R ) − ∆ ZI + · · · + δφ Iv ( t, x i ) 1( r + R ) ∆ ZI + · · · ,δχ I ( t, x i , r ) = ζ S I φ Iv ( t, x i ) 1( r + R ) − ZI + · · · + δc I ( t, x i ) + · · · . (2.18)From these expansions and using (2.12) we obtain δz Is = e i ( k Ii x i + c I ) ( iζ S I ( t, x i ) + δφ Is ( t, x i )) ,δz Iv = e i ( k Ii x i + c I ) ( iφ Iv δc I ( t, x i ) + δφ Iv ( t, x i )) , (2.19)where we have used that ¯ z Is = 0, or equivalently φ Is = 0, in the phases we are interestedin. This shows that, up to contact terms, δ (cid:104) S I (cid:105) = (cid:0) ∆ Z I − (cid:1) φ Iv δc I , and that 2 δφ Is is a source for Ω I while 2 ζ S I is a source for S I , consistent with equation (2.11) andthe discussion below it.In the next subsection we discuss aspects of the thermodynamics of our brokenphase black holes. This will give us the opportunity to define certain quantities thatwill appear later in the context of hydrodynamics.8 .1 Thermodynamics In this subsection we would like to consider the thermodynamics of the backgroundblack holes we are interested in. In order to do this we need to regularise the bulkaction (2.1) by adding suitable boundary terms which act as counter-terms [23, 24].The purpose of these terms is dual, the first is to render the total on-shell actionfinite. The second is to make the variational problem well defined, provided we havea unique way to fix the boundary conditions on our bulk fields.It is often the case that such terms are not unique and for the purposes of ourpaper it is enough to list the following terms S bdr = (cid:90) ∂M d x √− γ ( − K + 4 + R bdr ) − (cid:90) ∂M d x √− γ (cid:88) I [(3 − ∆ Z I ) ¯ Z I Z I + (3 − ∆ Y I ) ¯ Y I Y I ]+ 12 (cid:90) ∂M d x √− γ (cid:88) I [ 12∆ Z I − ∂ a ¯ Z I ∂ a Z I + 12∆ Y I − ∂ a ¯ Y I ∂ a Y I ] + · · · . (2.20)Further counter-terms [25] can be added but these will introduce extra contact termsin the retarded Green’s functions that we wish to compute from the bulk theory.In order to compute the free energy of the system we need to consider the Eu-clidean version of the total action I E = − iS tot . We then need to evaluate the valueof I E on the solution with the analytically continued time t = − iτ and the periodicidentification τ ∼ τ + T − . Since our system extends infinitely in the spatial fieldtheory directions, the total free energy W F E = T I E is not meaningful and we insteadconsider the free energy density w F E w F E = (cid:15) − T s − µ ρ , (2.21)where (cid:15) , s and ρ denote the energy density, entropy density and electric charge densityrespectively. Apart from the thermodynamic data T , µ , B and the explicit latticedata ψ Js , k Jsi our solutions also depend on the wavenumbers k Ii which are related tothe spontaneous breaking. Even though different values of c I in (2.14) yield differentsolutions, the free energy is independent of those in the spontaneous case, when φ Is = 0. Since the bulk U (1) symmetries which shift c I are global, the dual boundary theory does notpossess the corresponding local Noether charges and currents [10, 26]. Based on this fact, [10, 27, 28]argued that the corresponding gapless modes behave like phasons. δw F E = − ρ δµ − s δT + (cid:88) I w iI δk Ii − M δB . (2.22)where the electric charge and entropy densities can be computed from the black holehorizon data ρ = √ g (0) τ (0) a (0) , s = 4 π √ g (0) , (2.23)and M is the magnetisation. In order to compute the variation w iI of the free energywith respect to the wavenumber k Ii , we simply vary the total action S tot and use thebackground equations of motion to find w iI = ∂ k Ii w = (cid:90) ∞ dr √ g Φ I g ij k Ij . (2.24)Notice that in the spontaneous case this is coming entirely from the variation ofthe bulk action (2.1) and it is a finite number. Potential contributions from finitecounter-terms other than those listed in (2.20) would be possible but these wouldconstitute contact terms which we can ignore for our purposes.In order to obtain the electric magnetisation M ij of the dual field theory, we needto perform a straightforward variation of the bulk action (2.1) with respect to themagnetic field B . Apart from the electric magnetisation, our backgrounds are alsogoing to have a non-trivial thermal magnetisation M ijT . Similarly to the electric mag-netisation, this is not immediately obvious from the backgrounds in (2.14) since thehomogeneity of our solutions prevents the appearance of explicit heat magnetisationcurrents. In order to define it, we would need to introduce a larger background ansatzthan (2.14) in order to include NUT charges in our metric [29]. Instead of doing that,we simply give the expression for its value in terms of the background M ij = − Bε ij (cid:90) ∞ dr τ √ g = M ε ij , (2.25) M ijT = Bε ij (cid:90) ∞ dr τ √ g a t = M T ε ij . (2.26)Apart from the quantities that appear in the first variation of the free energy, wealso find it useful to introduce a set of susceptibilities through the second variationof the free energy δs = T − c µ δT + ξ δµ + (cid:88) I ν iI δk Ii , Here and below, √ g denotes the square root of the determinant of the spatial metric g ij . ρ = ξ δT + χ q δµ + (cid:88) I β iI δk Ii ,δw iI = − ν iI δT − β iI δµ + (cid:88) L w ijIL δk Lj . (2.27)An expression for the susceptibilities ν iI , β iI and w ijIL in terms of the background canbe obtained by simply varying e.g. equation (2.24). Moreover, the susceptibilities ν iI and β iI can also be found by varying the densities in equation (2.23) with respectto the spontaneous wavenumbers k Ii . Even though it is not obvious from the bulkexpressions that these two approaches lead to the same result, this is guaranteed bythe thermodynamic Maxwell relations. This observation will become important insection 3, when we derive the constitutive relations for the currents and the Josephsonequation in a derivative expansion. The first step in order to understand the subset of hydrodynamic fluctuations isto consider general gravitational perturbations around the background black holes(2.14). As we wish to study fluctuations which involve both spatial directions on theboundary, we need to write down the consistent ansatz δ ( ds ) = e − iω v EF + iq j x j (cid:2) δg µν ( r ) dx µ dx ν + 2( iω ) − U ζ i dtdx i (cid:3) ,δA = e − iω v EF + iq j x j (cid:2) δa µ ( r ) dx µ + ( iω ) − ( E i − a t ζ i ) dx i (cid:3) ,δφ I = e − iω v EF + iq j x j δφ I ( r ) , δχ I = e − iω v EF + iq j x j δχ I ( r ) ,δψ J = e − iω v EF + iq j x j δψ J ( r ) , δσ J = e − iω v EF + iq j x j δσ J ( r ) , (2.28)where we have also performed a separation of variables. We have introduced thecombination v EF = t + S ( r ) , (2.29)with S ( r ) such that close to the horizon at r = 0 it approaches S ( r ) ∼ πT ln r + · · · . Inthis case, the function v EF approaches the infalling Eddington Finkelstein coordinateand the perturbation is regular infalling by demanding that δg tt ( r ) = 4 πT r δg (0) tt + · · · , δg rr ( r ) = δg (0) rr πT r + · · · ,δg ti ( r ) = δg (0) ti + r δg (1) ti + · · · , δg ri ( r ) = δg (0) ri πT r + δg (1) ri + · · · ,δg ij ( r ) = δg (0) ij + · · · , δg tr ( r ) = δg (0) tr + · · · , δa i ( r ) = δa (0) i + · · · , a t ( r ) = δa (0) t + δa (1) t r + · · · , δa r ( r ) = 14 πT r δa (0) r + δa (1) r + · · · ,δψ J ( r ) = δψ J (0) + · · · , δφ I ( r ) = δφ I (0) + · · · ,δχ I ( r ) = δχ I (0) + · · · , δσ J ( r ) = δσ J (0) + · · · . (2.30)which are compatible with the equations of motion. In order to achieve regularity,the above need to be supplemented by − πT ( δg (0) tt + δg (0) rr ) = − πT δg (0) rt ≡ p ,δg (0) ti = δg (0) ri ≡ − v i ,δa (0) r = δa (0) t ≡ (cid:36) . (2.31)It is useful to note that at the current stage of the discussion, the 13 + 2 N Z + 2 N Y constants δg (0) tt , δg (1) ti , δg (0) ij , δa (0) i , δa (1) t , δψ J (0) , δφ I (0) , δχ I (0) , δσ J (0) , (cid:36) , p and v i areconstants of integration and therefore free. Moreover, we haven’t fixed a gauge choiceand coordinate system for our fluctuations. For example, the choice of δg rµ and δa r can be completely arbitrary, as long as we satisfy the regular boundary conditionsprescribed in (2.30) and (2.31). After doing so, all the remaining functions will satisfy9 + 2 N Z + 2 N Y second order ODEs in the radial direction as well as 4 constraintswhich come from diffeomorphism invariance and the Gauss constraint coming fromgauge invariance.In order to discuss the constraints which we need to impose, we choose to workin a radial foliation and we define the normal one form n = dr normal to constant r hypersurfaces. We now use our equations of motion in (2.2) to define L µ = E µρ n ρ and C = C ρ n ρ with E µν = L µν − g µν L ρρ . The four gravitational constraints aresimply L µ = 0 and the Gauss constraint is C = 0.The constraints that we need to satisfy can be imposed on any hypersurfacewith e.g. constant radial coordinate r . Close to the conformal boundary, they areequivalent to the Ward identities of charge conservation, diffemorphism and Weylinvariance of the dual conformal field theory. Similar to [11], we will derive an effectivehydrodynamic theory in section 3 by utilizing a subset of these constraints on a surfaceclose to the background black hole horizon at r = 0, in terms of the constants thatappear in (2.30) and (2.31).We now define the horizon currents δQ i (0) = 4 πT √ g (0) v i ,δJ i (0) = √ g (0) τ (0) (cid:16) iq i (cid:36) + iωg ij (0) δa (0) j + E i + v i a (0) t + F ij (0) v j (cid:17) . (2.32)12uilding on [29, 30], we derive the horizon constraints that the above currents shouldsatisfy iq i δQ i (0) = 2 πT iω √ g (0) g ij (0) δg (0) ij , (2.33a) iq i δJ i (0) = iω √ g (0) (cid:104) τ (0) (cid:18) a (0) (cid:16) δg (0) tt + p πT (cid:17) + δa (1) t − iω πT (cid:16) δa (1) t − δa (1) r (cid:17)(cid:19) + 12 τ (0) a (0) g ij (0) δg (0) ij + ∂ φ I τ (0) a (0) δφ I (0) + ∂ ψ J τ (0) a (0) δψ J (0) (cid:105) , (2.33b) iω (cid:18) − δg (1) ti − g (1) il v l + iq i ( δg (0) tr − δg (0) rr ) + i ω πT ( δg (1) ti − δg (1) ri ) + ζ i + iq k δg (0) ki (cid:19) + q v i + q i q j v j + iq i (cid:18) iω πT (cid:19) p − πT ζ i − τ (0) a (0) (cid:16) iq i (cid:36) + iωδa (0) i + E i (cid:17) + Ψ (0) J k Js i (cid:16) k Is J v j − iωδσ J (0) (cid:17) + Φ (0) I k Ii (cid:16) k Ij v j − iωδχ I (0) (cid:17) − F (0) ij (cid:0) √ g (0) (cid:1) − δJ j (0) = 0 . (2.33c)Close to the conformal boundary at r = ∞ , the expansion of our functions is δg tt ( r ) = · · · + δg ( v ) tt r + R + · · · , δg rr ( r ) = O ( r − ) , δg ti ( r ) = · · · + δg ( v ) ti r + R + · · · ,δg ri ( r ) = O ( r − ) , δg ij ( r ) = · · · + δg ( v ) ij r + R + · · · , δg tr ( r ) = O ( r − ) ,δa i ( r ) = δa ( v ) i r + R + · · · , δa t ( r ) = δa ( v ) t r + R + · · · , δa r ( r ) = O ( r − ) ,δψ J ( r ) = δψ J ( v ) ( r + R ) ∆ Y J + · · · , δφ I ( r ) = δφ I ( v ) ( r + R ) ∆ ZI + · · · ,δχ I ( r ) = ζ S I φ Iv ( r + R ) ZI − + · · · + δc I + · · · , δσ J ( r ) = δσ J ( v ) ( r + R ) Y J − + · · · , (2.34)where we have chosen to only show the undetermined terms involving the constantsof integration of the second order ODE’s we need to solve. We have also includedthe constants ζ S I which represent the sources for the operators S I we discussed insection 2 and which we need to fix. For the components which we are free to chooseby using diffeomorphism and gauge invariance, we only show their desired behaviourclose to the boundary as they offer no additional information as far as the constantsof integration are concerned.The remaining 9 + 2 N Z + 2 N Y constants of integration δg ( v ) tt , δg ( v ) ti , δg ( v ) ij , δa ( v ) t , δa ( v ) i , δψ J ( v ) , δφ I ( v ) , δc I and δσ J ( v ) together with the 13 + 2 N Z + 2 N Y coming from the It is relatively straightforward to check that the r and t components of the gravitational con-straints are equivalent in the r → N Z +2 N Y second order ODE’sand the 4 constraints. In the next section we construct solutions which correspondto the late time, long wavelength hydrodynamic modes of the boundary theory. In this section we study the hydrodynamic limit of the fluctuations that we introducedin subsection 2.2. In subsection 3.1 we discuss the construction of these modes upto second order in the derivative expansion. This allows us to write an effectivehydrodynamic theory for the conserved currents of the system and its gapless modesrelated to spontaneous breaking in the bulk in subsection 3.2.Having a complete effective theory for our fluctuations, in subsection 3.3 we ex-amine the gap induced for the spontaneous density waves sliding modes, by pertur-batively small deformations for the operators Ω I . To illustrate, we specialise to theisotropic case where we find that apart from a gap, the theory also develops reso-nance frequencies. In subsection 3.4 we study the retarded Green’s functions of theoperators in our theory at finite frequency and we give the precise way that the polesof subsection 3.3 influence the transport properties.We then move on to give more general, model-independent, Kubo formulas forsome of the transport coefficients in subsection 3.5. There, we also define heat andelectric current operators which decouple from the Goldstone modes, and we discusssome of their properties. Finally, in subsection 3.6 we give an algebraic equationwhose solutions yield the dispersion relations of our hydrodynamic modes. Eventhough we are not able to find the solutions in closed form, we can show that all ourmodes are purely diffusive at the order we are working. An interesting outcome ofour results is that, after correctly identifying the heat current, the thermodynamiccoefficients w iI all drop out from physically interesting quantities. In their infinite wavelength q i → Z I which break the global symmetries in the bulk. In order to keeptrack of our expansion we scale q i → εq i with ε a small number and expand thefrequencies and radial functions in the bulk according to ω = ε ω [1] + ε ω [2] + · · · δX ( r ) = δX [0] ( r ) + εδX [1] ( r ) + ε δX [2] ( r ) + · · · , (3.1)14here δX ( r ) can be any of the functions that appear in the ansatz for the perturbation(2.28).A key point of our construction is the leading piece of the ε expansion whichaccording to our earlier discussion has to reduce to δX [0] = DX b DT δT [0] + DX b Dµ δµ [0] + (cid:88) I ∂X b ∂c I δc I [0] . (3.2)The functions X b represent the background fields of the black hole in equation (2.14).In order to generate the perturbations which satisfy the correct boundary conditions(2.30), (2.31) and (2.34), at the same time with a simple partial derivative withrespect to T , µ and c I we also need to perform the perturbative coordinate andgauge transformation [30] t → t + δT [0] T − g ( r ) , A → A − δµ [0] d ( t + g ( r )) . (3.3)As expected, the boundary condition requirements for our perturbations do notuniquely fix g ( r ) in the bulk. It is enough to choose it such that close to the conformalboundary it vanishes sufficiently fast while close to the horizon at r = 0 it approaches g ( r ) → ln r/ (4 πT ) + g (1) r + · · · .Choosing the perturbations as in (2.28) leads to an inhomogeneous system of dif-ferential equations coming from the bulk equations of motion (2.2) at the perturbativelevel. It is clear from the above construction that the seed solution (3.2) satisfies thecorresponding homogeneous system of equations [11]. This suggests that we can addthem at each order in the ε expansion (3.1) and therefore consider the split, δX [ n ] = δ ˜ X [ n ] + DX b DT δT [ n ] + DX b Dµ δµ [ n ] + (cid:88) I ∂X b ∂c I δc I [ n ] , (3.4)with δ ˜ X [ n ] a solution to the inhomogeneous problem which is sourced by lower orderterms of the solution. Following closely the analysis of [11], we can show that theeigenmodes of the system necessarily have ω [1] = δT [0] = δµ [0] = 0. Therefore thevariation of the temperature and chemical potential starts at order O ( ε ).The next to leading part of the bulk solution δX [1] will only be driven by ashift in the background phases of the complex scalars according to δχ I = e iε q i x i δc I [0] .Moreover, since we are examining the equations of motion at order O ( ε ), it is only thefirst derivatives of the varying exponential that enter the source of the inhomogeneouspart δ ˜ X [1] . Effectively we can say that up to order ε we have χ I ≈ k Ii x i + c I + e iε q i x i δc I [0] ≈ ( k Ii + iε q i δc I ) x i + c I + δc I + · · · . (3.5)15he above implies that δ ˜ X [1] will simply be the change of the background solutionunder δk Ii = i εq i δc I . The same pattern will appear at all orders in the ε expansion,leading us to the further split of the solution to the inhomogeneous problem accordingto [11] δ ˜ X [ n ] = δ X [ n ] + i (cid:88) I ∂X b ∂k Ii q i δc I [ n − . (3.6)In the end, the whole solution is determined by the variations δT , δµ and δc I andso does the horizon fluid velocity v i , the local chemical potential (cid:36) , and the vectorpotential δa (0) j . More specifically, we can identify p = 4 π (cid:0) ε δT [1] + ε δT [2] + · · · (cid:1) , v i = ε v i [2] + · · · , (cid:36) = − (cid:0) ε δµ [1] + ε δµ [2] + · · · (cid:1) ,δg (0) ij = ε (cid:32) ∂g (0) ij ∂T δT [1] + ∂g (0) ij ∂µ δµ [1] + i (cid:88) I ∂g (0) ij ∂k Ii q i δc I [0] + ε δg (0)[2] ij + · · · (cid:33) ,δφ I (0) = ε (cid:32) ∂φ I (0) ∂T δT [1] + ∂φ I (0) ∂µ δµ [1] + i (cid:88) L ∂φ I (0) ∂k Li q i δc L [0] + ε δφ I (0)[2] + · · · (cid:33) ,δψ J (0) = ε (cid:32) ∂ψ J (0) ∂T δT [1] + ∂ψ J (0) ∂µ δµ [1] + i (cid:88) I ∂ψ J (0) ∂k Ii q i δc I [0] + ε δψ J (0)[2] + · · · (cid:33) ,δχ I (0) = δc I [0] + ε δχ I (0)[1] + · · · , δσ J (0) = ε δσ J (0)[1] + · · · . (3.7)The above identification will prove useful in the next subsection where we give theeffective theory of hydrodynamics that governs the fluctuations up to and including δX [1] in our expansion. An important point of our construction is the way that we choose to impose thegravitational and Gauss constraints. As we discussed in section 2, these constraintsshould be imposed at once on any hypersurface at e.g. constant radial coordinate r .From the dual field theory point of view, the natural choice for this hypersurface wouldbe near the conformal boundary as they become equivalent to the Ward identities ofdiffeomorphism, Weyl and global U (1) invariance. More specifically, the constraints L b = 0 and C = 0 give ∇ a J a = 0 ∇ a T ab = F ba J a + 12 (cid:32)(cid:88) J O Y J ∇ b ¯ y Js + (cid:88) I O Z I ∇ b ¯ z Is + c . c . (cid:33) , (3.8)16ith F = dA the field strength of the external source one-form A a and ¯ y Js , ¯ z Is are thesources for the complex scalar operators.Contracting the stress tensor Ward identity with a vector Λ b gives ∇ a (cid:104) ( T ab + A b J a ) Λ b (cid:105) = 12 T ab L Λ g ab + J a L Λ A a + 12 (cid:32)(cid:88) J O Y J L Λ ¯ y Js + (cid:88) I O Z I L Λ ¯ z Is + c . c . (cid:33) . (3.9)The thermal gradient ζ and electric field E perturbations enter the boundary metric g ab and external field A a according to δ (cid:0) ds (cid:1) = 2 ( iω ) − ζ i e − iω t + iq j x j dt dx i , δA = ( iω ) − ( E i − µ ζ i ) e − iω t + iq j x j dx i , (3.10)along with the source δz Is for the scalar field δz Is = e i ( k Ii x i + c I ) ( iζ S I + δφ Is ) e − iω t + iq i x i . (3.11)We are now going to make the choice Λ = ∂ t and perturbatively expand the contractedWard identity to give the electric charge and heat conservation ∂ a δJ a =0 ∂ a δQ a =0 (3.12)with δQ a = − δT at − µ δJ a . Equations (3.12) define two conserved currents at thelevel of first order perturbation theory. From the point of view of the effective theory,this is a good starting point in order to give a closed system of equations, providedthat we can express these currents in terms of the hydrodynamic variables δ ˆ µ , δ ˆ T and δ ˆ c I [11]. However, we will see soon that, in the phases we are interested in, thecurrent δJ aH which describes the transport of heat is different from δQ a . As we haveargued in the previous subsection, the time derivatives scale according to ∂ t ∝ O ( ε )while for the spatial derivatives we have ∂ i ∝ O ( ε ). This suggests that we need toconsider the charge densities up to order O ( ε ) and the transport currents up to order O ( ε ) in (3.12).We will write our theory in position space where all our functions will be denotedby hats. Moreover, from now on, we find it useful to define hatted thermodynamicquantities which are local functions of the dynamical temperature, chemical potentialand phasons. For example, we define ˆ w ≡ w ( T + δ ˆ T , µ + δ ˆ µ, k Ii + ∂ i δ ˆ c I ).In appendix A.2 we relate the boundary currents δJ a , δQ a to the horizon currentswe have defined in equation (2.32) in the ε -expansion. More specifically, we can write δ (cid:104) ˆ S I (cid:105) = (cid:104) Ω I (cid:105) δ ˆ c I , ˆ J i = δ ˆ J i (0) + δ ˆ j im ,δ ˆ Q i = δ ˆ Q i + δ ˆ q im = δ ˆ Q i (0) − (cid:88) I w iI ∂ t δ ˆ c I + δ ˆ q im , (3.13)where we have defined the divergence free magnetisation currents δ ˆ j im = − M ij ˆ ζ j + ∂ j ˆ M ij ,δ ˆ q im = − M ij ˆ E j − M ijT ˆ ζ j + ∂ j ˆ M ijT . (3.14)At this point it is important to identify the correct current δJ iH that describes thetransport of heat. For this reason, using the first law (2.22), we write the conservationequations as ∂ t ˆ ρ + ∂ i δ ˆ J i =0 ,∂ t (cid:32) T ˆ s + (cid:88) I w iI ∂ i δ ˆ c I (cid:33) + ∂ i δ ˆ Q i =0 ⇒ T ∂ t ˆ s + ∂ i (cid:32) δ ˆ Q i + (cid:88) I w iI ∂ t δ ˆ c I (cid:33) =0 . (3.15)The conservation equation (3.15) implies that up to magnetisation current contribu-tions, the currents that correctly describe the transport of heat and electric chargeare δ ˆ J iH = δ ˆ Q i + (cid:80) I w iI ∂ t δ ˆ c I = δ ˆ Q i (0) and δ ˆ J i = δ ˆ J i (0) . In terms of operators, we canwrite ˆ J iH = ˆ Q i + (cid:88) I w iI (cid:104) Ω I (cid:105) ∂ t ˆ S I . (3.16)At first sight, it might seem surprising that the horizon heat current correctly de-scribes the transport of heat. However, one might have expected this to happen sinceat the level of thermodynamics the entropy density of the system is determined bythe horizon. This also ties well with the common lore in holography that dissipationis captured by horizon physics.The above discussion suggests that the physically relevant current to discuss isˆ J iH rather than ˆ Q i . In the end, we would like to build our theory of hydrodynamicsaround the conserved currents ˆ J iH and ˆ J i and the light modes associated to theoperators S I . The corresponding triplet of sources is (cid:8) ζ i , E i , ξ S I (cid:9) for our choice of It would be interesting to go beyond linear hydrodynamics and examine whether demandingpositivity of entropy production (according to an appropriately defined entropy current) leads toconstraints on the transport coefficients defined below, as happens generally [31, 32], as well as inrelated contexts [15, 20, 33–35]. ξ S I = ζ S I − w iI (cid:104) Ω I (cid:105) ζ i . (3.17)Note that, as discussed in section 2.1, thermodynamically preferred phases satisfy w iI = 0, in which case ˆ J iH = ˆ Q i .Given these results, we are able to express the boundary transport currents interms of δT [1] , δµ [1] , δc I [0] and v i [2] at order O ( ε ). However, in [11] we have shownthat within perturbation theory we can choose to impose the constraint L i = 0 on ahypersurface close to the horizon. This allows us to integrate out the horizon fluidvelocity v i [2] and therefore obtain local expressions for the currents in terms of ourhydrodynamic variables δJ i = σ ijH (cid:16) ˆ E j − ∂ j δ ˆ µ (cid:17) + T α ijH (cid:16) ˆ ζ j − T − ∂ j δ ˆ T (cid:17) − (cid:88) I γ iI ∂ t δ ˆ c I δJ iH = T ¯ α ijH (cid:16) ˆ E j − ∂ j δ ˆ µ (cid:17) + T ¯ κ ijH (cid:16) ˆ ζ j − T − ∂ j δ ˆ T (cid:17) − (cid:88) I λ iI ∂ t δ ˆ c I . (3.18)For the specific holographic model we are considering, the transport coefficients canbe expressed in terms of horizon data and thermodynamic susceptibilities accordingto σ ikH = σ ik + 4 πρ s N ij (cid:0) B − (cid:1) jl N lk , α ikH = 4 πρ N ij (cid:0) B − (cid:1) jk , ¯ α ikH = 4 πρ (cid:0) B − (cid:1) ij N jk , ¯ κ ikH = 4 πT s (cid:0) B − (cid:1) ik ,γ iI = 4 πT ρ N ij (cid:0) B − (cid:1) jk η Ik , λ iI = 4 πT s (cid:0) B − (cid:1) ij η Ij , (3.19)where we have used the definitions σ ij = τ (0) s π g ij (0) , N ik = δ ik + Bρ ε ij σ jk , η Ii = 14 πT Φ (0) I k Ii , B ij = (cid:88) J Ψ (0) J k Jsi k Jsj + (cid:88) I Φ (0) I k Ii k Ij + τ (0) B ε ik ε jl g kl (0) − πρs Bε ij . (3.20)from appendix A.3, and indices in N are raised and lowered with the horizon metric g (0) ij .We now turn our attention to the pseudo gapless degrees of freedom δc I related tothe density waves in our system. In order to introduce pinning to our system we turnon a perturbative deformation δφ Is ∝ O ( ε ) in the background asymptotics (2.15).The constitutive relations (3.18) remain unchanged [11], while in appendix A.1 weintegrate the equation of motion (2.17) to obtain the effective Josephson relation θ I ∂ t δ ˆ c I + (cid:104) Ω I (cid:105) δφ Is δ ˆ c I + η Ii δ ˆ J iH − ∂ i ˆ w iI = (cid:104) Ω I (cid:105) ˆ ξ S I , (3.21)19ith the transport coefficient θ I = s Φ (0) I π . (3.22)Equation (3.21) holds for each capital index I separately, so we have a set of N Z Josephson relations. Along with the current conservation equations (3.15), it definesa closed system of equations for the dynamical fields δ ˆ T , δ ˆ µ and δ ˆ c I .Let us now make some comments on the above holographic results. From (3.18)–(3.20) we observe that, as in the case without spontaneous symmetry breaking [29],the horizon DC conductivities σ ikH , α ikH , ¯ α ikH , ¯ κ ikH are solely determined by B ij and σ ij ,along with other thermodynamic quantities. The coupling to the massless modes δ ˆ c I is determined by one extra quantity, η Ii . The reason is that the Josephson relation(3.21) only involves the heat current J H and not the U (1) current. The coupling tothe U (1) current has been considered from a hydrodynamic perspective in variouscontexts in [15, 18, 19, 36]. It would be interesting to find a holographic model whichrealizes that.In the following subsections we study the gap of hydrodynamic modes after turningon the pinning parameters δφ Is as well as the dispersion relations without pinning.Finally we compute the finite frequency retarded Green’s functions which will helpus extract the transport properties of the system. In order to identify the gapped modes in our effective theory we need to study per-turbations with the sources switched off and with wavevector q i = 0. They belong tothe Goldstone mode sector, which leads us to consider the ansatz δ ˆ T = 0 , δ ˆ µ = 0 , δ ˆ c I = δc I e − iδω g t . (3.23)This ansatz automatically solves the current conservation equations (3.15) while theJosephson relation (3.21) gives the matrix equation for the vector of amplitudes δc I (cid:88) M (cid:16) i ( M − ) LM + δ LM δω g (cid:17) δc M = 0 , (3.24)where we have defined the matrix M LM = (cid:104) (cid:104) Ω L (cid:105) δφ Ls (cid:105) − (cid:104) θ L δ LM − η Li λ iM (cid:105) = s π (cid:104) Ω L (cid:105) δφ Ls (cid:16) Φ (0) L δ LM − (cid:0) B − (cid:1) ij η Li η Mj (cid:17) . (3.25) We remind the reader that capital indices are not being summed over.
20n order for equation (3.24) to have non-trivial solutions, the matrix multiplyingthe vector of amplitudes should not be invertible. Equating the determinant of thismatrix to zero gives an algebraic equation which determines the N Z different valuesfor the gaps δω g in terms of the eigenvalues of the matrix M − .In order to illustrate the effect of the magnetic field on the gaps of the theory weconsider a simple case with N Z = N Y = 2 and which apart from the U (1) the modelhas a Z × Z symmetry which exchanges Z ↔ Z and Y ↔ Y . This allows us toconsider an isotropic background which can be achieved by choosing k Ii = k δ Ii , k Jsi = k s δ Ji ψ Js = ψ s , δφ Is = δφ s . (3.26)The symmetries of the model along with the choice of boundary parameters leads todata of integration in which the internal indices can be suppressed, allowing us towrite (cid:104) Ω I (cid:105) = (cid:104) Ω (cid:105) , Φ I (0) = Φ (0) , Ψ J (0) = Ψ (0) ,g (0) ij = δ ij G (0) , s = 4 πG (0) , σ ij = τ (0) δ ij ≡ σ δ ij . (3.27)This simplifies the quantities B ij = 4 π (cid:34)(cid:32) k s Ψ (0) π + k Φ (0) π + σ s B (cid:33) δ ij − ω c ε ij (cid:35) , (3.28)where we see that the magnetic field introduces an antisymmetric piece in the matrix B ij . In the above ω c = ρB/s , (3.29)can be identified with the cyclotron mode frequency [1, 2]. As a consequence, thematrix M will now have complex eigenvalues leading to the gaps δω ± g = − i π (cid:104) Ω (cid:105) δφ s s Φ (0) (cid:32) k Φ (0) / πk s Ψ (0) / π + σ B /s ± iω c (cid:33) . (3.30)for the pseudo-massless modes. It is easy to see that in zero magnetic field, thesemodes lie on the lower imaginary semi-axis and they agree with the expressions thatwere obtained in [11]. Moreover, due to the isotropy of the model and backgroundwe are considering, they also lie at the same point. However, the characteristicfrequency of our nearly gapless modes has a resonant frequency apart from a gap atfinite magnetic field. 21t is interesting to consider the extreme limits for the behaviour of the poles (3.30)of our simple example. In the limit where B is the smallest parameter we have aperturbatively small correction to the results of [11] δω ± g = − i π (cid:104) Ω (cid:105) δφ s s Φ (0) (cid:32) k Φ (0) k s Ψ (0) ± πi k Φ (0) k s Ψ (0) 2 ω c + O ( B ) (cid:33) . (3.31)We therefore see that for small magnetic fields the two modes split and they movehorizontally in opposite directions in the complex plane. In the opposite limit, where B is the largest scale in the system we have δω ± g = − i π (cid:104) Ω (cid:105) δφ s s Φ (0) + O ( B − ) , (3.32)and the two frequencies become degenerate once again, at the value which is given bythe k → ρ/B finite while taking B → ∞ . It is interesting to note that the expression (3.32)is the same with [10] but in a different thermal state. This situation is relevant toweak lattices and close to the phase transition at T ∼ T c .A third, distinct possibility which is relevant to weak lattices and magnetic fieldsis when σ B (cid:28) ρ and B (cid:29) k s Ψ (0) , B (cid:28) k Φ (0) , giving δω ± g = ± k (cid:104) Ω (cid:105) s ω c δφ s + · · · . (3.33)In this case we see that the resonant part dominates the poles related to the phaserelaxation. The expression (3.33) agrees with the prediction from hydrodynamics inthe pseudo-spontaneous regime once we identify the pinning frequency ω ∼ k (cid:104) Ω (cid:105) δφ s [17, 18, 37].Note that the full expression (3.30) holds for finite magnetic field; in particular,we nowhere assumed that B is perturbatively small. All the background quantitieshowever, including the horizon values Ψ (0) , Φ (0) , depend implicitly on B as well as k s . Thus, the scaling of δω g with B in the above expressions can be determinedanalytically (or, when not possible, numerically) from the properties of the groundstate. In particular, it is not straightforward to compare our results to [20], whichfound a B / scaling for large B using a holographic massive gravity model. To be precise, the regime where (3.31) is valid is σ B (cid:28) ρ and ω c (cid:28) k Φ (0) . In the regime σ B (cid:29) ρ and B (cid:29) k s Ψ (0) , k Φ (0) . The only exception is (3.31), since we expect the various background quantities to be continu-ously connected to the corresponding ones of the B = 0 state. .4 Finite frequency response In this subsection we study linear response at zero wave number q i = 0. In orderto achieve this we turn on all our sources with a finite frequency ω and look for asolution to the conservation law equations (3.15) and Josephson relation (3.21). Asuitable ansatz for this purpose is δ ˆ T = δT [1] e − iωt , δ ˆ µ = δµ [1] e − iωt , δ ˆ c I = δc I [0] e − iωt . (3.34)After combining the transport heat current (3.18) and the Josephson relation(3.21), we obtain δc I [0] = i (cid:88) K,L (cid:104)(cid:0) ω + i M − (cid:1) − (cid:105) I L Λ LK (cid:16) (cid:104) Ω K (cid:105) ξ S K − T η Kj ¯ α jkH E k − T η Kj ¯ κ jkH ζ k (cid:17) , (cid:0) Λ − (cid:1) I P ≡ (cid:104) Ω I (cid:105) δφ Is M I P = θ I δ I P − η Ii λ iP . (3.35)Plugging this solution back in the constitutive relations (3.15) we obtain the VEVsfor the scalar fields S I and the transport currents in terms of the sources ζ i , E i and ξ S I according to δ (cid:104) S I (cid:105) = (cid:104) Ω I (cid:105) δc I [0] ,δJ i = σ ijH E j + T α ijH ζ j + iω (cid:88) I γ iI δc I [0] ,δJ iH = T ¯ α ijH E j + T ¯ κ ijH ζ j + iω (cid:88) I λ iI δc I [0] . (3.36)In order to extract the retarded Green’s functions, we need to consider the derivativeof the VEVs with respect to the time dependent sources. Since we are only consideringlinear response, we can write δ (cid:104) C (cid:105) = ( iω ) − G CJ kH ζ k + ( iω ) − G CJ k E k + (cid:88) I G CS I ξ S I , (3.37)for any operator C in our theory. After a little algebra we obtain the expressions σ ik ≡ ( iω ) − G J i J k = σ ikH + ωT (cid:88) I,L,K γ iI (cid:104)(cid:0) ω + i M − (cid:1) − (cid:105) I L Λ LK η Kj ¯ α jkH ,T α ik ≡ ( iω ) − G J i J kH = T α ikH + ωT (cid:88) I,L,K γ iI (cid:104)(cid:0) ω + i M − (cid:1) − (cid:105) I L Λ LK η Kj ¯ κ jkH , Generally, the Green’s functions of two operators A and B satisfy G ˙ AB = − iωG AB + i (cid:104) [ A, B ] (cid:105) = − iωG AB + i ( χ AB − χ BA ). However, the therodynamics of our model leads to vanishing suscepti-bilities when A = S I and B is the electric or heat current. ¯ α ik ≡ ( iω ) − G J iH J k = T ¯ α ikH + ωT (cid:88) I,L,K λ iI (cid:104)(cid:0) ω + i M − (cid:1) − (cid:105) I L Λ LK η Kj ¯ α jkH ,T ¯ κ ik ≡ ( iω ) − G J iH J kH = T ¯ κ ikH + ωT (cid:88) I,L,K λ iI (cid:104)(cid:0) ω + i M − (cid:1) − (cid:105) I L Λ LK η Kj ¯ κ jkH ,G J i S K = −(cid:104) Ω K (cid:105) ω (cid:88) I,L γ iI (cid:104)(cid:0) ω + i M − (cid:1) − (cid:105) I L Λ LK ,G S K J i = (cid:104) Ω K (cid:105) ωT (cid:88) I,L (cid:104)(cid:0) ω + i M − (cid:1) − (cid:105) K I Λ I L η Lj ¯ α jiH ,G J iH S K = −(cid:104) Ω K (cid:105) ω (cid:88) I,L λ iI (cid:104)(cid:0) ω + i M − (cid:1) − (cid:105) I L Λ LK ,G S K J iH = (cid:104) Ω K (cid:105) ωT (cid:88) I,L (cid:104)(cid:0) ω + i M − (cid:1) − (cid:105) K I Λ I L η Lj ¯ κ jiH ,G S I S K = i (cid:104) Ω I (cid:105)(cid:104) Ω K (cid:105) (cid:88) L (cid:104)(cid:0) ω + i M − (cid:1) − (cid:105) I L Λ LK . (3.38)The non-trivial frequency dependence in the above thermoelectric conductivities in-cludes only the horizon quantities ¯ α ijH , ¯ κ ijH , due to the fact that the Goldstone couplesonly to the heat current, (3.21). Moreover, from the above expressions we see thatthe gaps of the previous subsection determine the poles of the retarded Green’s func-tions (3.38). From these expressions it is clear that the pseudo-gapless modes wediscussed in section 3.3 couple to the conserved currents of our system. It shouldbe emphasised that the Green’s functions in equation (3.38) attain the most generalform possible. In particular, G S I S K is simply controlled by the existence of a polerelated to the pseudo-spontaneous symmetry breaking (subject to the presence of thegap), while S I couple to the currents only through their time derivatives, giving anadditional factor of ω in the numerator. Thus, we expect to see the same structurein more general theories of this type.Given the time reversal symmetry of the theory, as a non-trivial check, we can seethat the expressions above satisfy the Onsager relations G CD ( ω ) (cid:12)(cid:12)(cid:12) B = ε C ε D G DC ( ω ) (cid:12)(cid:12)(cid:12) − B ,with ε C,D = ± C and D transform under time re-versal. In particular, we find that G J i J jH ( ω, (cid:12)(cid:12)(cid:12) B = G J jH J i ( ω, (cid:12)(cid:12)(cid:12) − B , G SJ i ( ω, (cid:12)(cid:12)(cid:12) B = − G J i S ( ω, (cid:12)(cid:12)(cid:12) − B ,G SJ iH ( ω, (cid:12)(cid:12)(cid:12) B = − G J iH S ( ω, (cid:12)(cid:12)(cid:12) − B , G J iH J jH ( ω, (cid:12)(cid:12)(cid:12) B = G J jH J iH ( ω, (cid:12)(cid:12)(cid:12) − B ,G J i J j ( ω, (cid:12)(cid:12)(cid:12) B = G J j J i ( ω, (cid:12)(cid:12)(cid:12) − B , G S I S K ( ω, (cid:12)(cid:12)(cid:12) B = G S K S I ( ω, (cid:12)(cid:12)(cid:12) − B . (3.39)24o show (3.39), it is enough to use the identities N ij (cid:12)(cid:12)(cid:12) B = N ji (cid:12)(cid:12)(cid:12) − B , B ij (cid:12)(cid:12)(cid:12) B = B ji (cid:12)(cid:12)(cid:12) − B , γ iI (cid:12)(cid:12)(cid:12) B = T η Ij ¯ α jiH (cid:12)(cid:12)(cid:12) − B , λ iI (cid:12)(cid:12)(cid:12) B = T η Ij ¯ κ jiH (cid:12)(cid:12)(cid:12) − B ,σ ijH (cid:12)(cid:12)(cid:12) B = σ jiH (cid:12)(cid:12)(cid:12) − B , ¯ α ijH (cid:12)(cid:12)(cid:12) B = α jiH (cid:12)(cid:12)(cid:12) − B , ¯ κ ijH (cid:12)(cid:12)(cid:12) B = ¯ κ jiH (cid:12)(cid:12)(cid:12) − B , [Λ − · ( ω + i M − )] I M (cid:12)(cid:12)(cid:12) B = [Λ − · ( ω + i M − )] M I (cid:12)(cid:12)(cid:12) − B . (3.40)which hold by construction.The explicit expressions (3.38) show that the limits ω → M → ω → χ IK σ ikDC = σ ikH , T α ikDC = T α ikH , T ¯ α ikDC = T ¯ α ikH ,T ¯ κ ikDC = T ¯ κ ikH , χ IK ≡ G S I S K = (cid:104) Ω I (cid:105) δφ Is δ I K , (3.41)with the rest of the correlators vanishing. However, when we first take δφ Is → σ ikDC = σ ikH + T (cid:88) I,K γ iI Λ I K η Kj ¯ α jkH , T α ikDC = T α ikH + T (cid:88) I,K γ iI Λ I K η Kj ¯ κ jkH ,T ¯ α ikDC = T ¯ α ikH + T (cid:88) I,K λ iI Λ I K η Kj ¯ α jkH , T ¯ κ ik ( ω = 0) = T ¯ κ ikH + T (cid:88) I,K λ iI Λ I K η Kj ¯ κ jkH ,G J i S K ( ω = 0) = −(cid:104) Ω K (cid:105) (cid:88) I γ iI Λ I K , G S K J i ( ω = 0) = (cid:104) Ω K (cid:105) T (cid:88) I Λ K I η Ij ¯ α jiH ,G J iH S K ( ω = 0) = −(cid:104) Ω K (cid:105) (cid:88) I λ iI Λ I K , G S K J iH ( ω = 0) = (cid:104) Ω K (cid:105) T (cid:88) I Λ K I η Ij ¯ κ jiH , (3.42)while G S I S K diverges as G S I S K ∼ i (cid:104) Ω I (cid:105)(cid:104) Ω K (cid:105) Λ I K ω . (3.43)
Given the results of the previous subsection we can proceed to obtain Kubo formulas,which can be taken as the fundamental definition of the corresponding transport co-efficients in a generic theory with the symmetry breaking pattern we are considering. In purely spontaneous phases the susceptibilities diverge [39], but here this divergence is regu-lated by the perturbative explicit source δφ Is .
25e first extract the transport coefficient Λ
I K asΛ
I K = 1 (cid:104) Ω I (cid:105)(cid:104) Ω K (cid:105) lim ω → lim δφ Is → (cid:0) − iω G S I S K (cid:1) , (3.44)which is a finite quantity, given by the combination of transport coefficients shownin (3.35) in our specific holographic model. We can then express γ iI and λ iI as γ iI = − (cid:88) K (cid:104) Ω K (cid:105) lim ω → lim δφ Ks → G J i S K (cid:0) Λ − (cid:1) K I ,λ iI = − (cid:88) K (cid:104) Ω K (cid:105) lim ω → lim δφ Ks → G J iH S K (cid:0) Λ − (cid:1) K I . (3.45)The order of the limits is important, as was also noticed in [18]. We first need takethe gap to zero in order to include the effects of the Goldstone modes in the lowfrequency regime, and then take ω → γ iI = (cid:88) K (cid:104) Ω K (cid:105) lim ω → lim δφ Ks → (cid:0) Λ − (cid:1) K I G S K J i , ˜ λ iI = (cid:88) K (cid:104) Ω K (cid:105) lim ω → lim δφ Ks → (cid:0) Λ − (cid:1) K I G S K J iH , (3.46)which are the time reversed versions of γ iI and λ iI and satisfy γ iI (cid:12)(cid:12)(cid:12) B = ˜ γ iI (cid:12)(cid:12)(cid:12) − B , λ iI (cid:12)(cid:12)(cid:12) B = ˜ λ iI (cid:12)(cid:12)(cid:12) − B , (3.47)In our specific holographic model, we have explicitly computed γ iI , λ iI , ˜ γ iI , ˜ λ iI , see (3.19)and (3.40), or (3.38). They are related by γ iI = ρT s N ij λ jI , ˜ γ iI = ρT s ˜ λ jI N ji , (3.48)since, as explained in section 3.2, only the heat current enters the Josephson relation.However, more generally, equations (3.45),(3.46) will hold in a generic theory in whichwe do not have explicit expressions for the low energy Green’s functions, as long asthe expressions (3.45), (3.46) remain finite as δφ Is → J i , ˜ J i and J iH , ˜ J iH which decouple from the Goldstone modes. This is satisfied aslong as we demand that the Green’s functions G J i S I , G J iH S I , G S I ˜ J i , G S I ˜ J iH vanish as ω → , ω g →
0, irrespective of the order of limits. Within the hydrodynamic regime,(3.45),(3.46) imply that J i = J i + (cid:88) I γ iI (cid:104) Ω I (cid:105) ∂ t S I , J iH = J iH + (cid:88) I λ iI (cid:104) Ω I (cid:105) ∂ t S I , J i = J i + (cid:88) I ˜ γ iI (cid:104) Ω I (cid:105) ∂ t S I , ˜ J iH = J iH + (cid:88) I ˜ λ iI (cid:104) Ω I (cid:105) ∂ t S I , (3.49)indeed satisfy G J i S I = 0 , G J iH S I = 0 , G S I ˜ J i = 0 , G S I ˜ J iH = 0 . (3.50)Note that these currents are related by J i (cid:12)(cid:12)(cid:12) B = ˜ J i (cid:12)(cid:12)(cid:12) − B , J iH (cid:12)(cid:12)(cid:12) B = ˜ J iH (cid:12)(cid:12)(cid:12) − B , (3.51)which implies that J i , J iH are not vector operators on backgrounds with B (cid:54) = 0, sincethey do not have definite transformation properties under time reversal. In otherwords, equation (3.50) shows that the Goldstone modes S I do not source J i , J iH ,but ˜ J i , ˜ J iH are the operators which do not source S I . Note however that the sums J i + ˜ J i , J iH + ˜ J iH define good vector operators.In a hydrodynamic theory with the constitutive relations (3.18), we can immedi-ately see that the combinations (3.49) simply remove the contributions of the Gold-stone modes δ ˆ c I . It is then straightforward to check that the corresponding Green’sfunctions satisfy( iω ) − G J i J k = σ ikH , ( iω ) − G J i J kH = T α ikH , ( iω ) − G J iH J k = T ¯ α ikH , ( iω ) − G J iH J kH = T ¯ κ ikH . (3.52)Thus, from a holographic perspective, the combinations (3.49) isolate the horizoncontribution to the electric and heat currents. As expected, the finite-frequencypoles related to the pseudo-gapless modes cancel out in the above Green’s functions,which turn out to be frequency-independent for low frequencies up to ω g .The Green’s functions for the time-reversed currents ˜ J i , ˜ J iH are simply the time-reversed versions of (3.52) and so they also satisfy Onsager relations similar to (3.39).This can be seen by combining (3.51) and (3.40).We can proceed further by recalling the relation (3.48) between the transportcoefficients entering in (3.49). We then see that the combinations J idec ≡ T s J i − ρ N ik J kH = T sJ i − ρ N ik J kH , ˜ J idec ≡ T s ˜ J i − ρ ˜ J kH N ki = T sJ i − ρ J kH N ki , (3.53)do not include contributions from the Goldstone modes, and can be solely expressedin terms of the original currents J i , J iH . As before, J idec + ˜ J idec is a well-defined vectoroperator. 27or the retarded Green’s function we find( iω ) − G J idec J jdec = ( T s ) σ ij − T sρ (cid:16) T N jk α ik + T N ik ¯ α kj (cid:17) + ρ N ik N jl T ¯ κ kl = ( T s ) σ ijH − T sρ (cid:16) T N jk α ikH + T N ik ¯ α kjH (cid:17) + ρ N ik N jl T ¯ κ klH = ( T s ) σ ij , (3.54)which turns out to be given by the horizon quantity σ ij defined in (3.20). Similarly( iω ) − G ˜ J idec ˜ J jdec ( ω ) = ( T s ) σ ij . (3.55)We thus observe that the horizon quantity σ ij defined in (3.20) corresponds to theconductivity of the part of the U (1) current which decouples from the heat current J H and the Goldstone modes. In the special case of time-reversal invariant back-grounds with B = 0, we have that N ik = δ ik = N ki , and thus J i = ˜ J i , J iH = ˜ J iH .Then both decoupled combinations (3.53) reduce to the current considered in [40,41].Furthermore, in the absense of a background lattice, the latter can be identified withthe incoherent current which decouples from the conserved momentum operator [42].Finally, note that all of the above results hold in the strong holographic latticelimit that we are considering in our paper, where the low frequency transport prop-erties are determined by the Goldstone modes and the momentum non-conservationpoles are outside our hydrodynamic regime. However, the explicit sources δφ Is alsorelax momentum apart from the massless modes with a relaxation rate ∼ ( δφ Is ) . So,had we not included such a background lattice, the momentum poles would dominateover the Goldstone mode poles in the hydrodynamic regime. In this subsection we wish to extract the dispersion relations of the hydrodynamicmodes in our system at zero pinning. Similarly to the previous section, we switch offall the sources and we look for solutions of the form δ ˆ T = δT e − iωt + iq i x i , δ ˆ µ = δµ e − iωt + iq i x i , δ ˆ c I = δc I e − iωt + iq i x i , (3.56)which solve the conservation law equations (3.15) and Josephson relation (3.21).Similarly to the previous subsection, the resulting system of equations reduces alinear system of equations for the vector of amplitudesˆ S ( ω, q i , B ) δT /Tδµ − iω δc L = 0 , (3.57)28here we have defined the matrixˆ S ( ω, q i , B ) = T ( − ωc µ − iq i q j ¯ κ ijH ) T ( − ωξ − iq i q j ¯ α ijH ) q i ( T ν iL − λ iL ) T ( − ωξ − iq i q j α ijH ) − ωχ q − iq i q j σ ijH q i (cid:0) β iL − γ iL (cid:1) q i T ( ν iI − η Ij ¯ κ jiH ) q i ( β iI − η Ij T ¯ α jiH ) − i (Λ − ) I L + ω − q i q j w ijIL = (cid:32) ( − ω X ( B ) − i Σ ( B )) αβ m ( B ) αL m (cid:48) ( B ) Lα ( − i Θ ( B ) + ω − W ( B )) IL (cid:33) , (3.58)where we used the notation of (3.35). The indices of the matrices we have definedabove take the values α, β = 1 , I, L = 1 , . . . , N Z . The dispersion relations ofthe hydrodynamic 2 + N Z modes ω = ω ( q i ) are then determined by demanding thatdet ˆ S = 0. The expressions for the dispersion relations are going to be rather compli-cated in general. However, we can obtain some of their interesting characteristics byclosely examining the expression (3.58) for the matrix ˆ S .Equations (3.40) also imply the following relations between the various submatri-ces in (3.58)( m ( B )) T = m (cid:48) ( − B ) , ( Θ ( B )) T = Θ ( − B ) , ( Σ ( B )) T = Σ ( − B ) , ( X ( B )) T = X ( − B ) = X ( B ) ( W ( B )) T = W ( − B ) = W ( B ) . (3.59)The above show that ˆ S ( ω, q i , − B ) = (ˆ S ( ω, q i , B )) T , (3.60)which further implies that the dispersion relations are independent of the sign of B ,since the determinant is invariant under transposition. However, the Kernel of ˆ S which solves equation (3.57) will depend on it and thus the actual modes will changeas we change the sign of B . By directly exploiting these properties, in Appendix Bwe show that the frequencies ω ( q i ) of our hydrodynamic modes are purely imaginary.Finally, we notice the the transformation q i → λ q i , ω → λ ω and δc I → λ − δc I isa symmetry property of equation (3.57). This shows that all our modes are diffusion-like with ω ( λ q i ) = λ ω ( q i ).In contrast, the effective field theory of [18], as well as the holographic modelof [20], both include a real quadratic part in the dispersion relation for the magne-tophonon mode in the purely spontaneous or pseudo-spontaneous symmetry breakingregime. In our case, the modes are purely diffusive as a result of being in the strongtranslational symmetry breaking regime. 29 Numerical checks
In this section we numerical confirm the results presented in section 3, and in partic-ular the formula for the dispersion relations of the hydrodynamic modes coming fromequation (3.58), the gap (3.30) and the optical conductivities (3.38). This is achievedby focusing on the model of [38], which is a truncation of the general bulk action (2.1)down to the four-dimensional Einstein-Maxwell theory coupled to six real scalars, φ , ψ , χ i and σ i with i = 1 , S = (cid:90) d x √− g (cid:16) R − V ( φ ) −
32 ( ∂φ ) −
12 ( ∂ψ ) − θ ( φ ) (cid:2) ( ∂χ ) + ( ∂χ ) (cid:3) − θ ( ψ ) (cid:2) ( ∂σ ) + ( ∂σ ) (cid:3) − τ ( φ, ψ )4 F (cid:17) , (4.1)where V ( φ, ψ ) = − φ , θ ( φ ) = 12 sinh ( δ φ ) ,τ ( φ, ψ ) = cosh( γ φ ) , θ ( ψ ) = ψ . (4.2)The variation of the above action gives rise to the following field equations of motion R µν − τ F µρ F ν ρ − g µν F ) − g µν V − ∂ µ φ∂ ν φ − ∂ µ ψ∂ ν ψ − (cid:88) i ( θ ∂ µ χ i ∂ ν χ i + θ ∂ µ σ i ∂ ν σ i ) = 0 , √− g ∂ µ (cid:0) √− g ∂ µ φ (cid:1) − ∂ φ V − ∂ φ τ F − θ (cid:48) (cid:88) i ( ∂χ i ) = 0 , √− g ∂ µ (cid:0) √− g ∂ µ ψ (cid:1) − ∂ ψ V − ∂ ψ τ F − θ (cid:48) (cid:88) i ( ∂σ i ) = 0 , √− g ∂ µ (cid:0) θ √− g ∂ µ σ i (cid:1) = 0 , √− g ∂ µ (cid:0) θ √− g ∂ µ χ i (cid:1) = 0 ,∂ µ ( √− g τ F µν ) = 0 . (4.3)The simplest solution to the above equations is the unit radius vacuum AdS , which isdual to a d = 3 CFT with a conserved U (1) charge. In this work we choose to placethe CFT at finite temperature and deform it by a chemical potential, an externalmagnetic field and a background lattice. Within this theory, we are interested inthermal states that correspond to density waves. Putting all the ingredients together,the solutions we are after are captured by the ansatz (2.14), which we rewrite herefor convenience ds = − U ( r ) dt + 1 U ( r ) dr + e V ( r ) dx dx + e V ( r ) dx dx , = a ( r ) dt + B x dx ,φ = φ ( r ) , χ I = k Ii x i ,ψ = ψ ( r ) , σ I = k Isi x i , (4.4)where I = 1 , i = 1 ,
2. For simplicity we choose k Ii = k i δ Ii , k Isi = k si δ Ii .Let us now move on to discuss the boundary conditions. In the IR, we demandthe presence of a regular Killing horizon at r = 0 by imposing the following expansion U ( r ) = 4 π T r + . . . , V i = V (0) i + . . . , a = a (0) r + . . . ,φ = φ (0) ( x ) + . . . , ψ = ψ (0) ( x ) + . . . , (4.5)which is specified in terms of 6 constants. In the UV, we demand the conformalboundary expansion U → ( r + R ) + · · · + W ( r + R ) − + . . . , V → log( r + R ) + · · · + W p ( r + R ) − + . . . ,V → log( r + R ) + . . . a → µ + Q ( r + R ) − + . . . ,φ → φ s ( r + R ) − + φ v ( r + R ) − + . . . , ψ → ψ s + · · · + ψ v ( r + R ) − + . . . . (4.6)Just like in [38], the scalar fields ( ψ, σ ) are taken to constitute the anisotropic Q-latticein which both translational invariance and the two U (1) ψ symmetries are explicitlybroken, while the density wave phase is supported by ( φ, χ ) and breaks the the two U (1) φ symmetries spontaneously. As such, the thermal states of interest correspondto taking ψ s (cid:54) = 0 and φ s = 0. Thus, this expansion is parametrised by 8 constants.Overall we have 14 constants appearing in the expansions, in comparison to the 11integration constants of the problem. Thus, for fixed γ, δ, B, k i , k si and temperaturesbelow a critical one T < T c , we expect to find a 3 parameter family of solutions,labelled by ψ s , µ, T .In figure 1 we plot the critical temperature, T c , as a function of k = k = k for a particular choice of parameters. This is obtained by considering linearisedfluctuations around the normal phase of the system ( φ = 0 , χ = 0) and exhibits theusual “Bell Curve” shape. We now move on to compute quasinormal modes for the backgrounds constructedabove. For simplicity, we focus only on isotropic backgrounds characterised by k = k ≡ k, k s = k s ≡ k s , V = V . We consider perturbations of the form δds = − U δh tt dt + 2 U δh t x i dtdx i + e V (cid:0) h dx + h dx + 2 h dx dx (cid:1) , (4.7)31 - / μ T / μ Figure 1: Plot of the critical temperature at which the background Q-lattice becomesunstable as a function of k for ( k s , k s , ψ s , γ, δ, µ, B ) = ( , , , , , , ). We seethat the most unstable mode corresponds to k = 0.together with ( δa t , δa , δa , δφ, δψ, δχ , δχ , δσ , δσ ), where the variations are takento have the form δf ( t, r, x ) = e − iωv ( t,r )+ iqx δf ( r ) , (4.8)with v EF the Eddington-Finkelstein coordinate defined as v EF ( t, r, x ) = t + (cid:90) r ∞ dyU ( y ) . (4.9)Compared to the analytic setup of the problem in section 3, we have chosen S in(2.29) such that S (cid:48) = U − , as well as a radial gauge in which all perturbations withan r index vanish. Such a gauge is not compatible with the way we constructed themodes in section 3, but the physical information of the quasinormal modes in theend should of course be the same. Note also that our choice for the momentum q i to point in the direction x is without loss of generality, because the background isisotropic. Plugging this ansatz in the equations of motion, we obtain 5 first orderODEs and 10 second order giving rise to 25 integration constants.Let us now discuss the boundary conditions that we need to impose on these fields.In the IR, we impose infalling boundary conditions at the horizon, which without ossof generality is set at r = 0 δh tt = c r + . . . ,δh t x = c + . . . , δh t x = c + . . . ,δh x x = c + . . . , δh x x = − c + . . . ,δh x x = c + . . . , δa t = c r + . . . ,δa x = c + . . . , δa x = c + . . . ,δφ = c + . . . , δψ = c + . . . , χ = c + . . . , δσ = c + . . . ,δχ = c + . . . , δσ = c + . . . , (4.10)where the constants c , c , c and c are not free but are fixed in terms of the oth-ers. Thus, for fixed value of q , the expansion is fixed in terms of 11 constants, ω, c , c , c , c , c , c , c , c , c , c .On the other hand, in the UV, the most general expansion with φ s = 0 is givenby δh tt = δh ( s ) tt + . . . ,δh tx = δh ( s ) t x + . . . , δh tx = δh ( s ) t x + . . . , ,δh x x = δh ( s ) x x + . . . , δh x x = δh ( s ) x x + · · · + δh ( v ) x x ( r + R ) + . . . ,δh x x = δh ( s ) x x + · · · + δh ( v ) x x ( r + R ) + . . . , δa t = a ( s ) t + . . . ,δa x = a ( s ) x + a ( v ) x ( r + R ) + . . . , δa x = a ( s ) x + a ( v ) x ( r + R ) + . . . ,δφ = δφ ( s ) ( r + R ) + δφ ( v ) ( r + R ) + . . . , δψ = δψ ( s ) + · · · + δψ ( v ) ( r + R ) + . . . ,δχ = δχ ( v )1 + . . . , δσ = δσ ( s )1 + · · · + δσ ( v )1 ( r + R ) + . . . ,δχ = δχ ( v )2 + . . . , δσ = δσ ( s )2 + · · · + δσ ( v )2 ( r + R ) + . . . . (4.11)For the computation of quasinormal modes, we need to ensure that we remove all thesources from the UV expansion up to a combination of coordinate reparametrisationsand gauge transformations [ δg µν + L ˜ ζ g µν ] → , [ δA + L ˜ ζ A + d Λ] → , (4.12)where the gauge transformations are of the form x µ → x µ + ˜ ζ µ , ˜ ζ = e − iωt + iqx ζ µ ∂ µ ,A µ → A µ + ∂ µ Λ , Λ = e − iωt + iqx ( λ + λ x ) , (4.13)for ζ µ , λ constants. This requirement demands that the sources appearing in (4.11)take the form δh ( s ) tt = 2 iω ζ − ζ , h ( s ) tx = iq ζ + iω ζ , δh ( s ) tx = iω ζ ,δh ( s ) x x = − ζ − iq ζ , δh ( s ) x x = − ζ ,δh ( s ) x x = − iqζ , δa ( s ) t = iµ ω ζ + iωλ ,δa ( s ) x = − iµ q ζ − iq λ + B ζ , δa ( s ) x = − B ζ ,δφ ( s ) = 0 , δψ ( s ) = 0 ,δσ ( s )1 = − k s ζ , δσ ( s )2 = − k s ζ , (4.14)where λ = Bζ . Therefore, the UV expansion is fixed in terms of 15 constants: ζ , ζ , ζ , ζ , λ and δh ( v ) x x , δh ( v ) x x , a ( v ) x , a ( v ) x , δφ ( v ) , δψ ( v ) , δσ ( v )1 , δσ ( v )2 , δχ ( v )1 , δχ ( v )2 .Overall, for fixed q, B , we have 26 undetermined constants, of which one can beset to unity because of the linearity of the equations. This matches precisely the 25integration constants of the problem and thus we expect our solutions to be labelledby q and B . We proceed to solve numerically this system of equations subject to theabove boundary conditions using a double-sided shooting method. Figure 2 showsthe dispersion relations for the four hydrodynamic quasinormal modes in our systemfor a particular choice of the background configuration. In the same figure, we alsoillustrate with dashed lines the dispersion relations fixed by the linear system 3.57.We see a good quantitative agreement in the limit q →
0. Note that all the modesthat we find are diffusive and purely imaginary. - - - - / μ I m [ ω ]/ μ Figure 2: The dispersion relations for the four diffusive modes in our system. Thedashed line represents the dispersion relations obtained from the linear system in(3.57). Here ( φ s , ψ s , T, µ, k, k s , γ, δ, B ) = (0 , , , , , , , , ). Note that, in order to evaluate the quasinormal modes using the analytic formula (3.58), weneed to compute the derivatives w iI and w ijIJ . In order to compute these correctly one needs toconsider backgrounds with general k Ii , i.e. k (cid:54) = k (cid:54) = k (cid:54) = k . .2 Pseudo-gapless modes and two-point fucntions In this subsection we outline the numerical computation of the pseudo-gapless modesas well as certain two-point functions involving the currents
J, Q in the presence ofpinning, φ s (cid:54) = 0. We perform a calculation similar to the one for quasinormal modes,but we now consider fluctuations with q = 0 around a background configuration thathas a small but finite source φ s (cid:54) = 0. Looking at the ansatz (4.7), it is consistent toset δh tt , δh x x , δh x x , δh x x , δa t , δφ, δψ = 0. We are thus left we 6 second order and2 first order equations for the remaining fluctuations, giving rise to 14 integrationconstants.The IR expansion close to the horizon ( r = 0) takes a similar form as above,namely δh t x = c + . . . , δh t x = c + . . . ,δa x = c + . . . , δa x = c + . . . ,δχ = c + . . . , δσ = c + . . . ,δχ = c + . . . , δσ = c + . . . , (4.15)where the constants c , c are fixed in terms of the others. We see that the expansionis fixed in terms of 7 constants, ω, c , c , c , c , c , c .On the other hand, the UV expansion changes slightly in comparison to (4.11)because φ s (cid:54) = 0. In particular, it is given by δh tx = δh ( s ) t x + . . . , δh tx = δh ( s ) t x + . . . ,δa x = a ( s ) x + a ( v ) x ( r + R ) + . . . , δa x = a ( s ) x + a ( v ) x ( r + R ) + . . . ,δχ = δχ ( s )1 + δχ ( v )1 ( r + R ) + . . . , δσ = δσ ( s )1 + · · · + δσ ( v )1 ( r + R ) + . . . ,δχ = δχ ( s )2 + δχ ( v )2 ( r + R ) + . . . , δσ = δσ ( s )2 + · · · + δσ ( v )2 ( r + R ) + . . . . (4.16)Once again, we remove all the sources from the UV expansion apart from an externalelectric field E and a temperature gradient ζ in the x direction, up to a combina-tion of coordinate reparametrisations and gauge transformations. This is done byimposing the following constraints on the sources in (4.16) δh ( s ) tx = iω ζ + ζi ω , δh ( s ) tx = iω ζ , δa ( s ) x = B ζ + ( E − µ ζ ) i ω , δa ( s ) x = − B ζ ,δσ ( s )1 = − k s ζ , δσ ( s )2 = − k s ζ , δχ ( s )1 = − k ζ , δχ ( s )2 = − k ζ . (4.17)35et us first consider the case of the pseudo-gapless modes by setting ( E, ζ ) = (0 , ζ , ζ , a ( v ) x , a ( v ) x ,δσ ( v )1 , δχ ( v )1 , δσ ( v )2 , δχ ( v )2 . Overall, we have 15 undetermined constants, one of whichcan be set to unity because of the linearity of the equations. This matches preciselythe 14 integration constants of the problem and thus we expect to find a discrete setof solutions, labelled by B . We proceed to solve numerically this system of equationssubject to the above boundary conditions using a double-sided shooting method aim-ing to identify the two pseudo-gapless modes of equations (3.30). Note that the twomodes have equal imaginary parts and opposite real parts. In figure 3 we plot thereal and imaginary part of these modes as a function of the pinning parameter, φ s ,and the external magnetic field, B , and we compare with the analytic formulas whichare depicted with dashed lines. We see that the numerical and analytic calculationsare in good agreement. The reader is reminded that the analytic computation isperturbative in φ s , but exact in B . × - × - × - × - × - × - B | R e [ ω g ] | × - × - × - × - B | I m [ ω g ] | × - × - × - ϕ s R e [ ω g ] × - × - × - - - - - - ϕ s I m [ ω g ] Figure 3: In panel (a) and (b) we plot the real and imaginary part of the gap, ω g , asfunctions of the magnetic field B for φ s = 10 − . In panel (c) and (d) we plot the realand imaginary part of ω + g as functions of the magnetic field φ s for B = 1 / B and perturbative in φ s . Here ( ψ s , T, µ, k, k s , γ, δ ) = (4 , , , , , , ).We finally consider the computation of the conductivities. From (4.17) we see that,for fixed ( E, ζ ), we have 7 constants in the IR and 8 in the UV. Comparing with the14 integration constants in the problem, we expect to find a 1-parameter family of36olutions labelled by ω . Using the linearity of the equations we set ( E, ζ ) = (1 ,
0) or(
E, ζ ) = (0 ,
1) depending on which source we want to keep. The diffusion currentsare then given by δ ˆ J x = E − (cid:18) µ + iQω (cid:19) ζ + a ( v )1 + iBωζ ,δ ˆ J x = a ( v )2 − iBωζ − M ζ ,δ ˆ Q x = − µa ( v )1 + 3 i k s ψ s ω δσ ( v )1 + 12 iδ kφ s ω δχ ( v )1 − E (cid:18) µ + iQω (cid:19) − i ζω (cid:18) W + 3 S V φ s + 32 ik s ψ s ω − µQ + iµ ω (cid:19) + ζ (cid:0) k δ φ s + 3 k s φ s ψ s + 2 k s ψ s − k s ψ s ω + 4 B (cid:1) + iBω a ( v )2 − iBωµζ ,δ ˆ Q x = − µa ( v )2 + 3 i k s ψ s ω δσ ( v )2 + 12 iδ kφ s ω δχ ( v )2 − iBω E + i ζω B (cid:18) µ + iQω (cid:19) + ζ (cid:0) k δ φ s + 3 k s φ s ψ s + 2 k s ψ s − k s ψ s ω + 4 B (cid:1) − iBω a ( v )1 + iBωµζ − M E − M T ζ . (4.18)Carrying out the numerical shooting computation, we calculate the (1 ,
1) and(1 ,
2) components of the two-point functions ( iω ) − G J i J j , ( iω ) − G J i Q j , ( iω ) − G Q j J i ,( iω ) − G Q i Q j . For fixed B and φ s , these quantities are plotted in figure 4 with solidlines. In order to compare our numerics with the analytic results of section 3, we usethe definition (3.16) to write( iω ) − G J i J j = σ ij , ( iω ) − G J i Q j = T α ij − (cid:88) I w jI (cid:104) Ω I (cid:105) G J i S I , ( iω ) − G Q j J i = T ¯ α ij + (cid:88) I w iI (cid:104) Ω I (cid:105) G S I J j , ( iω ) − G Q i Q j = T ¯ κ ij − (cid:88) I w jI (cid:104) Ω I (cid:105) G J iH S I + (cid:88) I w iI (cid:104) Ω I (cid:105) (cid:32) G S I J jH − iω (cid:88) K w iK (cid:104) Ω K (cid:105) G S I S K (cid:33) , (4.19)and thus obtain analytic expressions using (3.38), which are depicted in figure 4with dashed lines. We see that the two are in good quantitative agreement at smallfrequencies. The reader is reminded that in this calculation we have set ζ IS = 0 andwe only included sources in the x direction; thus we can not compute the (2 ,
1) and(2 ,
2) components of the the two-point functions.37 .0 0.2 0.4 0.6 0.8 1.07678808284868890 10 ω σ ω ( i ω ) G Q J ω ( i ω ) G J Q ω ( i ω ) G Q Q - - - - - -
26 10 ω σ - - - - - - ω ( i ω ) G Q J - - - - - ω ( i ω ) G J Q - - - - - ω ( i ω ) G Q Q Figure 4: Plots of the components (1 ,
1) and (1 ,
2) of the thermoelectric conductivitiesas functions of the frequency. The dashed lines correspond to the analytic formulas(3.38). Here ( φ s , ψ s , T, µ, k, k s , γ, δ, B ) = (10 − , , , , , , , , ). In this paper we constructed the effective theory of hydrodynamics which capturesholographic phases in which translations are broken explicitly and spontaneously. We38ave significantly extended the construction of [11] to include an arbitrary number N Z of gapless degrees of freedom emerging from spontaneous density waves and wealso included a background magnetic field.A holographic model which incorporates the two Goldstone modes arising fromspontaneous breaking of translations in magnetic fields, along with the coupling tothe heat current was studied in [20] and a complex quadratic dispersion relation wasfound. In our setup, the strength of the explicit breaking is large compared to thewavelength of the hydrodynamic fluctuations. In section 3.6 we analytically derivedan equation whose roots yield the dispersion relations of the hydrodynamic modesgoverning our system. Despite not being able to write down the dispersion relationsof all of our 2 + N Z hydrodynamic modes in closed form, we prove that they arepurely imaginary and diffusive, unlike [20].In our construction we have also included the corresponding N Z perturbativedeformation parameters which pin down the density waves and introduce N Z gaps inour theory. Interestingly, we have shown that apart from the gap, the magnetic fieldfield causes the corresponding poles to move off the imaginary axis due to resonanceeffects. In section 3.4 we computed the retarded Green’s functions of the operatorsrelevant to the hydrodynamic description of the system. As one might expect, thepoles due to pinning have a direct effect on the transport properties of our system ascan be seen from the explicit form of the Green’s functions in equation (3.38).Finally, an important byproduct in our work is the identification of the correctcurrent in (3.16) which describes the transfer of entropy as can be seen by the con-servation equation in the last line of (3.15). Given this definition, the variation of thefree energy w iI with respect to the wavenumbers k Ii drops out of the correspondingGreen’s functions (3.38). Moreover, the gaps and the resonance frequencies whichcan be found by solving the eigenvalue problem equation (3.24) are also independentof w iI .There are various open questions which one could further explore. It would beinteresting to consider second order hydrodynamic perturbation theory and examinewhat the second law implies for the transport coefficients in phases with sponta-neous and explicit symmetry breaking. Additionally, it is important to examine howtransport in such phases is constrained from purely field theoric considerations, suchas the Ward identities, and also investigate the possible experimental significanceof the decoupled/incoherent currents we defined in this paper. Finally, it would beenlightning to move away from homogeneity and explore what kind of novel effectsinhomogeneous models with similar symmetry breaking patterns might exhibit.39 cknowledgements We would like to thank Blaise Gout´eraux for useful discussions. AD is supported bySTFC grant ST/T000708/1. CP is supported by the European Union’s Horizon 2020research and innovation programme under the Marie Sk(cid:32)lodowska-Curie grant agree-ment HoloLif No 838644. The work of VZ was supported by the China PostdoctoralScience Foundation (International Postdoctoral Fellowship Program 2018), the Na-tional Natural Science Foundation of China (NSFC) (Grant number 11874259), andis supported by the European Research Council (ERC) under the European Union’sHorizon 2020 research and innovation programme (grant agreement No.758759).
A Perturbations in the bulk
In this appendix we derive the constitutive relations (3.18) for the transport partof the heat and electric currents along with the Josephson relations (3.21) for thedensity wave degrees of freedom. In order to do this, we solve for the perturbation δχ I through its equation of motion (2.17). We only need to do this up to second orderin our ε expansion (3.1), which we carry out in Appendix A.1. Then, in AppendixA.2 we derive the constitutive relations for the currents by relating the field theoryand horizon currents densities of our holographic model. A.1 Perturbations for χ I After perturbatively expanding the equation of motion (2.17), we have − ∂ t (cid:16) √− g Φ I g tt g ij k Ii δg tj (cid:17) − ∂ r (cid:16) √− g Φ I g rr g ij k Ii δg rj (cid:17) + ∂ j (cid:16) δ (cid:0) √− g Φ I g ij (cid:1) k Ii (cid:17) + ∂ µ (cid:16) √− g Φ I g µν ∂ ν δχ I (cid:17) = 0 . (A.1)From the form of the solution close to the conformal boundary at r → ∞ , we can infera relation between the sources ζ S I of the operators S I and their vevs δ (cid:104) S I (cid:105) . This willessentially give a Josephson type of equation for the variable δ ˆ c I through equation(3.13). In the next subsections we will solve equation (A.1) in an ε expansion. A.1.1 Field theory interpretation at order O ( ε ) At order O ( ε ) we obtain the equation −√ g Φ I k Ii g ij ζ j [1] − ∂ r (cid:16) √ g Φ I U g ij k Ii δg rj [1] (cid:17) + iq j δ (cid:0) √ g Φ I g ij (cid:1) [0] k Ii + ∂ r (cid:16) √ g Φ I U ∂ r δχ I [1] (cid:17) − iω [1] ∂ r (cid:16) √ g Φ I U ∂ r S δc I [0] (cid:17) = 0 . (A.2)40e integrate this equation for δχ I [1] while insisting on the near horizon behaviour(2.30). After doing so, we obtain the asymptotic behavior δχ I [1] = r ZI − (2∆ Z I − φ Iv (cid:104) w jI ζ j [1] + √ g (0) Φ (0) I (cid:16) g ij (0) k Ii v j [1] − iω [1] δc I [0] (cid:17) (A.3)+ iq i (cid:0) ν iI δT [0] + β iI δµ [0] (cid:1)(cid:105) + · · · . (A.4)Demanding that the operator S I is not sourced at this order, we must have √ g (0) Φ (0) I (cid:16) iω [1] δc I [0] − g ij (0) k Ii v j [1] (cid:17) = w jI ζ j [1] + iq i (cid:0) ν iI δT [0] + β iI δµ [0] (cid:1) . (A.5) A.1.2 Field theory interpretation at order O ( ε ) At order O ( ε ) we obtain the equation −√ g Φ I k Ii g ij ζ j [2] − ∂ r (cid:16) √ g Φ I U g ij k Ii δg rj [2] (cid:17) + iq j δ (cid:0) √ g Φ I g ij (cid:1) [1] k Ii − q i q j √ g Φ I g ij δc I [0] + ∂ r (cid:16) √ g Φ I U ∂ r δχ I [2] (cid:17) − iω [2] ∂ r (cid:16) √ g Φ I U ∂ r S δc I [0] (cid:17) = 0 , (A.6)where we have used that δT [0] = δµ [0] = ω [1] = 0 shown in subsection A.3 below.Following similar steps as above, we obtain the asymptotic expansion δχ I [2] = r ZI − (2∆ Z I − φ Iv (cid:104) w jI ζ j [2] + √ g (0) Φ (0) I (cid:16) g ij (0) k Ii v j [2] − iω [2] δc I [0] (cid:17) + iq i (cid:32) ν iI δT [1] + β iI δµ [1] − i (cid:88) J w ijIJ δc J [0] q j (cid:33)(cid:105) + · · · . (A.7)This result, along with equation (3.13), allows us to write the Josephson relation(3.21). A.2 Constitutive relations for the thermoelectric currents
In this subsection we will relate the horizon current densities (2.32) to the boundaryquantities δJ i and δQ i that appear in the current conservation equation (3.12).The bulk electric current is defined as δJ ibulk = √− g τ δF ir . (A.8)The equations of motion (2.2) imply ∂ r δJ ibulk = ∂ t (cid:0) √− g τ δF ti (cid:1) + ∂ j δ (cid:0) √− g τ F ji (cid:1) . (A.9)41ollowing [43], for any vector Λ µ in the bulk we can define the bulk two-form G µν = − ∇ [ µ Λ ν ] − τ Λ [ µ F ν ] ρ A ρ − (cid:0) Λ ρ A ρ − f (cid:1) τ F µν , (A.10)where Λ µ F µν = ∂ ν f + β ν , with β a 1-form and f a globally defined function. Afterusing the equations of motion (2.2), its divergence can be brought to the form ∇ µ G µν = V Λ ν + 2 ∇ ν ∇ ρ Λ ρ − ∇ µ ∇ ( µ Λ ν ) + 12 τ F νρ β ρ − A ρ L Λ ( τ F νρ ) − τ F νρ A ρ ∇ µ Λ µ + (cid:32)(cid:88) I G I ∂ ν φ I ∂ ρ φ I + (cid:88) J W J ∂ ν ψ J ∂ ρ ψ J (cid:33) Λ ρ + (cid:32)(cid:88) I Φ I ∂ ν χ I ∂ ρ χ I + (cid:88) J Ψ J ∂ ν σ J ∂ ρ σ J (cid:33) Λ ρ . (A.11)We now consider Λ µ = ∂ t , and a general perturbation around the backgroundansatz (2.14) (not necessarily of the form (2.28)). The bulk heat current is definedas δQ ibulk = √− g G ir = U √ g g ij (cid:18) ∂ r (cid:18) δg jt U (cid:19) − ∂ j (cid:18) δg rt U (cid:19)(cid:19) − a t δJ ibulk = U / √ g (cid:104) K it + U / g ij ∂ t δg rj (cid:105) − a t δJ ibulk , (A.12)where we have used the result of Appendix B of [30] for the extrinsic curvaturecomponent K i t = 12 U / g ij (cid:20) ∂ r (cid:18) δg jt U (cid:19) − ∂ j (cid:18) δg rt U (cid:19) − ∂ t δg rj U (cid:21) . (A.13)Writing ˜ t µν = − K µν + Xδ µν + Y µν , where X = 2 K + · · · and Y are additonal termsthat come from the counterterms, we recognize ˜ t µν as the field theory stress tensor,when evaluated on the boundary. Evaluating (A.12) at the boundary, this gives δQ ibulk (cid:12)(cid:12)(cid:12) ∞ = − (cid:16) r − t it + µ δJ ibulk (cid:12)(cid:12)(cid:12) ∞ (cid:17) , (A.14)where t µν = r ˜ t µν . Note that the contribution from Y it , as coming from (2.20),and contribution from the term involving a time derivative are subleading even inthe precense of sources. This result matches the expression for the boundary heatcurrent obtained from the variation of the action in the presence of the sources as in(2.28) δS = (cid:90) d x √− h (cid:20) r − t µν δg µν + r − J µ δA µ (cid:21) , (A.15)42here h µν = g µν − n µ n ν and n is the unit norm normal vector. Furthermore, equation(A.11) implies the radial dependence ∂ r δQ ibulk = ∂ j (cid:0) √− gG ji (cid:1) + ∂ t (cid:0) √− gG ti (cid:1) − √− g ∂ i ∂ t log √− g + 2 √− gg iρ ∇ µ ∂ t g µρ − √− g τ F iρ ∂ t A ρ + 12 ∂ t (cid:0) √− g A ρ τ F iρ (cid:1) − √− g (cid:32)(cid:88) I G I ∂ i φ I ∂ t φ I + (cid:88) J W J ∂ i ψ J ∂ t ψ J (cid:33) − √− g (cid:32)(cid:88) I Φ I ∂ i χ I ∂ t χ I + (cid:88) J Ψ J ∂ i σ J ∂ t σ J (cid:33) . (A.16) A.2.1 The boundary currents at order O ( ε ) Expanding the radial evolution for the electric current (A.9) to order O ( ε ) we obtain ∂ r δJ ibulk [1] = − τ √ gg lk g ij ε kj B ζ l [1] + iq j δ (cid:16) τ √ gg jk g il (cid:17) [0] ε kl B , (A.17)which can integrate from the horizon up to the conformal boundary at infinity to find δJ i ∞ [1] = δJ i (0)[1] − M ij ζ j [1] + iq j (cid:0) ∂ T M ij δT [0] + ∂ µ M ij δµ [0] (cid:1) = √ g (0) τ (0) g ij (0) (cid:16) − iq j δµ [0] + E j [1] + v j [1] a (0) t + Bε jl g lk (0) v k [1] (cid:17) − M ij ζ j [1] + iq j (cid:0) ∂ T M ij δT [0] + ∂ µ M ij δµ [0] (cid:1) . (A.18)For the radial evolution of the heat current, after expanding equation (A.16) weobtain ∂ r δQ ibulk [1] = √ gτ ε ij B ( E j [1] − a t ζ j [1] ) − iq j δ (cid:0) √− g τ a t F ji (cid:1) [0] + iω [1] √ g g ij (cid:88) I Φ I k Ij δc I [0] . (A.19)Integrating from the horizon to infinity we obtain δQ i ∞ [1] = δQ i (0)[1] + iω [1] (cid:88) I w iI δc I [0] − M ij E j [1] − M ijT ζ j [1] + iq j (cid:0) ∂ T M ijT δT [0] + ∂ µ M ijT δµ [0] (cid:1) = 4 πT √ g (0) g ij (0) v j [1] + iω [1] (cid:88) I w iI δc I [0] − M ij E j [1] − M ijT ζ j [1] + iq j (cid:0) ∂ T M ijT δT [0] + ∂ µ M ijT δµ [0] (cid:1) . (A.20)43 .2.2 The boundary currents at order O ( ε ) Using the fact that δT [0] = δµ [0] = ω [1] = 0 (shown in subsection A.3 below), weproceed to compute the currents at next order in O ( ε ).The radial evolution equation (A.9) for the electric current gives ∂ r δJ ibulk [2] = − τ √ gg lk g ij ε kj B ζ l [2] + iq j δ (cid:16) τ √− gg jk g il (cid:17) [1] ε kl B , (A.21)which can be integrated to give the expression δJ i ∞ [2] = δJ i (0)[2] − M ij ζ j [2] + iq j (cid:32) ∂ T M ij δT [1] + ∂ µ M ij δµ [1] + iq l (cid:88) I ∂ k Il M ij δc I [0] (cid:33) = √ g (0) τ (0) g ij (0) (cid:16) − iq j δµ [1] + E j [2] + v j [2] a (0) t + Bε jl g lk (0) v k [2] (cid:17) − M ij ζ j [2] + iq j (cid:32) ∂ T M ij δT [1] + ∂ µ M ij δµ [1] + iq l (cid:88) I ∂ k Il M ij δc I [0] (cid:33) . (A.22)For the heat current we have ∂ r δQ ibulk [2] = √ gτ ε ij B ( E j [2] − a t ζ j [2] ) − iq j δ (cid:0) √− g τ a t F ji (cid:1) [1] + iω [2] √ g g ij (cid:88) I Φ I k Ij δc I [0] . (A.23)Integrating from the horizon to infinity we obtain δQ i ∞ [2] = δQ i (0)[2] + iω [2] (cid:88) I w iI δc I [0] − M ij E j [2] − M ijT ζ j [2] + iq j (cid:32) ∂ T M ijT δT [1] + ∂ µ M ijT δµ [1] + iq l (cid:88) I ∂ k Il M ijT δc I (0) (cid:33) = T s v i [2] + iω [2] (cid:88) I w iI δc I [0] − M ij E j [2] − M ijT ζ j [2] + iq j (cid:32) ∂ T M ijT δT [1] + ∂ µ M ijT δµ [1] + iq l (cid:88) I ∂ k Il M ijT δc I (0) (cid:33) . (A.24) A.3 Horizon vector constraint
In this subsection, following [11], we use the vector constraint (2.33c) in order to showthat δT [0] = δµ [0] = ω [1] = 0, as well as solve for the horizon fluid velocity v i [2] in termsof the zero modes and the sources . At O ( ε ), the vector constraint (2.33c) gives B ij v j [1] + iq i √ g (0) (cid:0) ρ δµ [0] + s δT [0] (cid:1) + iq k Bε ij g jk (0) τ (0) δµ [0] − iω [1] (cid:88) I Φ (0) I k Ii δc I [0] = 0 , (A.25)44here we have defined B ij = (cid:88) J Ψ (0) J k Jsi k Jsj + (cid:88) I Φ (0) I k Ii k Ij + τ (0) B ε ik ε jl g kl (0) − πρs Bε ij . (A.26)We now note that the boundary currents δJ i ∞ and δQ i ∞ are of order O ( ε ), and so theWard identities (3.15) give iω [1] (cid:32) T − c µ ξξ χ q (cid:33) (cid:32) δT [0] δµ [0] (cid:33) = 0 . (A.27)We now consider the two possibilities for ω [1] : ω [1] (cid:54) = 0: Provided the matrix of susceptibilities in (A.27) is invertible, as isgenerically the case, we deduce from (A.27) that δT [0] = δµ [0] = 0. However, we canthen combine (A.5) and (A.25), leading to √ g (0) (cid:32)(cid:88) J Ψ (0) J k Jsi k Jsj + τ (0) B ε ik ε jl g kl (0) − πρs Bε ij (cid:33) v j [1] − (cid:88) I k Ii w jI ζ j [1] = 0 . (A.28)In order to find quasinormal modes we set the sources to zero ζ i [1] = 0, which thenleads to v j [1] = 0. This in turn leads to the trivial perturbation with δc I [0] = 0 aswell. ω [1] = 0: In this case, (A.27) contributes nothing new. However, at next order in ε , the continuity equations will lead to another version of (A.27), but with ω [1] → ω [2] .The only way this relation can avoid conflicting with the combination of (A.5) and(A.25) is if δT [0] = δµ [0] = 0.We now solve the horizon vector constraint at order O ( ε ) to write T s v j [2] = iω [2] (cid:88) I λ jI δc I [0] + T ¯ κ jiH (cid:0) ζ i [2] − iq i T − δT [1] (cid:1) + T ¯ α jkH (cid:0) E k [2] − iq k δµ [1] (cid:1) , (A.29)where we have defined σ ij = τ (0) s π g ij (0) , N ik = δ ik + Bρ ε ij σ jk , η Ii = 14 πT Φ (0) I k Ii , ¯ α ikH = 4 πρ (cid:0) B − (cid:1) ij N jk , ¯ κ ikH = 4 πT s (cid:0) B − (cid:1) ik , λ jI = T ¯ κ jiH η Ii , (A.30) As explained in [11], we can alternatively get this system by considering the horizon scalarconstraints (2.33a)-(2.33b) at order O ( ε ). Assuming that the matrix multiplying v j [1] in (A.28) is invertible in generic backgrounds. B ij is given in (A.26) and indices in N are raised and lowered with the horizonmetric g (0) ij . The expressions (A.22) and (A.24) for the currents contain the horizonfluid velocity v i [2] . Substituting (A.29), leads to the constitutive relations (3.18) pre-sented in the main text. Essentially the explicit lattice has allowed us to integrateout the fluid velocity, and this was conveniently done by solving the constraints atthe black hole horizon. B Details on dispersion relations
We will find convenient to split the mode vector which solves (3.57) according to | v (cid:105) = (cid:32) δT /Tδµ (cid:33) , | v (cid:105) = − iω ( q i ) ... δc L ... . (B.1)Then for the background with B → − B there is a different mode with | ˜ v (cid:105) and | ˜ v (cid:105) but with the same dispersion relation ω ( q i ). This can be justified by using thetransformation property (3.60) and the comment below it.We can write − ( ω X ( B ) + i Σ ( B )) | v (cid:105) + m ( B ) | v (cid:105) =0 m (cid:48) ( B ) | v (cid:105) + (cid:0) − i Θ ( B ) + ω − W ( B ) (cid:1) | v (cid:105) =0 , (B.2)while for the time reversed configuration with B → − B we have − ( ω X ( − B ) + i Σ ( − B )) | ˜ v (cid:105) + m ( − B ) | ˜ v (cid:105) =0 m (cid:48) ( − B ) | ˜ v (cid:105) + (cid:0) − i Θ ( − B ) + ω − W ( − B ) (cid:1) | ˜ v (cid:105) =0 . (B.3)From the above systems and after using (3.59) we obtain the relation iω = ω ¯ ω (cid:16) (cid:104) ˜ v | X | v (cid:105) + (cid:104) v | X T | ˜ v (cid:105) (cid:17) + (cid:104) ˜ v | W | v (cid:105) + (cid:104) v | W T | ˜ v (cid:105)(cid:104) ˜ v | Σ | v (cid:105) + (cid:104) v | Σ T | ˜ v (cid:105) + (cid:104) ˜ v | Θ | v (cid:105) + (cid:104) v | Θ T | ˜ v (cid:105) , (B.4)showing that iω has to be a real number.For B = 0, the matrices X , W , Σ and Θ are symmetric. We also know that thevectors | v (cid:105) and | v (cid:105) coincide with the vectors | ˜ v (cid:105) and | ˜ v (cid:105) . This observation showsthat if the matrices X and W are positive definite, then iω > eferences [1] S. A. Hartnoll, P. K. Kovtun, M. Muller, and S. Sachdev, “Theory of theNernst effect near quantum phase transitions in condensed matter, and indyonic black holes,” Phys. Rev.
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