Hydrodynamic magneto-transport in charge density wave states
Andrea Amoretti, Daniel Arean, Daniel K. Brattan, Nicodemo Magnoli
PPrepared for submission to JHEP
IFT-UAM/CSIC-21-1
Hydrodynamic magneto-transport in charge density wave states
Andrea Amoretti, , Daniel Are´an, , Daniel K. Brattan, Nicodemo Magnoli. , Dipartimento di Fisica, Universit`a di Genova, via Dodecaneso 33, I-16146, Genova, Italy I.N.F.N. - Sezione di Genova, via Dodecaneso 33, I-16146, Genova, Italy Departamento de F´ısica Te´orica, Universidad Aut´onoma de Madrid, Campus de Canto-blanco, 28049 Madrid, Spain Instituto de F´ısica Te´orica UAM/CSIC, Calle Nicol´as Cabrera 13-15, 28049 Madrid, Spain
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
In this paper we study the dynamical properties of charged systems im-mersed in an external magnetic field and perturbed by a set of scalar operators break-ing translations either spontaneously or pseudo-spontaneously. By combining hydro-dynamic and quantum field theory arguments we provide analytic expressions for allthe hydrodynamic transport coefficients relevant for the diffusive regime in terms ofthermodynamic quantities and DC thermo-electric conductivities. This includes themomentum dissipation rate. We shed light on the role of the momentum dissipationrate in the transition between the pseudo-spontaneous and the purely explicit regimesin this class of systems. Finally, we clarify several relations between the hydrodynamictransport coefficients which have been observed in the holographic literature of chargedensity wave models. a r X i v : . [ h e p - t h ] J a n ontents The stability of many time evolving processes relies on dissipation precluding a run-away behavior. In particular, momentum dissipation plays a key role in systems such asdisordered materials or phases with pseudo-spontaneous spatial ordering. The analysisof the transport properties in systems at non-zero charge densities where translationsare broken either spontaneously or pseudo-spontaneously, commonly referred as chargedensity wave systems, can be traced back to [1–3]. Particularly relevant in order tomake contact with experimental setups is the study of such systems in the presenceof an external magnetic field [4, 5]. In recent times, the discovery of this type ofordering in strongly coupled materials such as high temperature superconductors (see e.g. [6, 7]) has caused a revival of the experimental and theoretical approaches to thetopic of pseudo-spontaneous spatial ordering. In particular, hydrodynamic descriptionsof these systems both at zero [8, 9] and non-zero [10, 11] external magnetic field havebeen developed in the last years, together with several holographic models [12–26].In this work we study (2 + 1)-dimensional systems at finite temperature andchemical potential in the presence of an external magnetic field, where a scalar operator– 1 –ccounts for the breaking of (spatial) translation invariance. More specifically, weconsider systems satisfying the following one-point function Ward identities ∂ µ (cid:104) T µν (cid:105) = F νµ (cid:104) J µ (cid:105) − (cid:0) ∂ ν Φ i (cid:1) (cid:104) O i (cid:105) , ∂ µ (cid:104) J µ (cid:105) = 0 , (1.1)where T µν is the stress tensor, F µν an external electromagentic field strength, and J µ a U (1) charge current. The terms O i and Φ i are respectively a scalar operator and itssource. They transform under the two-dimensional Euclidean group and when non-zerowill therefore break spatial translation invariance. At the background level we assumethat the source Φ i and vev O i are time independent.Different regimes for these systems can be defined by the relative values of Φ i and O i . Whenever the source Φ i is vanishing, but the vev (cid:104) O i (cid:105) is non-zero in the groundstate, we say that translations are broken spontaneously. Fluctuations of the scalarsabout the background vev are then Goldstone bosons. This should be compared againstthe explicit situation where the O i acquire a non-zero vev only because the Φ i are non-zero. Within the explicit regime we can further distinguish the pseudo-spontaneousand truly explicit cases. In the former the vev (cid:104) O i (cid:105) is much larger than its source andfluctuations of the scalar can then be interpreted as pseudo-Goldstone bosons. Thesepseudo-Goldstone bosons have a small, non-zero mass which is related to the “pinningfrequency” [8] (denoted ω ). On the other hand, a truly explicit case occurs when themagnitudes of source and vev are at least comparable. This happens for example inthe models of [27], where only the source is non-zero and the vev is vanishing.In both the spontaneous and explicit regimes the conservation laws (1.1) implyWard identities involving the two point functions of the system. It has been known fora long time [28] that for charged fluids in an external magnetic field, the thermo-electricconductivities are inter-related in such a way that knowing the electric conductivity atnon-zero frequency is sufficient to determine the thermo-electric and thermal conduc-tivities. We shall demonstrate that this structure generalises to the systems describedabove and, as a consequence, we find non-trivial constraints on the low frequency ACconductivities.The results from the Ward identities provide important constraints when we sub-sequently construct a complete description of the hydrodynamics of this class of modelsin the diffusive sector to first order in fluctuations and derivatives. We use the approachof [29] to determine analytically, to all orders in the magnetic field, the hydrodynamictransport coefficients which appear in the constitutive relations relevant for the diffu-sive sector. This method [29] ensures that hydrodynamics always reproduces the DCconductivities of the system (including the thermal conductivities). The resulting an-alytic expressions are one of our key results and allow us to perform an analysis of thebehaviour of these systems in various regimes without resorting to numerics.– 2 –ne of the most important consequences of the Ward identities for hydrodynamicsconcerns the types of dissipation that can appear in the hydrodynamic equations ofmotion. Generically, when translation invariance is broken, one allows for explicitdissipation terms, i.e. ∂ t P i ∼ − Γ ij P j + F ij J j − (cid:0) ∂ i Φ j (cid:1) O j , ∂ t O i ∼ − Ω ij O j , (1.2)where P i is the spatial momentum. The first equation contains the momentum dissipa-tion tensor Γ ij , whose precise hydrodynamic definition we shall give later. The secondexpression contains a phase relaxation tensor Ω ij which accounts for the spreading outthat occurs for a massive scalar field as it evolves (it is not protected by a conservationlaw). As a precise consequence of the Ward identities coming from (1.1) we find thatΓ ij = 0. We shall demonstrate however the emergence of an effective Γ ij , given in termsof the phase relaxation Ω ij and the pinning frequency ω , in regimes where the scalarfield is effectively kinetic (strongly explicit regime).With this result to hand, we are also in a position to clarify and extend a keyobservation of [17], further discussed in [24, 30, 31], namely that in the limit of smallpinning frequencies the phase relaxation coefficients (longitudinal and Hall) are pro-portional to the square of the pinning frequency. We also suggest a way to identifythe constant of proportionality to be the Goldstone susceptibilities [17, 24, 31] at lowpinning frequency, independently of the value of the magnetic field.The outline of the paper is as follows: in section 2 we discuss the consequences ofthe Ward identities for the correlators of the system we have outlined above (withoutrestricting ourselves to any particular regime). In section 3 we introduce the hydrody-namic description and compute the AC correlators. At the end of this section we derivenovel and important results for the transport properties of the system that constitutethe main outcome of this paper. Finally, in section 4 we draw some conclusions anddiscuss future directions. Consider a generic (2 + 1)-dimensional system at non-zero temperature ( T ) and non-zero chemical potential ( µ ) in the presence of an external magnetic field ( B ). Weassume that this system contains a scalar operator that transforms under two copies ofthe two-dimensional Euclidean group - one which represents spatial symmetry and theother a redundant shift symmetry. These scalars will either spontaneously gain a vevor be explicitly sourced in such a way that this double Euclidean symmetry is brokendown to a diagonal subgroup in the ground state. We will only consider one such– 3 –air of translation breaking scalars, but the generalisation to more is straightforward.Additionally we assume that spatial rotation invariance and spatial parity invarianceare unbroken (except in this latter case by terms proportional to the magnetic field).We are interested in the response of this theory to perturbations of charge density,temperature and the translation breaking scalars. Consequently, we now consider theWard identities following from the conservation equations for the (almost) conservedstress tensor ( T µν ) and U (1) charge current ( J µ ) [32], ∂ µ (cid:104) T µν (cid:105) = F νµ (cid:104) J µ (cid:105) − (cid:0) ∂ ν Φ i (cid:1) (cid:104) O i (cid:105) , ∂ µ (cid:104) J µ (cid:105) = 0 , (2.1)where i = 1 , F µν is abackground field strength and Φ i and (cid:104) O i (cid:105) are the sources and vevs for the translationbreaking scalars. We shall be interested in situations where the background sourcesΦ i and vevs (cid:104) O i (cid:105) are spatially homogeneous and consider two cases: spontaneous andexplicit breaking of translation invariance. In the former situation Φ i = 0 in the groundstate but (cid:104) O i (cid:105) (cid:54) = 0. In the latter - explicit - case Φ i (cid:54) = 0.In the subsequent sections we will discuss the consequences of these Ward identitiesfor AC transport in the diffusion sector. Hence, as our concern will primarily be inthe regime described by diffusive hydrodynamics, we will deal with heat rather thanspatial momentum ( P i ) conservation. Therefore we define the spatial canonical heatcurrent ( Q i ) by (cid:104) Q i (cid:105) = (cid:104) P i (cid:105) − µ (cid:104) J i (cid:105) , (2.2)where i indexes spatial directions. It is important to note that in the non-linear hydro-dynamic theory the canonical heat current is generally not the one that enters into theentropy conservation equation; this will be relevant when we discuss the constitutiverelations as the leading term in the spatial heat current will differ from sT . We wish to obtain the Ward identities that follow from (2.1) when considering theconsequences of broken translational symmetry and unbroken U (1) gauge invariance.We follow a Fourier transform convention where for a function f ( t, (cid:126)x ) f ( t, (cid:126)x ) = (cid:90) d kdω (2 π ) f ( ω, (cid:126)k ) e − i ( ωt − i(cid:126)k · (cid:126)x ) . (2.3)– 4 –onsequently, at arbitrary frequency and zero wavevector, the desired Ward identitiesare iω (cid:104) Q i Q j (cid:105) = − (cid:0) iωµδ ik − F ik (cid:1) (cid:104) Q k J j (cid:105) − iω ( χ ππ − µn ) δ ij , (2.4a) iω (cid:104) Q i J j (cid:105) = − (cid:0) iωµδ ik − F ik (cid:1) (cid:104) J k J j (cid:105) − iωnδ ij , (2.4b) F ij = B(cid:15) ij , (cid:15) = 1 , where n is the electric charge density and χ ππ is the momentum susceptibility whichthe reader can either define through the contact term in the Ward identity (2.4a) orthrough the thermodynamic susceptibility we discuss in section 3. There is a thirdWard identity involving the scalar which requires a little care, as the stress energytensor in the presence of external sources must be carefully identified before varying itwith respect to the scalar source. Following the procedure described in [22, 32, 33] andnormalising the scalar operator in such a way that ∂ i (cid:104) O j (cid:105) ∝ δ ij , we obtain: iω (cid:104) Q i O j (cid:105) = − (cid:0) iωµδ ik − F ik (cid:1) (cid:104) J k O j (cid:105) + δ ij . (2.4c)From these three Ward identities, one can see that knowing the correlators (cid:104) J i J j (cid:105) and (cid:104) J i O j (cid:105) to arbitrary frequency is equivalent to knowing all of the correlators in(2.4). This follows from the ladder structure, e.g. knowing (cid:104) J i O j (cid:105) we can solve (2.4c)for (cid:104) Q i O j (cid:105) . Similarly, (cid:104) J i J j (cid:105) gives (cid:104) Q i J j (cid:105) through (2.4b), and subsequently (cid:104) Q i Q j (cid:105) through (2.4a).We define the following AC transport quantities( σ, α, κ ) ij ( ω ) = 1 iω (cid:0) (cid:104) J i J j (cid:105) , (cid:104) Q i J j (cid:105) , (cid:104) Q i Q j (cid:105) (cid:1) , ( γ, θ ) ij ( ω ) = ( (cid:104) J i O j (cid:105) , (cid:104) Q i O j (cid:105) ) , ξ ij ( ω ) = − iω (cid:104) O j O i (cid:105) , (2.5)with the first three being the standard thermo-electric conductivities. There is an unfor-tunate but deliberate overlap between the typical notation for the transport coefficientsthat will appear in the hydrodynamic constitutive relations and the corresponding ACconductivities. We shall indicate the latter, and also their DC limits, with an explicitargument. The former will always appear without an argument. Using our assumptionthat translation invariance is broken homogeneously, the AC transport tensors of (2.5)can be decomposed with respect to SO (2) rotation invariance into longitudinal andHall parts,( σ, α, κ, γ, ϑ, ξ ) ij ( ω ) = ( σ, α, κ, γ, ϑ, ξ ) (L) ( ω ) δ ij − ( σ, α, κ, γ, ϑ, ξ ) (H) ( ω )( F − ) ij , (2.6) This normalization is typically used in constructing effective field theories for phonons [34] and itwill be employed later in the hydrodynamic derivation (see formula (3.3)). – 5 –here we have employed the inverse of the field strength in our decomposition as thisintroduces an overall 1 /B factor. We use the field strength F ij to decompose our tensorsrather than the more typical two-dimensional Levi-Civita tensor (cid:15) ij because we wishto emphasise that we are considering a theory that preserves spatial parity invariancemicroscopically. Any parity breaking is due to the presence of a fixed magnetic field B . The Hall conductivities σ (H) et cetera will be smooth in the vanishing magneticfield limit with our choice of decomposition (2.6). This should be compared to theexplicit case (2.11) where the limit as B → F ij instead.Using (2.6) and substituting into (2.4) gives relations between the AC transportcoefficients for arbitrary frequency. We can thus take σ ij ( ω ) and γ ij ( ω ) to be the“independent” transport terms; all others, with the special exception of ξ ij ( ω ) whichdoes not appear in the Ward identities, can be derived from them using (2.4). Ofparticular interest are the low frequency expansions of these independent terms ( σ ij ( ω )and γ ij ( ω )): σ (L) ( ω ) = − iB (cid:0) α (H) (0) + µn (cid:1) ω + κ (L) (0) B ω + O ( ω ) , (2.7a) σ (H) ( ω ) = σ (H) (0) + κ H (0) − µ (2 χ ππ − µn ) B ω + O ( ω ) , (2.7b) γ (L) ( ω ) = i (cid:0) µ + θ (H) (0) (cid:1) B ω + O ( ω ) , (2.7c) γ (H) ( ω ) = γ (H) (0) − iθ (L) (0) ω + O ( ω ) , (2.7d)where σ (L) (0) = α (L) (0) = γ (L) (0) = 0 , σ (H) (0) = − n ,α (H) (0) = χ ππ − µn , γ (H) (0) = 1 . (2.8)The values obtained in (2.8) are a pure consequence of symmetry and the definitionof the canonical heat; for any system regardless of its microscopic formulation thatsatisfies (2.4), the displayed DC conductivities will take this form. Further, as theseexpansions are purely consequences of the Ward identities, they must hold for arbitrarymagnetic fields and temperatures.It was noted in [29] that the thermo-electric and thermal conductivities are re-sponsible for the O ( ω ) and O ( ω ) terms in the AC electric conductivity. As we haveshown, this remains true even when spatial translation invariance is spontaneously bro-ken by scalar operators e.g. if one knows for example σ (cid:48)(cid:48) (L) (0), then one can determine κ (L) (0) from (2.7a). Moreover, the DC electric and thermo-electric conductivities are– 6 –till constrained by symmetry, such that κ (L) (0) and κ (H) (0) are the system dependentDC conductivities in the suite of thermo-electric conductivities. The expressions forthe AC longitudinal and Hall electric conductivities at low frequency (2.7) are alsothe same as those in the absence of the spontaneously broken translation invariance(modulo replacement of the thermal conductivities by their system dependent values).Additional correlators involving the scalar are also relevant to the low frequencybehaviour of the system. As we can see from (2.7) these correlators form a distinct setat low frequency. Symmetry dictates the value of the γ (L) (0) = 0 and γ (H) (0) = 1, see(2.8), which are the zero frequency limits of the scalar-electric current correlators. Thisleaves θ (L) (0) and θ (H) (0) to carry the system dependent information which appears at O ( ω ) in the low frequency expansions of γ (L) ( ω ) and γ (H) ( ω ). Completely in oppositionto what we will find in the explicit case, the scalar-scalar correlator ( ξ ij ( ω )) plays norole in (2.7). In the explicit case, the translation breaking scalars have a non-zero source in thebackground which we denote by s i . We choose for this source to be linear in the spatialcoordinates in accordance with [32, 34] such that ∂ i s j = ϕ δ ij . Assuming as before that ∂ i (cid:104) O j (cid:105) = δ ij the 2-pt Ward identities at arbitrary frequency and zero wavevector canbe derived from (2.1) in the same way as in the spontaneous case [32]: iω (cid:104) Q i Q j (cid:105) = − (cid:0) iωµδ ik − F ik (cid:1) (cid:104) Q k J j (cid:105) + ϕ (cid:104) Q i O j (cid:105) − iω ( χ ππ − µn ) δ ij , (2.9a) iω (cid:104) Q i J j (cid:105) = − (cid:0) iωµδ ik − F ik (cid:1) (cid:104) J k J j (cid:105) + ϕ (cid:104) J i O j (cid:105) − iωnδ ij , (2.9b) iω (cid:104) Q i O j (cid:105) = − (cid:0) iωµδ ik − F ik (cid:1) (cid:104) J k O j (cid:105) + ϕ (cid:104) O i O j (cid:105) + δ ij , (2.9c)Unlike the spontaneous case, we can see that three correlators, (cid:104) J i J j (cid:105) , (cid:104) J i O j (cid:105) and (cid:104) O i O j (cid:105) are required to specify the arbitrary frequency behaviour of the system; com-pared to just the first two for the spontaneous case as all other correlators can bederived from them through (2.9). Clearly the scalar-scalar correlator will now play anintegral role in DC transport.We define the following new AC quantities( (cid:36), ϑ ) ij ( ω ) = 1 iω ( (cid:104) J i O j (cid:105) , (cid:104) Q i O j (cid:105)(cid:105) ) , ζ ij ( ω ) = 1 iω (cid:18) (cid:104) O j O i (cid:105) − ϕ δ ji (cid:19) , (2.10)which replace γ ij , θ ij and ξ ij in (2.5). Notice the introduction of an iω factor intothe definitions of (cid:36) and ϑ in the explicit case in comparison to the spontaneous case.The definition of the standard thermo-electric conductivities are the same as in the– 7 –pontaneous case. Moreover, we now decompose the tensor transport quantities withrespect to SO (2) rotational invariance according to( σ, α, κ, (cid:36), ϑ, ζ ) ij ( ω ) = ( σ, α, κ, (cid:36), ϑ, ζ ) (L) ( ω ) δ ij + ( σ, α, κ, (cid:36), ϑ, ζ ) (H) ( ω ) F ij . (2.11)This decomposition introduces an additional factor of B into the Hall conductivitiescompared to (2.6). This has been done because in the explicit case the Hall conductiv-ities are smooth in the B → σ ij ( ω ) and (cid:36) ij ( ω ) at arbitrary frequency, allows us to solve for α ij ( ω ). Similarly, from(2.9c), given (cid:36) ij ( ω ) and ζ ij ( ω ), we can obtain ϑ ij ( ω ). Finally, using these previouslydetermined quantities we can solve for κ ij ( ω ) through (2.9a). Therefore, σ ij ( ω ), (cid:36) ij ( ω )and ζ ij ( ω ) are the independent AC transport quantities, as everything else can bedetermined in terms of them, and they have the following form at low frequency σ (L) ( ω ) = − ϕ ζ (L) (0) B + O ( ω ) , (2.12a) σ (H) ( ω ) = n + ϕ ζ (H) (0) B + O ( ω ) , (2.12b) (cid:36) (L) ( ω ) = − ϕζ (H) (0) + O ( ω ) , (2.12c) (cid:36) (H) ( ω ) = ϕζ (L) (0) B + O ( ω ) , (2.12d)with the low frequency expansion of ζ ij ( ω ) unconstrained. Corrections up to andincluding O ( ω ) of (2.12a) and (2.12b) involve at most the first and second derivativesof ζ ij ( ω ) evaluated at zero frequency, the DC thermal conductivities and (cid:36) (L) ( ω ) and (cid:36) (H) ( ω ) plus their first derivatives evaluated at zero frequency.One can see from (2.12) the crucial role played by the scalar operator in lowfrequency transport. It is entirely responsible for the now ( c.f. the spontaneous case)non-zero longitudinal terms σ (L) (0) and (cid:36) (L) (0). Moreover it shifts the DC Hall termsfrom their symmetry dictated values.With these facts taken care of we are now in a position to make our first importantobservation. In much of the holographic literature (see e.g. [17] and references therein),when considering the hydrodynamics of systems where translation invariance is bro-ken solely by scalar operators of the kind discussed above, the effective momentumconservation equation entering the hydrodynamic description was assumed to have theform ∂ t P i + ∂ j T ij = − Γ ij P j + F iµ J µ − (cid:0) ∂ i Φ j (cid:1) O j , (2.13)– 8 –here Γ ij is some putative momentum dissipation tensor. Notice however that if wetake this equation as a starting point, we can then demonstrate that the correlationfunctions for a system with a constant Γ ij satisfy Ward identities of the form (cid:0) iωδ ik − Γ ik (cid:1) (cid:104) Q k Q j (cid:105) = − (cid:0) µ (cid:0) iωδ ik − Γ ik (cid:1) − F ik (cid:1) (cid:104) Q k J j (cid:105) + ϕ (cid:104) Q i O j (cid:105)− iω ( χ ππ − µn ) δ ij , (2.14a) (cid:0) iωδ ik − Γ ik (cid:1) (cid:104) Q k J j (cid:105) = − (cid:0) µ (cid:0) iωδ ik − Γ ik (cid:1) − F ik (cid:1) (cid:104) J k J j (cid:105) + ϕ (cid:104) J i O j (cid:105)− iωnδ ij , (2.14b) (cid:0) iωδ ik − Γ ik (cid:1) (cid:104) Q k O j (cid:105) = − (cid:0) µ (cid:0) iωδ ik − Γ ik (cid:1) − F ik (cid:1) (cid:104) J k O j (cid:105) + ϕ (cid:104) O i O j (cid:105) + δ ij . (2.14c)These should be compared to (2.9). As a direct consequence, for any system comingfrom (2.1), we have Γ ij = 0 . (2.15)In summary, even if the effective hydrodynamic description of the momentum conser-vation equation (2.13) could somehow have differed from the one-point function Wardidentities (2.1), for the resultant hydrodynamic correlation functions to satisfy (2.9) itmust be the case that (2.15) holds. A nonzero Γ features in the hydrodynamic descrip-tion of several holographic models in the literature where the breaking of translationsis mostly explicit ( e.g. [27, 35–39]). In the next section we will show within a concretehydrodynamic setup how an effective Γ ij emerges when one enters the explicit regimewhere the dynamics of the pseudo-Goldstone bosons freeze out. Hydrodynamics is an effective approach to the description of interacting systems validat large distances and late times. It consists of conservation equations and accompa-nying constitutive relations which express charge currents in terms of the differences inthermodynamic parameters from global equilibrium. Quasihydrodynamics is the part-ner theory where some of these initially conserved charges are not conserved, but decay.Recent work has shown that the quasihydrodynamic description of certain systems canbe valid in regimes where the non-conservation of a hydrodynamic charge is relativelystrong [29] i.e. one where it does not vanish in the ground state. In what follows weshall apply quasihydrodynamics to the systems discussed above.Our model will consist of the typical ingredients for describing the hydrodynamicsof a charged fluid in an external magnetic field (see e.g. [10]). In the pseudo-spontaneous– 9 –ase we must include two pseudo-Goldstone bosons arising from the breaking of atranslational+shift symmetry down to its diagonal subgroup. To start this procedurerequires us to supply the form of the currents and the associated conservation equationsin global thermodynamic equilibrium. In particular we need to understand how thetranslation breaking modes contribute to the thermodynamics.The free energy ( F ) of our (pseudo-)Goldstone bosons ( δO i =1 , ), including linearand quadratic orders in the field, must take the form [40, 41] F = (cid:90) d x (cid:20) P l (cid:0) ∂ i δO i + ∂ i δO j ∂ i δO j (cid:1) + K ∂ i δO i ) + G (cid:0) ∂ i δO j ∂ i O j + k δO i δO i (cid:1) + ( δP l − δs tr ) ∂ i δO i − δs curl (cid:15) ij ∂ i δO j (cid:21) , (3.1)where P l is the background lattice pressure [21, 40, 41], δP l its fluctuation with respectto chemical potential and temperature, K is the bulk modulus, G the shear modulusand k a small mass term (or equivalently a large inverse correlation length) for theboson. We define the “pinning” frequency in terms of k to be ω = Gk χ ππ . (3.2)In writing (3.1) we have made use of spatial rotational invariance, and spatial isotropyfor the underlying lattice [8, 10]. We have included δs tr and δs curl as the sources forthe (pseudo-)Goldstone bosons, corresponding to the operators λ tr = ∂ i δO i and λ curl = (cid:15) ij ∂ i δO j , that will enter our hydrodynamic description. We choose these operatorsas they respect rotational symmetry and become invariant under the emergent shiftsymmetry, δO i → δO i + a i , as k → k = 0 the action (3.1) enjoysboth a shift and a spatial translation symmetry. We assume our scalar effectivelydescribes the situation where this joint symmetry is broken to a diagonal subgroupsuch that the following Poisson bracket holds: (cid:8) δO i ( t, (cid:126)x ) , P j ( t, (cid:126)y ) (cid:9) = (cid:0) δ ji + ∂ j δO i ( t, (cid:126)x ) (cid:1) δ ( (cid:126)x − (cid:126)y ) , (3.3)where P i is the generator of spatial translations. Of the two terms on the right handside the latter is the usual one for translating a vector field by a constant shift. Theformer however indicates that the bosons have a non-zero “charge” under translations.This is important as it subsequently implies non-conservation of the spatial momentumfor our model ∂ t P i = (cid:8) P i , H (cid:9) = − ω χ ππ δO i + higher derivatives (3.4)– 10 –here H is the Hamiltonian of the full theory. Physically, even in the absence of otherdissipative effects, because the boson is not massless ( k (cid:54) = 0) our system will losemomentum. The appearance of the k dependent term in (3.4) is responsible for anon-zero pinning frequency.Secondly, another consequence of the commutation relations (3.3) is the “Joseph-son relation”. This gives the time evolution of the scalar in the Hamiltonian formalism[8, 42]. In particular, in the presence of a source velocity the Hamiltonian density, tofirst order in fluctuations, can be taken to have the form H = P i v i + . . . . Therefore ∂ t δO i = (cid:8) δO i , H (cid:9) (3.3) = v i + higher derivatives . (3.5)This expression will enter our hydrodynamic description as the lowest order in deriva-tives effective equation for the evolution of the scalar operators δO i . In the effectiveapproach we will have to supplement the right hand side with additional higher deriva-tive terms as one increases the derivative order of hydrodynamics. In the explicit casewe must also take account of the fact that the phonons can relax [8, 14]. This is donethrough the introduction of a phase relaxation term Ω ij δO j on the right hand side of(3.5) i.e. ∂ t δO i = − Ω ij δO j + v i + higher derivative terms . (3.6)On the assumptions we have made in detailing the thermodynamics of our trans-lation breaking scalars, we should ask ourselves what happens as the pinning frequencybecomes large. As the pinning frequency increases it will become harder and harder toexcite the translation breaking scalars. Of course, there is still a Josephson relation,and the Ward identities will continue to hold as their applicability does not dependon the value of k . Nevertheless, in the range of parameters where the phonon modeis extremely heavy, it becomes non-dynamical. In this case one can neglect the timederivative in the Josephson relation (3.6) which subsequently constraints δO i in termsof the fluid velocity v i . Substituting the solution to this constraint into the equationof motion for the momentum density (3.4) one finds: ∂ t P i = − ω χ ππ (cid:0) Ω − (cid:1) ij P j + ... , (3.7)where the ellipsis represents terms not relevant for this discussion. Equation (3.7)is exactly the equation of motion one should expect for the momentum density in asystem which exhibits explicit momentum relaxation if one identifies Γ ij ≡ ω (Ω − ) ij .This shows that in the purely explicit limit (meaning when k is large), one recovers (as– 11 –xpected) an effective description in terms of a standard momentum dissipation rateΓ, with the latter being expressed in terms of the “old” phonon variables ω and Ω ij .The correlator will look like the standard correlator of charged hydrodynamics with anexternal magnetic field and momentum dissipation [43].The observations above, (2.14) and (3.7), apply regardless of the specifics of thehydrodynamic system. From this point onward we shall make a simplifying assumption- namely, we will neglect the lattice pressure P l and its thermodynamic derivatives δP l in (3.1). In fact P l can be set to zero if one restricts the analysis to thermodynamicallystable vacua [21, 41]. Thermodynamic derivatives of the lattice pressure, δP l , might benon-zero even in these configurations but they do not appear in the set of correlatorsthat we consider in this work [41]. We shall then show how one can apply the Wardidentities to the hydrodynamic correlators to evaluate the hydrodynamic transportcoefficients that appear in the constitutive relations.With these assumptions at hand, we find that the contribution of δO i to thespatial stress tensor, obtained by applying the Noether procedure on the symmetry δO i → δO i + a i , has the form T Oij = − (cid:0) K∂ k δO k δ ij + G∂ i δO j (cid:1) , (3.8)when the source terms vanish. This expression can be made symmetric while preservingthe translational Ward identity, as expected for a theory which has spatial rotationinvariance, by subtracting the term ∆ T Oij = G (cid:0) ∂ j δO i − δ ij ∂ k δO k (cid:1) which is divergencefree in the “ i ” index. The spatial part of the stress tensor can also be re-expressedin terms of λ tr = ∂ i δO i and λ curl = (cid:15) ij ∂ i δO j at the cost of leaving an explicitly anti-symmetric term in the expression. This is achieved by adding ∆ T Oij to (3.8), T Oij = − (( K + G ) λ tr δ ij + Gλ curl (cid:15) ij ) . (3.9)The latter expression (3.9) will appear at leading order in our constitutive relation forthe spatial stress tensor of the full theory ( T ij ).We can also derive the static susceptibility for our theory. It has the form χ = ∂ T s ∂ µ s ∂ T n ∂ µ n χ ππ χ ππ χ tr = (cid:126)k Gk +( K + G ) (cid:126)k
00 0 0 0 0 χ curl = (cid:126)k G ( k + (cid:126)k ) , (3.10)– 12 –here s is the entropy density and we work in the basis of sources s A = ( T, µ, (cid:126)v, s a )with (cid:126)v the spatial velocity of the fluid. To derive the susceptibilities for the Goldstonebosons ∂ s b λ a one need simply rewrite the static equations of motion following from(3.1) including the displayed non-zero source terms. We recall that if one considers thelattice pressure to be non-zero (non-thermodynamically stable vacua), additional termswould appear in (3.10) [21, 40, 41]. These terms do not influence the results obtainedin this paper. With the susceptibilities (3.10) to hand we are in a position to begin building thehydrodynamic constitutive relations. We start in the spontaneous case, where thereis no source term for the scalars in the background. The constitutive relations wederive are to first order in derivatives and fluctuations of temperature T + δT , chemicalpotential µ + δµ and scalar sources δs tr and δs curl assuming that the spatial velocity v i vanishes in equilibrium. To lay out our notation for the transport coefficients it is easiest to first consider theconstitutive relations for the fluctuations of the vector currents corresponding to heatand electric charge flow which take the form δQ i = ( χ ππ − µn ) v i + α ij (cid:0) δE j + F jk v k − ∂ j δµ (cid:1) − κ ij ∂ j δTT − θ ij ∂ j δs tr − ι ij (cid:15) kj ∂ k δs curl , (3.11) δJ i = nv i + σ ij (cid:0) δE j + F jk v k − ∂ j δµ (cid:1) − α ij ∂ j δTT − γ ij ∂ j δs tr + γ ij (cid:15) kj ∂ k δs curl , (3.12) F ij = B(cid:15) ij , (cid:15) = 1 , to order one in derivatives and fluctuations where δE i is an electric field fluctuation and (cid:15) ij is the two dimensional Levi-Civita tensor. The tensor transport coefficients can bedecomposed with respect to SO (2) rotational and microscopic spatial parity invariance e.g. ( σ, α, κ, γ, θ, ι, . . . ) ij = ( σ, α, κ, γ, θ, ι, . . . ) (L) δ ij + ( σ, α, κ, γ, θ, ι, . . . ) (H) F ij . (3.13)The above decomposition of the transport coefficients appearing in the hydrodynamicconstitutive relations is with respect to F ij and not its inverse; unlike the AC transportcoefficients (2.6). This reflects the fact that these terms should be smooth as we take– 13 – →
0. We also note that while permitted by the gradient expansion, ι (L) and β (H) shall not appear in the diffusive sector of the spontaneous case (this will be differentin the explicit case where their value will be fixed).With these definitions to hand we can also write down the Josephson conditionto first order in derivatives and fluctuations (starting from (3.5)). Upon also imposingthe Onsager conditions we find ∂ t δO i + J iO = 0 ,J iO = − v i + γ ij (cid:0) δE j + F jk v k − ∂ j δµ (cid:1) − (cid:0) θ (L) δ ij − ι (H) F ij (cid:1) ∂ j δTT − χ tr (cid:0) ξ tr , (L) δ ij + ξ tr , (H) F ij (cid:1) ∂ j δs tr − χ curl (cid:0) ξ curl , (L) δ ij + ξ curl , (H) F ij (cid:1) (cid:15) kj ∂ k δs curl , (3.14)where we have employed spatial rotation invariance to decompose the terms propor-tional to derivatives of the scalar sources into longitudinal and Hall parts. This hasbeen done to make clear constraints imposed by the Onsager relations. In (3.14) wehave, a little erroneously, treated ξ tr , (H) and ξ curl , (H) as distinct variables. From theOnsager relations however one finds X (H) ≡ ξ tr , (H) G + K = − ξ curl , (H) G . (3.15)Similarly, in [20] spatial rotation invariance implies X (L) ≡ ξ tr , (L) G + K = − ξ curl , (L) G . (3.16)We shall henceforth collect these transport coefficients into a new tensor structure X ij .Last, but not least, we must supply the constitutive relation for the fluctuation ofthe spatial stress tensor. This has the form δT ij = (cid:18) nδµ + ( χ ππ − µn ) δTT − ( G + K ) χ tr δs tr (cid:19) δ ij − Gχ curl δs curl (cid:15) ij − ησ ij − ζ∂ k v k δ ij , (3.17) σ ij = 12 ( ∂ i v j + ∂ j v i ) − ∂ k v k δ ij , (3.18)The first two terms in (3.17) are the standard thermodynamic contributions to thestress tensor from charge and heat. The second two terms are contributions from theGoldstone boson (3.9). The second line of terms in (3.17) are all first order in derivativesand standard for a relativistic fluid. – 14 – .1.2 The AC correlators With the constitutive relations to hand we can employ the Martin-Kadanoff procedureto determine the finite frequency response of the system. Of particular interest are therelations that result when we substitute these expressions into the Ward identities.The hydrodynamic correlators must satisfy the Ward identities at arbitrary frequency.From this substitution we immediately learn, by expanding the resultant expressionsat large frequency and zero wavevector, that α ij = − µσ ij , κ ij = µ σ ij , θ ij = − µγ ij . (3.19)These constraints are the usual relations one finds for a system which has Lorentzcovariance (even if the ground state of the theory breaks Lorentz invariance) given ourchoice of heat current and normalisations of the transport coefficients. The coefficients ι (L) and ι (H) remain unconstrained. Fortunately neither of these coefficients enter intothe diffusive sector for spontaneous breaking. In the explicit case, where they do enter,we will fix them using the Ward identities.The relativistic constraints on the hydrodynamic transport coefficients (3.19) canreadily be seen from the Ward identities at large frequency. At low frequency we canuse the Ward identities of (2.7) and the AC hydrodynamic correlators to determine thehydrodynamic transport coefficients in terms of the DC values of those same correlators.We find for the incoherent electric charge conductivitiesΞ σ (L) = χ ππ κ (L) (0) , (3.20a)Ξ σ (H) = nB (cid:16) B κ (L) (0) + (cid:0) κ (H) (0) + µ n (cid:1) (cid:17) − µχ ππ B + χ ππ B (cid:0) κ (H) (0) + 5 µ n (cid:1) − µnχ ππ B (cid:0) κ (H) (0) + µ n (cid:1) , (3.20b)Ξ ≡ B κ (L) (0) + (cid:0) κ (H) (0) + µ n − µχ ππ (cid:1) , (3.20c)where the lack of argument on σ (L) and σ (H) indicates that these are transport coeffi-cients of the constitutive relations ( e.g. (3.11)), while κ (L) (0) and κ (H) (0) are DC valuesfor the thermal conductivities. Additionally we have expressions for the hydrodynamictransport coefficients γ (L) , γ (H) , X (L) and X (H) which we shall present in a compactifiedform shortly (see (3.28)). As per magneto-transport without broken translation invari-ance we see that in the spontaneous case, generically, there is a non-zero incoherentHall conductivity.Having imposed all the constraints implied by the Ward identities we can nowunderstand the response of our system to time varying fields. For example, the electricconductivities at non-zero frequency in the spontaneous case have almost exactly the– 15 –ame form as those for the dyonic black hole in terms of the momentum susceptibility χ ππ and the DC thermal conductivities. In particular σ (L) ( ω ) = iωχ ππ ( γ − iγ c ω + ω ) B (( ω + iγ c ) − ω ) , (3.21) σ (H) ( ω ) = − (cid:18) n + ω ω c Bχ ππ (( ω + iγ c ) − ω ) (cid:19) , (3.22)where the cyclotron frequency and decay rate are ω c = Bχ ππ (cid:0) µ n − µχ ππ + κ (H) (0) (cid:1) Ξ , γ c = B χ ππ κ (L) (0)Ξ , (3.23)respectively. The longitudinal conductivity expressed in terms of the cyclotron fre-quency and decay rate has the same form as found in [28], the key difference between[29] and [28] being a different identification of the cyclotron frequency and cyclotrondecay rate. The Hall conductivity differs between the two because in [29] there is nosimple expression for n in terms of ω c .As was noted previously, it is only necessary to know (cid:104) J i J j (cid:105) and (cid:104) J i O j (cid:105) at arbitraryfrequency to determine all other correlators that appear in the Ward identities. Forthis latter correlator, its value at non-zero frequency is γ (L) ( ω ) = ω (cid:0) Bωω c θ (L) (0) − ( γ c ( ω + iγ c ) + iω ) (cid:0) θ (H) (0) + µ (cid:1)(cid:1) B (( ω + iγ c ) − ω ) , (3.24) γ (H) ( ω ) = ω ω c (cid:0) θ (H) (0) + µ (cid:1) + B (cid:0) iω ( γ − iγ c ω + ω ) θ (L) (0) + ( ω + iγ c ) − ω (cid:1) B (( ω + iγ c ) − ω ) , (3.25)where θ (L) (0) and θ (H) (0) are the longitudinal and Hall components of the zero frequencylimit of (cid:104) Q i O j (cid:105) . We remind the reader that this is the system dependent informationnecessary to specify the zero frequency behaviour of any system satisfying our assump-tions.The AC correlator involving two Goldstone bosons is separated from the othercorrelators as it does not enter into the Ward identities (unlike in the explicit case).Consequently, its values are not generally constrained by the Ward identities and DCdata. Instead, we can only learn information about this correlator by specifying thatwe are in the hydrodynamic regime. The expression is somewhat complex so it isworthwhile to introduce some new notation. Letˆ σ = σ (L) − σ (H) F , ˆ γ = γ (L) − γ (H) F , ˆΓ = F · ˆ n , ˆ I = + F · ˆ γ , ˆ n = 1 χ ππ ( n − F · ˆ σ ) , (ˆ κ, ˆ θ )(0) = ( κ, θ ) (L) (0) − ( κ, θ ) (H) (0) F − , (3.26)– 16 –hen the Goldstone-Goldstone correlator takes the formˆ X ( ω ) = ˆ X − iωχ ππ ˆ I · ˆ I · ˆΛ − ( ω ) , ˆΛ( ω ) = − iω · (cid:16) − iω + ˆΓ (cid:17) . (3.27)Simultaneously, we can rewrite our other correlators in this unified notation (whichwill be particularly useful for the explicit case). As such, the independent transportcoefficients appearing in the hydrodynamic constitutive relations areˆ σ = ˆΦ − (cid:0) n ˆ κ (0) + ( µn − χ ππ ) F − (cid:1) , (3.28a)ˆ γ = ˆΦ − (cid:16) µ ( µn − χ ππ ) F − − χ ππ ˆ θ (0) − ˆ κ (0) (cid:17) , (3.28b)ˆ X = ˆΦ − (cid:16) µ ( µn − χ ππ ) ˆ X (0) − µ F − − ˆ X (0) · F · ˆ κ (0)+ 2 µ ˆ θ (0) − ˆ θ (0) · F · ˆ θ (0) (cid:17) , (3.28c)ˆΦ = F · ˆ κ (0) − (cid:0) µ n − µχ ππ (cid:1) , (3.28d)which allows us to rewrite (3.21) and (3.24) asˆ σ ( ω ) = ˆ σ − iωχ ππ ˆ n · ˆ n · ˆΛ − ( ω ) , (3.29a)ˆ γ ( ω ) = ˆ γ + iω ˆ n · ˆ I · ˆΛ − ( ω ) , (3.29b)with ˆΛ( ω ) given in (3.27). Distinctly from the spontaneous case, and recalling that due to (2.15) there is nomomentum relaxation rate (Γ ij = 0), in the explicit case there is the potential for anon-zero phase relaxation term Ω ij in the Josephson relation, namely ∂ t O i + J iO = − Ω ij O j , (3.31) ∂ t P i + ∂ j T ij = − nE i + F ij J j − ω χ ππ O i . (3.32)We can employ SO (2) spatial rotation invariance and microscopic spatial parity invari-ance to decompose the phase relaxation tensor intoΩ ij = Ω (L) δ ij + Ω (H) F ij , (3.33) One can instead assume the most general thing possible ( i.e. Γ ij (cid:54) = 0) for the equations of motion,so that ∂ t P i + ∂ j T ij = − Γ ij P j − nE i + F ij J j − ω χ ππ O i . (3.30)However, the comparison of the previous equations against the Ward identities (2.9) at large frequencyimplies Γ ij = 0, in accordance with (2.15). – 17 –here the coefficients must be determined from data.Modulo these modifications to the (non-)conservation equations, the situation isnot so different from the spontaneous case. In particular the constitutive relationsare unmodified. However, to employ our formalism we shall have to assume that thepinning frequency, and consequently k , is sufficiently small such that the dynamicsof the scalar modes are of the pseudo-Goldstone type and hence they are long-livedenough to enter our effective hydrodynamic description. This can be seen post-hoc bycomparing AC correlators to real data. We can once again employ the Martin-Kadanoff procedure and determine the con-straints that result when we substitute the correlators into the Ward identities. First,by comparing the 1-pt Ward identity (2.1) with (3.32) we find ϕ = ω χ ππ , (3.34)which fixes the source of the scalar ϕ in terms of the pinning frequency, or using (3.2) interms of the mass of the phonon k . Additionally, from expanding the Ward identitiesat large frequencies we learn that ι ij = µγ ij , (3.35)which constrain transport coefficients that were unconstrained in the spontaneous case.Given these new identifications we find that the independent AC correlators, fromwhich all others can be derived using (2.9), areˆ σ ( ω ) = ˆ σ + (cid:16) χ ππ ˆ n · ˆ n · (cid:16) − iω + ˆΩ (cid:17) + χ ππ ω (ˆ n · ˆ γ · (2 + F · ˆ γ ) + iω ˆ γ · ˆ γ ) (cid:1) ˆΞ − ( ω ) , (3.36a)ˆ (cid:36) ( ω ) = (cid:16) ˆ n · ˆ I + (cid:16) iω − ˆΓ (cid:17) · ˆ γ (cid:17) ˆΞ − ( ω ) , (3.36b)ˆ ζ ( ω ) = 1 ω χ ππ (cid:16) iω − ˆΓ (cid:17) ˆΞ − ( ω ) , (3.36c)ˆΞ( ω ) = (cid:16) − iω + ˆΓ (cid:17) (cid:16) − iω + ˆΩ (cid:17) + ω ˆ I · ˆ I , (3.36d)where ˆΩ = Ω (L) − Ω (H) F . (3.37)To arrive at (3.36) we have imposed the constraints (3.19), (3.35) and (2.15).– 18 –gain, we can extract the incoherent conductivities (and other hydrodynamic trans-port coefficients) in terms of the DC conductivities of our system using the low fre-quency Ward identities. One point of note is that, in opposition to what is found forthe spontaneous case, the electric and thermo-electric DC conductivities contain infor-mation about the system beyond what is dictated by symmetry. We can therefore usethem to constrain the hydrodynamic transport coefficients entering the constitutive re-lations entirely in terms of the DC electric, thermo-electric and thermal conductivities.For example, the independent transport coefficients appearing in the hydrodynamicconstitutive relations for the currents are σ (L) = −
12 Tr (cid:104) ˆΨ − (cid:0) n (ˆ κ (0) · F · ˆ σ (0) − ˆ α (0) · F · ˆ α (0)) − n ˆ κ (0)+ 2 n ( χ ππ − µn ) ˆ α (0) − ( χ ππ − µn ) ˆ σ (0) (cid:1)(cid:3) , (3.38a) σ (H) = 12 Tr (cid:104) F − ˆΨ − (cid:0) n (ˆ κ (0) · F · ˆ σ (0) − ˆ α (0) · F · ˆ α (0)) − n ˆ κ (0)+ 2 n ( χ ππ − µn ) ˆ α (0) − ( χ ππ − µn ) ˆ σ (0) (cid:1)(cid:3) , (3.38b) γ (L) = −
12 Tr (cid:104) ˆΨ − ( n ˆ κ (0) + µ ( µn − χ ππ ) ˆ σ (0) + (2 µn − χ ππ ) ˆ α (0) − ˆ κ (0) · F · ˆ σ (0) + ˆ α (0) · F · ˆ α (0))] , (3.38c) γ (H) = 12 Tr (cid:104) F − ˆΨ − ( n ˆ κ (0) + µ ( µn − χ ππ ) ˆ σ (0) + (2 µn − χ ππ ) ˆ α (0) − ˆ κ (0) · F · ˆ σ (0) + ˆ α (0) · F · ˆ α (0))] , (3.38d)ˆΨ = χ ππ + µ ( µn − χ ππ ) F · ˆ σ (0) + 2 ( µn − χ ππ ) F · ˆ α (0)+ nF · ˆ κ (0) + ( F · ˆ α (0)) − F · ˆ κ (0) · F · ˆ σ (0) , (3.38e)(ˆ σ, ˆ α, ˆ κ )(0) = ( σ, α, κ ) (L) (0) + ( σ, α, κ ) (H) (0) F . (3.38f)Notice the redefinition of ˆ κ compared to the spontaneous case. These expressions interms of the thermo-electric conductivities are the most convenient for comparisonagainst any putative experiment, as they do not require any specification of the DCvalues of correlators involving the scalar. However, the interested reader can use theWard identities (2.12) to switch out dependence on σ (L) , (H) and α (L) , (H) for (cid:36) (L) , (H) and ζ (L) , (H) if they so choose.Finally, to completely specify the correlators in the explicit case we need expressionsfor the phase relaxation rates. When performing the above identification procedure forthe hydrodynamic transport coefficients one additionally findsΩ (L) = ω χ ππ (cid:104) ˆΨ − (cid:0) ˆ κ (0) + 2 µ ˆ α (0) + µ ˆ σ (0) (cid:1)(cid:105) , (3.39a)Ω (H) = − ω χ ππ (cid:104) F − ˆΨ − (cid:0) ˆ κ (0) + 2 µ ˆ α (0) + µ ˆ σ (0) (cid:1)(cid:105) . (3.39b)– 19 –hese expressions are valid at arbitrary magnetic fields and pinning frequencies, in-cluding the B → ω >
0, so long as we remain within the hydrodynamicregime. Importantly, the limits of the trace terms in (3.39) as ω → B >
0, and thus the leading dependence on ω of the phase relaxationcoefficients at small pinning frequency is ω . This becomes important when comparingthe explicit correlator as ω → (L) = χ ππ ω X (L) + O ( ω ) , Ω (H) = χ ππ ω X (H) + O ( ω ) , (3.40)for arbitrary values of the magnetic field. The first identification was already discussedin [17, 24, 30, 31]. Here we have generalized this relation to include Ω (H) (which ispresent only at non-zero magnetic field) and most importantly we have avoided anyordering ambiguities compared to the B = 0 case as in that situation ω → k → In this paper we have developed a formalism for computing the diffusion character-istics of (2 + 1)-dimensional charged fluids in an external magnetic field with brokentranslation invariance. We have worked at order one in derivatives and fluctuations andconsidered a breaking mechanism where a scalar operator acquires a spatially modu-lated vev. We observed that the system does not include any fundamental momentumdissipation effects except as an effective description when the dynamics of the trans-lation breaking scalars are frozen out. We have supplied analytic expressions for thehydrodynamic transport coefficients appearing in the constitutive relations in terms ofexperimentally measurable quantities: the DC thermo-electric transport coefficients.It would be interesting to use our formalism to more precisely understand thebehaviour of certain holographic models with (pseudo-)spontaneous breaking of trans-lation invariance [12–25]. This necessarily will require us to consider non-zero latticepressures (as these systems are unstable) [21, 40, 41]. The resultant hydrodynamicsis more complicated but we do not expect any significant divergences from what wehave found here and are currently pursuing this study. In this context, it would alsobe worthwhile to perform a more systematic scan of the parameter space of these holo-graphic models in an effort to elucidate the bounds on our hydrodynamic approach,particularly in light of current programmes examining the convergence radius of hydro-dynamics [44–46]. – 20 –e might also consider modifying our action to include terms that can generatethe effective actions described in [10] i.e. time dependencies for the scalar of the form (cid:15) ij O i ∂ t O j . Indeed, as discussed in [10], there should be a limit of our analytic expres-sions where the magnetoplasmon is pushed out of the hydrodynamic regime. Whatis not clear to us is whether, when we take this limit, the effective action describingthe scalar fluctuations in our system includes the desired kinetic term. We are alsoexploring this issue.The applications of the approach of [29] are only in the initial phases of exploration.A fundamental question to address in this direction is to apply the method to modelswhich realise translation symmetry breaking in a different way from the class analysedhere, e.g. via a spatially modulated charge density profile, and study if in this case thecorrect hydrodynamic theory is still the one described in this work.An additional potential extension is to consider generalising our formalism to(3 + 1)-dimensions and tackling anomalous transport. A comprehensive approach tosuch systems without momentum dissipation and assuming a vanishing spatial velocityin the ground state is discussed in recent work [47]. However it is expected that theseanomalous fluids will generally have a non-zero velocity in the “laboratory frame”. Thishas presented several problems in the literature as without momentum dissipation thissystem is inherently unstable. Additionally, in formulations that incorporate a non-zero velocity in the ground state, there are known problems in satisfying the Onsagerrelations [48]. Acknowledgments
We would like to thank Blaise Gout´eraux for carefully reading an earlier version ofthe manuscript. D. A. is supported by the ‘Atracci´on de Talento’ programme (2017-T1/TIC-5258, Comunidad de Madrid) and through the grants SEV-2016-0597 andPGC2018-095976-B-C21. This work has been partially supported by the INFN Sci-entific Initiative SFT: “Statistical Field Theory, Low-Dimensional Systems, IntegrableModels and Applications”.
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