Magneto-transport from momentum dissipating holography

We obtain explicit expressions for the thermoelectric transport coefficients of a strongly coupled, planar medium in the presence of an orthogonal magnetic field and momentum-dissipating processes. The computations are performed within the gauge/gravity framework where the momentum dissipation mechanism is introduced by including a mass term for the bulk graviton. Relying on the structure of the computed transport coefficients and promoting the parameters to become dynamical functions, we propose a holography inspired phenomenology open to a direct comparison with experimental data from the cuprates.


Introduction and main results
Holographic models featuring massive bulk gravitons constitute a promising framework to obtain universally relevant information on strongly coupled thermodynamical and transport properties of systems which do not conserve momentum. This has a direct impact on condensed matter studies where disorder, lattices or heavy degrees of freedom exchange momentum with the light degrees of freedom responsible for transport.
We follow the gauge/gravity way to unveil the structure of thermoelectric transport in planar, strongly coupled systems in the presence of an orthogonal magnetic field and disorder which could be valid, or at least furnish robust insight (e.g. suggesting precise experimental hints), also beyond the holographic framework.
It is well known on general grounds that holographic models at sufficiently high temperature admit a hydrodynamical description [1]. Therefore, in order to analyze the structure of the transport coefficients we are interested in, we can follow the inspiring hint coming from the hydrodynamic analysis performed in [2]. There it was shown that the thermoelectric transport properties of a hydrodynamic system at non-zero charge density, in the presence of some generic mechanism of momentum dissipation and immersed in an external magnetic field are completely determined in terms of the thermodynamical properties, the momentum dissipation rate τ −1 and the charge-conjugation symmetric conductivity σ Q . We show explicitly that for the holographic models that we study this fact remains true also outside the regime of validity of hydrodynamics 1 . In this sense, our results allow to extend the hydrodynamical analysis. As we will comment, this even leads to interesting relations with a distinct framework to study transport properties based on a Boltzmann treatment.
To convey the idea of the general aim at stake, it is instructive to think of the universal shear viscosity over entropy density ratio η/S. This is surely one of the most important holographic results in momentum conserving models [3,4]. Its universality motivates us to pursue the quests for its counterparts within a more articulate framework where specific momentum dissipating phenomena tend to obscure or impede universal claims. To be more specific, in the absence of momentum conservation, one cannot think in terms of viscosity [5], however diffusion bounds for conserved quantities (e.g. charge and heat) analogous to that concerning η/S can be possibly formulated [6,7]. The general idea of attempting to tackle the strange metal puzzles relying on universal results constitute our main background motivation.
In the same spirit, it has been recently proposed that holography supplemented with phenomenological inputs can furnish clear and consistent predictions for scaling properties of various quantities [8] (e.g. the linear in T resistivity and the quadratic Hall angle). The present paper builds on these grounds and constitutes a first step towards instilling in holography the recent experimental suggestion that the momentum dissipation rate is related to the competition of temperature and magnetic field [9], where α and η are dimensionless parameters (the latter not to be confused with the shear viscosity) and ρ is the charge density. The first main result of the paper consists in obtaining explicit expressions for all the thermoelectric transport coefficients in the model-independent language already advocated, namely in terms of thermodynamical quantities, the critical conductivity σ Q and the momentum dissipation rate τ −1 . The explicit expressions that we obtain for the transport coefficients are The complete derivation of these formulae and further comments are given in later sections.
Here we just observe that they are suitable to be compared to and/or integrated with phenomenological data. On this observation we build our second main result. We start investigating the possibility of considering the transport coefficients (1.2), (1.3), (1.4) independently of their specific holographic origin and test directly their possible wider applicability against experimental results inspired by the analysis of [8].
We underline that the parameters σ Q and τ are completely external and undetermined by the holographic model. Under appropriate assumptions, we could think of attributing to them particular behaviors (e.g. with respect to the temperature T ) inspired by experimental information. This constitutes a departure from the holographic setup and it basically amounts to assume the set of transport coefficients as a starting point to be confronted with measurements.
The most ambitious (and theoretically interesting) attitude would be to formalize and attempt to keep under control all the necessary complications of promoting the model parameters to become, in some sense to be specified, dynamical functions. It is tantalizing to think that possibly this could be a viable way of building effective field theories from holography. Without pretending to reach this goal now, we can however start to check if this extra freedom we allow ourselves (even prior to definitively firm arguments) could nevertheless permit to encompass the experimental data with a few assumptions.
In the phenomenological spirit just described, we are able to incorporate the experimental hint (1.1) and compare with the scaling of the transport coefficients described in the experimental literature. In particular, we extend the analysis of [8] to the entire set of thermoelectric transport coefficients and compare our outcomes with the hydrodynamical results of [2].
The paper is structured as follows. In Section 2 we first introduce the holographic model proposed in [10] and describe the dyonic black brane solutions needed to account for a boundary theory immersed in a magnetic field. Sections 3 and 4 constitute the computational core of the paper and they are dedicated to the analysis of thermodynamical and transport properties respectively. The reader interested in the physical outcome (in spite of the specific computational details) could jump directly to Section 5 where we set the arguments for the applicability of the formulae for the transport coefficients beyond the specific setup where they were obtained. Specifically, we compare and extend the hydrodynamical analysis performed in [2] with particular attention paid to the electromagnetic duality of the bulk model and its effects to the boundary theory. In Section 6 we make contact with the phenomenology; namely we supplement the holographic model with phenomenological information and compare the resulting set of scalings for the transport coefficients with the experimental literature. Eventually in Section 7 we give our concluding remarks suggesting many interesting directions of further investigation.

The holographic model
The model we consider features massive gravitons in the bulk. Its first consideration within the holographic context is due to [10] and then it has been further analyzed in [7,8,[11][12][13][14]. Since we are interested in describing a magnetic field, we generalize the dyonic black hole holographic models [15,16] to include massive gravity.
The gravitational action is where β is a parameter having the dimension of a mass squared, L is the AdS 4 radius, κ 4 is the gravitational constant in 3 + 1 spacetime dimensions and F µν ≡ ∂ [µ A ν] is the field strength of the gauge field A µ . We also supplemented the bulk action with the usual Gibbons-Hawking counter-term, expressed in terms of the metric (g b ) µν induced on the radial shell corresponding to the UV cut-off and the trace K of the extrinsic curvature K µν of the same manifold 2 . The Gibbons-Hawking counter-term is necessary to have a well defined bulk variational problem. The mass term for the gravitons is proportional to β and it is defined in terms of the trace (indicated with small square brackets in (2.1)) of the two matrices K µ ν and (K 2 ) µ ν defined as follows where f µν is a fixed, non-dynamical metric. This fiducial metric f µν controls the way in which the graviton mass potential breaks the diffeomorphism invariance. Since we want to discuss holographic systems dual to an isotropic boundary field theory where translational invariance is broken and, consequently, momentum is non-conserved, we need to consider the fiducial metric f µν = diag{0, 0, 1, 1} , (2.3) as described in [10]. Indeed, the non-trivial diagonal entries correspond to the boundary spatial directions. The trivial ones instead correspond to the radial and time directions along which we want to preserve bulk diffeomorphism invariance. The former being justified by the adoption of the holographic approach itself (where the holographic coordinate parametrizes the boundary theory renormalization flow), the latter being associated to energy conservation and, technically, to the construction of thermal solutions and thermal fluctuations in terms of the metric fields (see [17,18]). We do not discuss here the stability of the massive gravity model and questions related to ghost modes for which we refer to [10,19] and references therein. We want to discuss the effects due to the presence of an external magnetic field B orthogonal to the plane xy. In particular its consequences on the thermoelectric transport coefficients in the holographic system (2.1) at non-zero chemical potential µ. To include the constant magnetic field B we adopt the following ansatz for the background metric g µν and the gauge field A µ Substituting this ansatz within the equations of motion derived from (2.1), we obtain the following black brane solution where we have denoted with z h the horizon location defined by the vanishing of the emblackening factor, namely f (z h ) = 0. The definition of ρ is actually substantiated by the explicit analysis of the thermodynamics that we perform in Section 3. For the sake of practical convenience, we introduced γ ≡ κ 4 /q.

Thermodynamics
As discussed in the previous section, the black brane solution (2.5) corresponds to a planar dyonic black hole having both electric and magnetic charges. From the boundary theory standpoint, B represents a magnetic field perpendicular to the spatial manifold xy which enters the boundary thermodynamical quantities; as usual in gauge/gravity, these are derived from the bulk on-shell action as we now show in detail. The temperature T and the entropy density S are the easiest thermodynamical quantities to compute since they are determined from the horizon data, namely In order to compute the energy density E, the pressure P , the charge density ρ and the magnetization M, we need to evaluate explicitly the Landau potential Ω which, according to the holographic dictionary, is identified with the on-shell bulk action. Not surprisingly, the bulk action (2.1) when naively evaluated on the solution (2.5) is divergent and therefore needs to be renormalized. The standard renormalization process consists in regularizing the action by means of a UV cut-off z UV and supplementing it with appropriate counter-terms. These can necessarily be written in terms of boundary fields but (proceeding as in [12]), once evaluated on the solution (2.5), can be explicitly expressed as follows where g b is the metric induced on the z = z UV shell. With this counter-term in place, the Landau potential Ω assumes the following form We have denoted with V the boundary spatial volume. Once the Landau potential is known, the other thermodynamical quantities follow easily by means of standard thermodynamical relations. We explicitly obtain 3

Transport coefficients
We compute analytically the whole set of thermoelectric DC transport coefficients for the boundary theory corresponding to the bulk model (2.1). To this aim, we employ the method first illustrated in [20] and subsequently applied in [8,14,[21][22][23]. Such an approach enforces and extends the so-called "membrane paradigm" [24] to momentum-dissipating systems and relies on the analysis of quantities which do not evolve along the holographic direction from the IR to the UV.

Electric conductivity
We consider linearized fluctuations around the bulk background solution (2.5). Following [20], to the purpose of computing the linear response to a "pure" electric field (i.e. in the absence of a thermal gradient), one considers the following ansatz for the fluctuating fields where i = x, y; we henceforth adopt small Latin letters to refer to spatial boundary indices 4 . The vector E i introduced in the ansatz corresponds to an external electric field perturbing the system. The quantityJ x, y is a boundary spacetime index) is conserved along the holographic direction as a direct consequence of the Maxwell equation for the fluctuations. Indeed, recalling the ansatz (4.1), we obtain The capital indices refer to the bulk spacetime and the arrow means that we consider just the spatial components. The quantitiesJ i are radially conserved and explicitly given bȳ 4 Note that the magnetic field mixes the x and y fluctuation sectors and therefore all the components along these directions in (4.1) must be switched on.
where ǫ ij = −ǫ ji = 1. To obtain (4.6) we have again referred to the ansatz (4.1) and considered just up to the linear order in the fluctuating fields. We remind the reader that the boundary indices are raised and lowered with the flat boundary Minkowski metric.
To study the near-horizon behavior of the fluctuating fields and demand regular infrared behavior, it is convenient to adopt the Eddington-Finkelstein coordinates, namely leaving all the other bulk coordinates untouched. Skipping the details (which can be found in the analogous computation described in [14]), the infrared regularity requirement amounts to having the following asymptotic behaviors To avoid clutter, we relegated the explicit expressions of the equations of motion for the linearized fluctuations in Appendix A.1. It is however important to recall thath zi (z) is governed by an algebraic equation and therefore expressible in terms of the other fluctuating fields (hence it does not demand further IR requirements). An explicit infrared asymptotic analysis returns which we report explicitly for the sake of completeness and to underline that its B → 0 limit is consistent with previous results obtained directly at B = 0 in [14]. It is essential to observe that, turning the attention to the near-boundary asymptotics, one actually identifiesJ(z = 0) with the electric current J i of the boundary theory, Moreover, beingJ radially conserved, it can be evaluated in the IR and then expressed exclusively in terms of the near-horizon asymptotic data, namelyJ(z = 0) =J(z = z h ). The electric conductivity matrix is Hence its entries are directly read from the explicit expression of the electric currents; this yields and . (4.14) Given the isotropy of the setup, we have σ xx = σ yy and σ xy = −σ yx .

Thermoelectric response
We want now to compute the thermoelectric conductivities α xx and α xy . Considering the system at non-zero electric field and zero thermal gradient, these two conductivities are defined by the following relation where Q i is the heat current in the i-direction, which can be related to the boundary stress-energy tensor T µν and to the electric current J µ by the identity Q i = T ti − µJ i . In order to apply the same procedure used for the electric conductivity in the previous section, we need to define in the gravitational system (2.1) a quantityQ i which is radially conserved and that, choosing the appropriate UV boundary conditions for the functions appearing in the ansatz (see [14,20]), can be identified at the boundary with the heat current Q i . As illustrated in [20], the quantity we are after is where k = ∂ t . The proof that this quantity is radially conserved relies on the fact that k is a Killing vector for the gravitational action (2.1) (see [20] for more details). Evaluating the expression (4.16) on the ansatz (4.1) at the linear order in the fluctuations we obtain (4.17) The proof thatQ corresponds to the heat current at the boundary is straightforward. In fact, the third term in (4.16) vanishes at z = 0, the second term is equal 5 to −µJ i , and the first term coincides at the boundary with the ti component of the holographic stress-energy tensor, namely . (4.18) Exploiting its radial conservation, we can computeQ i at z = z h and express the heat current only in terms of horizon data. Finally, using the definition of the thermoelectric conductivities (4.15), we obtain: and . (4.20) However, as illustrated in [2,25], in order to properly define the thermoelectric response in the presence of a magnetic field, one has to subtract to the heat current the contribution due to the magnetization current. This implies that the off-diagonal components of the thermoelectric conductivity have to be defined as and, recalling the explicit expression of the temperature T and of the magnetization M derived in Section 3, we obtain (4.22) From now on, we will refer to α sub xy as the off-diagonal component of the thermoelectric conductivity and we will indicate it with α xy .

Thermal conductivity
The thermal conductivitiesκ xx andκ xy are defined in terms of the heat current generated by the presence of an external thermal gradient ∇ i T in the following way As illustrated in [20], in order to study the holographic model (2.1) in the presence of an external thermal gradient and at zero electric field we have to consider the following ansatz for the fields of the theory where s i can be proven to be equal to the quantity − ∇ i T T in the boundary field theory (see [20]). The linearized equations of motion for this ansatz can be found in Appendix A.2.
The computation of the thermal conductivities is now straightforward. As in the previous section, we compute the radially conserved quantity (4.16) on the ansatz (4.24) at the linear order in the fluctuations. Also in this case, as long as we consider the DC response,Q i can be proven to coincide with the heat current in the boundary field theory 6 . Finally, computing the quantityQ i at the horizon z = z h , and considering the definition of the thermal conductivities (4.23) we obtain Also in this case we need to subtract the contribution due to the magnetization current [2,25] from the off-diagonal conductivityκ xy , namelȳ (4.26)

Universal thermoelectric transport coefficients
The behaviors of the transport coefficients found in the previous section depend on the specific form of the thermodynamical quantities obtained in the massive gravity model in Section 3. Nevertheless, these transport coefficients can be cast in a form which aspires to be universal, at least in the holographic framework. Such a claim is corroborated by some preliminary results obtained in more general holographic models 7 [28]. In order to proceed and write the full set of transport coefficients in terms of the thermodynamical quantities and the generic physical parameters σ Q and τ , we need the explicit expression of these latter for the model at hand. As far as σ Q is concerned, reminding ourselves that it is the charge-conjugation symmetric conductivity, it is not affected by the presence of a magnetic field; we can then safely assume that it has the 6 Actually, considering the ansatz (4.24) there are some additional technical difficulties in proving this statement due to the fact that the quantity ∇ t k i differs from the holographic stress-energy tensor T ti by terms linear in the time coordinate t. However, as proven in [20], these terms do not contribute to DC transport properties. 7 This being in line with the interesting idea (still to be stringently tested) that massive gravity could furnish the paradigmatic framework to account for momentum non-conservation in holography. The connection between massive gravity and solutions with spatially modulated sources has been studied in [26]. Moreover, it is relevant to mention that the analyses involving explicit spatially dependent sources allow for a direct implementation of disorder in holography [27]. same form as in the B = 0 case [12][13][14], namely σ Q = 1/q 2 . The characteristic time of momentum dissipation τ , as found in [11] through a hydrodynamic analysis, assumes in massive gravity the following explicit form We underline that, strictly speaking, the quantity (5.1) can be interpreted as a dissipation characteristic time only in the hydrodynamic regime, namely when momentum is slowly dissipated [29]. Nevertheless, in full analogy to the B = 0 case [8,[12][13][14], the explicit and exact expressions for the holographic transport coefficients can still be written in terms of τ (as defined in (5.1)) also outside the hydrodynamic regime. We find this useful both for phenomenological reasons (see Section 6) and to make contact with the hydrodynamic analysis of [2]. So, allowing for a slight abuse of language, we will refer to (5.1) as a "dissipation characteristic time" also in circumstances when the hydrodynamic approximation is not valid and a dissipation characteristic time is not rigorously defined.
It amounts just to a matter of algebra to verify that the transport coefficients derived in Section 4 can be expressed in the general form given in (1.2), (1.3), (1.4). We highlight once more that the transport coefficients (1.2), (1.3), (1.4) are expressed as functions of the entropy density S, the charge density ρ, the magnetic field B, the charge-conjugation symmetric conductivity σ Q and the ratio τ E+P only. They do not depend on any detail of the specific model that has been used to derive them. We therefore advance the proposal that they can have universal relevance. Within the holographic context, such a claim could be corroborated by the comparison of these formulae with the hydrodynamic prediction (as done below) and with the corresponding results obtained in other holographic models [28]. More generally, a careful comparison with the phenomenology and real experiments must be pursued. In a later section we start addressing this wide question (without the pretension of being conclusive).

Comparison with the hydrodynamic prediction
In order to make a precise comparison between our results and the hydrodynamic analysis of [2], it is useful to define two characteristic frequencies of the system: the damping frequency γ d and the cyclotron frequency ω c : In terms of these frequencies, the transport coefficients (1.2), (1.3), (1.4) can be written as follows Comparing the previous formulae with the result given in (3.37) of [2], we find that the electric and Hall conductivities assume exactly the same form predicted in hydrodynamics. The holographic formula is however valid also outside the hydrodynamic regime and, in this sense, extends the hydrodynamic results. On the other hand, the explicit expressions for the thermoelectric conductivities and the thermal conductivities do not match completely. The holographic results have additional pieces which however vanish in the hydrodynamical regime; this fact was observed already in the B = 0 analysis [13,14].

Bulk electromagnetic duality and its consequences from the boundary perspective
The 3 + 1 dimensional bulk Lagrangian (2.1) enjoys electromagnetic self-duality. From the boundary viewpoint this implies that the equilibrium states corresponding to two bulk solutions connected by the electromagnetic duality can be mapped into each other, which practically means that the information regarding the thermodynamics and and the transport can be interpreted in two dual ways. Though, this does not correspond to a boundary electromagnetic duality; actually, from the holographic dictionary it emerges clearly that the bulk electromagnetic duality exchanges the boundary magnetic field with the charge density. The physical relevance of these duality arguments is connected to the possible description of the theory in terms of dual degrees of freedom and is related to the ubiquitous particle/vortex dualities of critical or near-to-critical systems. Indeed, as noted in [2], in the limit ρ, τ −1 , B ≪ T 2 , and ρ ∼ B , the hydrodynamic transport coefficients enjoy the above mentioned duality (a priori of any gauge/gravity argument), namely whereρ =σ −1 is the resistivity matrix,θ ≡ −ρ ·α is the Nernst coefficient matrix and κ =κ − Tα ·ρ ·α is the thermal conductivity matrix at zero electric current 8 As just argued, in a gauge/gravity context, the map (5.6) becomes particularly transparent as a direct consequence of bulk electro-magnetic duality. So the transport coefficients (5.3)-(5.5) that we have obtained holographically naturally satisfy (5.6) in any dynamical regime; both within and outside the hydrodynamic approximation.
The validity of (5.6) also far from criticality has interesting consequences. One can move away from criticality either lowering the temperature or increasing the magnetic field. This statements can be made precise referring to the charge density ρ; lowering the temperature means actually to consider a regime where the ratio T / √ ρ turns to be small. Then it is possible to appreciate that the two ways out of criticality just mentioned are dual in the sense of the map (5.6) and correspond roughly speaking to spoil criticality by means of a strong magnetic field or a strong charge density. Such considerations are relevant because the T / √ ρ ≪ 1 regime presents interesting features which admit an account in terms of a Boltzmann description [13]. In other words, the strongly coupled medium holographically connected to massive gravity presents a set of transport coefficients which outside criticality can be fitted by those obtained from a quasi-particle treatment in the Boltzmann framework. In particular, they respect the Wiedemann-Franz law. Relying on (5.6), one can extend similar arguments to the dual picture and observe a dual Wiedemann-Franz law associated to (vortex-like) degrees of freedom at strong magnetic field [30].

Holographic phenomenology
In this section we discuss the possible universal predictions based on the thermoelectric transport coefficients (1.2), (1.3), (1.4) explicitly computed with gauge/gravity techniques.
We want to express the behavior of the holographic model (2.1) as independently as possible of its specific details. In line with this aim, the transport coefficients (1.2), (1.3), (1.4) were expressed in terms of the thermodynamical quantities (computed in Section 3) and in terms of the critical conductivity σ Q and the momentum dissipation rate τ −1 . These latter quantities represent generic "phenomenological parameters" whose value is not predicted within the model itself 9 .
and similarly for the other transport matrices. 9 To clarify this idea of extending the results beyond the model used to obtain them, consider for example the rate τ −1 . For the specific model at hand it is related to the graviton mass β. However, expressing all the physical results directly in terms of τ −1 , corresponds to a model-independent formulation where We must also recall that, as usual in bottom-up holographic models (i.e. not derived as consistent low-energy effective theories of UV complete string setups), we have no direct control of the microscopic degrees of freedom. Hence, it is particularly natural to exploit the bottom-up holographic model as a simple example grasping essential features of a whole class of strongly coupled theories. Such logic leap should be carefully tested against experimental data. When successful, it can inform us about universal characteristics and shed light on the mysterious behavior of the transport properties in strongly correlated materials such as the cuprates (see [31] for a wide and precise experimental report). Previous phenomenological analyses along these lines have been performed in [6,8,32,33].
In [8] the formulae (1.2) for the electric conductivities are considered assuming that, in a regime with weak magnetic field, both the thermodynamics and the momentum dissipation rate do not depend explicitly on the magnetic field. It was then noted that, expanding σ xx and σ xy at low magnetic field, the electric conductivity follows an inverse Matthiessen's rule, namely In [8] it was also noted that in order to fit the experimental scalings of the conductivity and Hall angle measured in the cuprates, namely ρ xx ∼ σ −1 xx ∼ T and tan θ H ∼ 1/T 2 , the two conductivities σ Q and σ L must have the following scalings in temperature 10 where σ 0 Q and σ 0 L are dimensionless parameter which do not depend on T . Inspired by phenomenological intuition, we have also supposed that the charge density ρ is constant in temperature. In addition, near the quantum critical point, the DC conductivity must be dominated by the charge-conjugation symmetric part, namely σ Q ≫ σ L . As illustrated in the previous section (and previously noted in [8]), this phenomenological result based on holographic intuition coincides with what one gets from the hydrodynamical analysis [2] since the electric conductivities (1.2) obtained from these two approaches are completely equivalent.
τ −1 is regarded as a parameter accounting for an a priori unspecified momentum relaxing process. 10 We have chosen to express the scalings in temperature as a function of the dimensionless quantity T / √ ρ, considering the system at fixed charge density.
A slightly different point of view is that followed in [33], where the authors base their analysis on three assumptions: the theory is time-reversal invariant, it is at zero charge density but it is not charge-conjugation symmetric 11 and all the conductivities are determined by the quantum critical scalings. Building on these assumptions, the analysis of [33] first sets the values of three critical exponents by fitting the resistivity, the Hall angle and the Hall Lorentz ratio κxy T σxy (which has to be linear in T ) with experimental data, then determines the scalings of the other transport coefficients.
Since many experiments are done at finite charge density, we prefer to work along the lines of [8] and determine what it implies on the thermoelectric and thermal transport coefficients. Specifically, we will determine the scalings for the same transport coefficients discussed in [33], namely the the resistivity ρ xx , the Hall angle tan θ H , the Hall Lorentz ratio L xy , the magneto-resistance ∆ρ ρ = ρxx(B)−ρxx (0) ρxx (0) , the Seebeck coefficient s = αxx σxx , the Nernst coefficient ν = 1 B αxy σxx − s tan θ H , the thermal conductivity κ xx and the thermal magneto-resistance ∆κ . As already argued before, we regard the transport coefficients (1.2), (1.3), (1.4) as (possibly) universal functions of the magnetic field B, the charge density ρ, the entropy S, the charge-conjugation symmetric conductivity σ Q and the ratio τ E+P . Then, once the values of the charge density and the magnetic field are set, we need to fix the scalings of three quantities in order to determine the behavior of all the transport coefficients 12 . Note that the same approach cannot be followed in the hydrodynamical analysis of [33] because the hydrodynamical expressions are not always writable in terms of the ratio τ E+P . In order to discuss the consequences of the proposal of [8] extended to the whole set of thermoelectric transport coefficients, we consider the following phenomenologically inspired inputs where R ≡ and (1.4), in order to analyze the consequences of the proposal of [8], we expand the transport coefficients at the first leading order in the dimensionless ratio R and at weak magnetic field, namely B/ρ 0 ≪ 1. First we note that, in order for the leading term in the expansion of the Hall Lorentz ratio L xy to be linear in temperature (as needed to match the experiments [31]), we have to set δ = 1, namely the proposal of [8] combined with the input of experimental data forces the entropy to scale linearly in temperature near the quantum critical point With this assumption, at leading order in the dimensionless ratio R and B/ρ 0 , we find the following scalings Some of the temperature scalings derived with this approach are in accordance with the analysis of [33]. There are however three discrepancies: the magneto-resistance for which we find a B 2 T −3 scaling instead of B 2 T −4 and the Nernst coefficients and the Seebeck coefficients for which the authors of [33] found behaviors of the type T −3/2 and −T 1/2 respectively. This discrepancies are due to the fact that, as opposed to the assumptions made in [33], in the present analysis the charge density is non-zero and the entropy has to scale linearly in temperature (instead of T 2 as predicted in [33]). However experiments do not seem to be conclusive on these questions and we prefer to postpone any stringent comment on the possible connection between the present analysis and the experimental data to future discussions [28,30]. We want nonetheless to stress that, as it is evident from the universal formulae (1.2), (1.3) and (1.4), the behavior of the transport coefficients and that of the thermodynamical quantities are intimately related. If the universality of the transport formulae is confirmed, any proposal on the mechanism which determines the transport properties of the cuprates (at finite charge density) has to keep into account also the correct behavior for the thermodynamical quantities.
A natural improvement of this phenomenological analysis based on the holographic formulae (1.2), (1.3) and (1.4) is to keep into account the competition between different scales and systematically analyze all the various regimes of the system. We refer in particular to the recent proposal of [9], where, based on the experimental hint that in the normal phase of the cuprates [34][35][36] the magneto-resistance qualitatively appears more B-linear at low temperature and more B-squared at high temperature, the authors suggest that the magnetic field and the temperature influence the transport properties in the strange metals by competing to set the scale of the momentum dissipation rate. Consequently they propose the following form for the momentum dissipation rate where α and η are dimensionless parameters relating the momentum dissipation rate directly to the temperature and magnetic field respectively. In this case one has to discuss separately the weak T / √ ρ ≫ B/ρ and strong T / √ ρ ≪ B/ρ magnetic field regime.
Some preliminary results where the proposal of [8] is upgraded to keep into account the form of the momentum dissipation rate (6.10) 13 seem to predict the correct scalings for the magneto-resistance at low and at high temperature. However, since we are dealing with several different scales, to test properly the whole phase diagram in this case requires a careful analysis which we postpone to future studies [30].

Discussion
Relying on a membrane paradigm for holographic models featuring massive gravity, we analytically computed all the thermoelectric transport coefficients for a strongly coupled 2 + 1 dimensional system in the presence of a mechanism for momentum dissipation and a magnetic field perpendicular to the plane where the system lives. The transport coefficients can be expressed in a model-independent fashion in terms of the thermodynamical variables and two generic parameters accounting for two independent contributions to the electric conductivity. These corresponding to momentum conserving and momentum dissipating processes which combine to give the total electric conductivity according to an inverse Matthiessen's rule.
The model-independence of the expressions for the transport coefficients suggests by itself a possible general relevance of the formulae both within and beyond the holographic realm. Inspired by this observation (and extending the previous analysis of [8] to the whole set of thermoelectric coefficients), we enriched the holographic outcomes with phenomenological information. Specifically, we considered phenomenological behaviors obtained from experimental data for the critical conductivity σ Q and the momentum dissipation rate τ −1 . The set of transport coefficients supplemented with this phenomenological information allows us to derive the scaling properties in T for all the transport quantities and compare them back with measurements. Furthermore, by means of comparison to measurements of the Hall Lorentz ratio, we fixed the scaling properties of the entropy of the system to be linear in T . What emerges from this analysis is a coherent picture which is openly exposed to be contrasted against experiments. The definitive experimental test is however articulate and we postpone to subsequent work any conclusive claim. This constitutes one of the main future prospects of the present analysis. The aim being to systematically study all the various dynamical regimes corresponding to different hierarchies of the three scales T , B and the charge density ρ.
Another interesting point to be further developed regards the coincidence of the results on the transport outside the hydrodynamical regime with a Boltzmann treatment (shown in [13] and here extended also to vortex-like degrees of freedom at strong magnetic field). This can be read in two ways. One can either suspect the presence of "emergent" quasiparticle degrees of freedom in the far-from-critical holographic medium or could argue on the universality of both the holographic and Boltzmann treatments which, in some sense to be specified, lead naturally to equivalent results. This latter point is quite speculative so far, the former instead can be put under precise scrutiny. A direct look at the intimate structure of the medium can be obtained by studying the quasi-normal-mode spectrum and its features [30]. This constitute one of our main future perspectives which can be pursued both numerically and, possibly, analytically 14 .

Acknowledgments
A particular thanks goes to Alessandro Braggio, Nicola Maggiore and Nicodemo Magnoli for collaboration at an early stage of the project. A.A. wants to thank Richard Davison and Jan Zaanen for very helpful discussions. D.M. is very grateful to Alejandra Castro, Miller Mendoza and Sauro Succi for very nice and insightful discussions.