Thermo-electric transport in gauge/gravity models with momentum dissipation

We present a systematic definition and analysis of the thermo-electric linear response in gauge/gravity systems focusing especially on models with massive gravity in the bulk and therefore momentum dissipation in the dual field theory. A precise treatment of finite counter-terms proves to be essential to yield a consistent physical picture whose hydrodynamic and beyond-hydrodynamics behaviors noticeably match with field theoretical expectations. The model furnishes a possible gauge/gravity description of the crossover from the quantum-critical to the disorder-dominated Fermi-liquid behaviors, as expected in graphene.


Introduction and motivation
An accurate physical description of real condensed matter systems usually requires to include mechanisms for momentum relaxation. An important consequence being the finiteness of DC transport coefficients. The presence of impurities is a generic instance where translational invariance is broken resulting in momentum dissipation. Even more generically, a background lattice implies that, because of Umklapp scattering processes, momentum is conserved only modulo reciprocal lattice vectors.
In general it is impossible to over-estimate the value of improved descriptions of impurities effects on the transport phenomena. Indeed these are directly connected to the effects of disorder which are ubiquitous and important across the whole condensed matter context. With this in mind, we mean to realize a gauge/gravity model that describes a crossover from a weak-disorder quantum-critical regime to a disorder-dominated Fermiliquid-like regime. A crossover of this sort actually is expected in graphene [1]. At the outset, however, it is important to underline that the applicability of a thermo-electric, momentum dissipating model in gauge/gravity is significantly wider. In fact, apart from impurity disorder, the presence of the lattice also breaks translational invariance. As far as the present analysis is concerned, we focus on the effects of disorder neglecting the presence of a lattice. This corresponds to interpreting the momentum dissipation as exclusively due to non-dynamical impurities. Physically this possibly matches with the expectations for graphene as long as the effects of phonons can be neglected which is actually the physical range we are interested in 1 .
Describing momentum dissipating effects in the gauge/gravity framework is not an easy task. Actually all the early works applying the gauge/gravity correspondence to model condensed matter systems do not include momentum dissipation and therefore feature a delta function at ω = 0 in the real part of the transport coefficients [3]. At present, several ways to introduce momentum relaxation in AdS/CFT are known. One possible approach is to consider spatially modulated backgrounds which directly simulate, for example, a lattice potential [4][5][6][7]. Another viable way consists in analyzing circumstances where a few light charged excitations scatter and dissipate their momentum on a bath of heavy neutral degrees of freedom [8][9][10].
Recently it was proposed to introduce momentum relaxation in holography in an effective way (i.e. without a precise dynamical mechanism of momentum dissipation in mind) by using massive gravity [11]. Indeed the bulk graviton mass breaks explicitly the diffeomorphism invariance of the gravitational action which in turn implies that the stressenergy tensor of the dual field theory is not conserved and momentum can be dissipated. This effective mechanism is theoretically extremely interesting since, in contrast to the methods featuring explicit spatial modulations, allows us to obtain quantitative information about correlators and physical observables without the need to resort to complicated numerical methods that typically involve the solution of systems of coupled partial differential equations. 2 In [11] the massive gravity model originally introduced in [13] has been considered within the holographic framework. In the bulk model the graviton mass is introduced by coupling the dynamical metric with a fixed fiducial metric that breaks diffeomorphism invariance. The way in which diffeomorphisms are broken depends on the particular choice of the fiducial metric.
In this paper (inspired by [11] and subsequent articles) we analyze a massive gravity model where diffeomorphism invariance is broken in such a way that the dual field theory at the boundary conserves the energy but dissipates momentum. In the condensed matter framework this kind of mechanism occurs, for example, in the presence of elastic electron scattering due to fixed impurities. The same model has been studied also in [14,15]; specifically, in [14], by analyzing the poles of the correlation functions in the hydrodynamic limit (namely T τ −1 , where momentum is an almost conserved quantity), it was proven that massive gravity is the dual gravitational realization of a system in which the conservation law for the stress-energy tensor is where τ −1 is the momentum dissipation rate determined in terms of the graviton mass and the equilibrium thermodynamical quantities.
In [15] a universal analytical formula for the DC electrical conductivity in holographic massive gravity models was found. Comparing this expression with the electrical conductivity for a general hydrodynamic theory including the effects of impurity scattering (obtained in [16]), it was noted that the two expressions agree provided that the scattering rate τ −1 assumes the specific form , (1.2) found in [14]. We will be later more precise about the explicit expression of the scattering rate. For now it is sufficient to say that S, E and P are respectively the entropy density the energy density and the pressure of the system at equilibrium and that β is a parameter related to the bulk graviton mass which dually accounts for the "strength" of momentum dissipation.
Concerning the holographic massive gravity model at hand, some natural and important questions arise. The hydrodynamic theory considered in [16] for a relativistic model with scattering due to impurities provides us universal expressions for the full set of thermo-electric transport coefficients in terms of the momentum dissipation rate τ −1 and of the thermodynamical quantities of the system. Is then the hydrodynamic regime of massive gravity completely equivalent to this theory? In other words, given the expression for the scattering rate (1.2) and the thermodynamical quantities of the system, do all the transport coefficients agree with those obtained in [16] from the hydrodynamical analysis? Moreover, since massive gravity introduces the dissipation of momentum in an effective way, what is the physical character of the model when the hydrodynamical approximation ceases to hold? And, relatedly, is it possible in this non-hydrodynamic regime to understand anything concerning the possible microscopic processes giving rise to the effective mechanism of momentum relaxation?
As regards the microscopic realization of massive gravity, some progress has been attained in [17][18][19], where it was investigated how massive gravity can be derived from general relativity in AdS, and in [20] where it was proven that a perturbative background lattice provides a mass for the graviton. Nevertheless the ultimate answer about the microscopic origin of these massive gravity models is still hazy. We attempt to participate to such a debate from a rather phenomenological perspective. This primarily requires a full characterization of the system's behavior within the hydrodynamic regime (as a consistency check) and outside hydrodynamics (this to actually appreciate its proper peculiarities). To this end, it is important to realize what can be understood about massive gravity and its holographic dual interpretation by analyzing the full set of thermo-electric transport coefficients. Since we are not able to find an analytical expression for the static Seebeck coefficient s DC and the static thermal conductivityκ DC as it has been done for the DC electrical conductivity σ DC in [15], we have proceeded relying on numerical methods.
The proper definition of the thermo-electric transport coefficients within massive gravity has to be considered carefully. We have addressed technical difficulties which arise in the holographic renormalization procedure concerning the need of finite boundary counterterms in order to avoid unphysical features in the transport coefficients. Such possibility is a crucial test that massive gravity has to pass in order to be considered a sound holographic model. Note in fact that massive gravity models are obviously considered in a fully bottom-up and phenomenological spirit, at least as long as a consistent microscopic derivation of the bulk model is lacking. For this reason a systematic check of the consistency of the dual phenomenological picture as a whole is always in order. To a similar purpose in [15] a careful study of the equilibrium thermodynamical consistency of holographic models with massive gravitons has been considered. We here proceed the investigation with an analogous attitude to the full thermo-electric linear response. On top of that, as the system at hand features a coupled thermo-electric dynamics, the extraction of the pure electrical and pure thermal response requires an attentive analysis of the intertwined linear response of the system 3 .
At sufficiently high temperature T where the diffusion rate τ −1 is the smallest energy scale of the system and the hydrodynamic limit is satisfied, we find that not only σ DC but the full set of transport coefficients agree with those predicted by the hydrodynamic theory analyzed in [16]. Specifically, they acquire the following form: where σ, s and κ are respectively the electric conductivity, the Seebeck coefficient and the thermal conductivity (at zero electric field); in addition, ρ is the charge density, µ is the chemical potential and q is a free parameter of the gravitational Lagrangian.
In the hydrodynamical regime the diffusion rate τ −1 decreases with the temperature as T −1 . However, in the low-T region the diffusion rate increases and eventually the hydrodynamic picture and expressions (1.3) ceases to be valid. Remarkably, in this nonhydrodynamical regime the transport coefficients computed numerically are in agreement with those computed (in the same conditions) in [1] for Dirac fermions with fermionfermion interaction and a dilute density of charged impurities using the Boltzmann approach, namely: This fact strongly suggests that the system in this non-hydrodynamic regime has a "ballistic" behaviour, i.e., to a first approximation, its phenomenology appears to admit a description in terms of charged quasi-particles 4 . Furthermore, by analyzing the transport coefficients (1.4) and considering the specific expressions of the thermodynamical quantities of holographic massive gravity model, we find that in this limit the system has some features in common with the disorder-dominated Fermi-liquid regime. In fact the Wiedemann-Franz law is satisfied and the electric conductivity is temperature independent while the thermal conductivityκ DC goes linearly to 0 as T → 0 and is proportional to the heat capacity. The paper is organized as follows. In Section 2 we review the standard analysis of the thermo-electric response of a holographic model without momentum dissipation and admitting asymptotically AdS 4 Reissner-Nordström solution. We systematically consider its holographic renormalization; the expert reader can however jump directly to Section 3 where the momentum dissipating system is addressed. There the massive gravity model of interest is defined and studied in depth. Again, particular attention is paid to the precise renormalization procedure and definition of the transport coefficients. In Section 4 we present a detailed account of the numerical results and describe the phenomenological picture which arises from them. We comment on the presence and the physical significance of different regimes where the system admits either a hydrodynamic or a ballistic-like description. Particular attention is paid to the relation of our model to the physics of the crossover between a quantum-critical and a Fermi-liquid regime expected in dirty graphene. Eventually Section 5 contains concluding remarks and an outline of many interesting future prospects.

Thermo-electric transport without momentum dissipation
In the present Section we review the thermo-electric transport in a simple system without momentum dissipation, namely we discuss the holographic dual of the well-known 4dimensional Einstein-Hilbert-Maxwell model on a Reissner-Nordström AdS black hole. This review is meant to recapitulate tidily the details of the holographic renormalization and the definition of transport coefficients in the standard momentum-conserving systems. We will then be able to highlight in later sections the differences one encounters in treating massive gravity.

Bulk solution
Consider the simplest 4-dimensional gravitational model admitting asymptotically AdS charged black hole solutions, namely an Einstein-Hilbert-Maxwell theory. This corresponds to the action where we have already included the Gibbons-Hawking boundary term, which is expressed in terms of the induced metric (g b ) µν and the extrinsic curvature K on the surface at z = z U V . Actually z U V represents a UV cutoff that will be sent to zero in the final step of the holographic renormalization procedure. As it is well known (see for example [36]) the Gibbons-Hawking term is necessary in order to have a well-defined bulk variational problem. In the action (2.1) Λ = −6 is the dimensionless cosmological constant measured in units of the AdS 4 radius L; κ 4 and q are respectively the gravitational and Maxwell coupling constants and their dimension is [κ 4 ] = 1 and [q] = 0. From the action (2.1) we get the Einstein and Maxwell equations where we have introduced the ratio of the gravitational and Maxwell couplings, namely γ ≡ κ 4 q . Being the equations of motion (2.2) insensitive to an overall rescaling of the action (2.1), they depend only on γ and not on the individual couplings. It is worth noticing that for the simple model at hand γ could be rescaled away by means of a field redefinition 5 .
The model admits the following black-brane solution (see for example [21]): where z is the radial coordinate running from z U V at the UV radial shell to z h at the black hole horizon. Of course, in the limit of vanishing cut-off, the radial UV shell is identified with the conformal boundary of the asymptotic AdS geometry. We recall that the coefficients of the leading and subleading near-boundary terms of the bulk gauge vector are respectively mapped to the dual chemical potential µ and charge density ρ ≡ µ/(q 2 z h ) of the corresponding global current in the boundary theory. Eventually, the black hole temperature (which coincides with that of the boundary theory) and the other thermodynamical quantities, such as the energy density E and the pressure P , can be derived in the standard holographic way (see for instance [3,28]). One obtains (2.6)

Fluctuations
We consider vector fluctuations on the homogeneous and isotropic background specified by (2.3), (2.4) and (2.5). Without spoiling the generality of the treatment, the fluctuating fields that we study are the gauge field fluctuations along the x spatial direction, namely a x , and the vector mode of the metric, h tx ; these are the relevant fluctuations in order to analyze the thermoelectric transport (see below). We further assume harmonic temporal dependence and isotropic spatial dependence (null momentum) for the fluctuations. The fluctuation dynamics is governed by the Einstein and Maxwell equations (2.2) which assume the following explicit form To actually solve the differential problem governing the fluctuation dynamics, we need to specify appropriate boundary conditions at the horizon; we consider in-falling boundary conditions which are those needed to compute retarded correlators of the dual theory [26]. From (2.9) we have that the gauge field fluctuations can be analyzed and solved without considering the metric fluctuations which are later determined by means of (2.8) upon substituting the solution for a x . Therefore we have to impose the in-falling boundary conditions at the horizon on the gauge field alone, Since the equation (2.9) is homogeneous, we can rescale the parameter b 0 to 1, as a consequence a x and h tx are completely determined in terms of the frequency ω and the background quantities. As we will see, this is not the case for massive gravity. There we face a system of two coupled equations where the ratio of the two leading IR coefficient of the fluctuation fields is physically relevant. We will later discuss more in detail this important point.

Renormalization of the fluctuation action
In order to compute the correlators to be plugged into the Kubo formulae for the transport coefficients, we need to consider the on-shell bulk action expanded at the second order in the fluctuations. The gauge/gravity prescription identifies the boundary value of the bulk fluctuation fields with the dual sources. The correlators of interest are then obtained taking appropriate functional derivatives of the on-shell action with respect to these sources. This entire procedure represents the gauge/gravity version of the standard field theory paradigm to derive correlation functions.
In general the bulk on-shell action for the fluctuating field is divergent and needs to be properly renormalized. The holographic renormalization procedure consists in considering a regularized action to be integrated up to a near-boundary radial cut-off; then, appropriate boundary counter-terms are considered and eventually the limit of zero cut-off defines the renormalized action. The boundary counter-terms make the on-shell action finite once the UV cut-off goes to zero. They must respect the symmetries of the boundary theory and provide a well-defined bulk variational problem. As mentioned before, in (2.1) we have already added the Gibbons-Hawking boundary terms to the bulk action; this provides a well-defined bulk variational problem for the fields. Then (see for instance [3]) the only well-behaved boundary term needed in order to render the on-shell action finite is Eventually, the limit of vanishing cut-off is considered and (as we are interested in the linear response or, said otherwise, to two-point correlators) only the quadratic part of the action in the fluctuating field is retained. The renormalized quadratic action is defined as (2.12) Once we have obtained a finite on-shell action, it is perfectly legitimate to ask ourselves whether finite counter-terms could also be added. Such finite counterterms would lead to ambiguities in the definition of the renormalized action 6 . We state once more that the counter-terms have to respect all the symmetries of the boundary theory 7 , the power counting and the definition of the bulk variational problem. This latter characteristic amounts to avoid introducing boundary terms containing radial derivatives. The former symmetry requirements impede us to consider terms as a i a i which would brake the boundary gauge symmetry. The power-counting criterion instead forbids us to consider F ij F ij which is allowed by the symmetries but would force us to introduce new dimensionful parameters. We further notice that a Chern-Simons term is always trivial on our background solutions as a consequence of spatial rotational invariance. Such arguments exhaust all the possibilities as far as the gauge field is concerned. Turning our attention to the metric, we are allowed to consider two kinds of terms: a boundary cosmological constant and a term proportional to the boundary Ricci scalar. The first actually appeared in (2.11); the latter is null as the manifold transverse to the radial coordinate z is flat Minkowski space-time upon which we are considering homogeneous configurations in the space coordinates (i.e. null momentum).
From an asymptotic study of the equations of motion we have that the boundary expansions of the fields a x and h tx are To have an example where finite counter-terms can be added to the bulk action of a holographic model and have an impact on the resulting physics, see [34]. 7 As a general feature, the correlators satisfy Ward identities related to the symmetries of the model. In a generating functional framework, such identities (as the correlators themselves) are obtained by appropriate functional derivatives of the generating functional itself and of the expectation values of the various quantities in the theory. Counterterms (either finite or not) in the QFT action which respect the symmetries of the original theory affect both the Ward identities and the correlators in a consistent way [32]. and consequently the renormalized quadratic on-shell action for the model at hand is given by where we have Fourier transformed with respect to the time coordinate. We anticipate that, as opposed to the model just analyzed in which there are no finite boundary counter-terms which can be added to the regularized action, in the massive gravity case, as we will see, the explicit breaking of diffeomorphism invariance allows us to add to the action non-trivial finite counter-terms. These may (and actually do) affect the physical quantities and, in particular, the transport coefficients.

Review and definition of the transport matrix
The generic transport coefficient C XY is defined through the Kubo formula where X, Y indicate the (here unspecified) physical sources (e.g. E or ∇T ) while the correlator G is the Green function obtained through functional differentiation of the quadratic on-shell action S (2) with respect to the sources a (0) and h (0) . We are interested in computing the thermo-electric transport coefficients which relate at linear order the heat flow Q x and the electric current J x to the electric field E x and the gradient of the temperature ∇ x T in the following way: where σ is the electric conductivity, s is the Seebeck coefficient andκ is the thermal conductivity at vanishing electric field 8 . The connection between the bulk field fluctuations and the fluctuations of the physical quantities (i.e. between A, h and E, ∇T ) is the following [3,27]: In particular, as explained in [3], in order for this identification to be valid the theory must be invariant at least under temporal diffeomorphisms. Indeed to relate the fluctuation h tx to a thermal gradient one relies on a temporal diffeomorphism "gauge" transformation. This is related to the fact that in the framework of thermal quantum field theory, the imaginary period of the complexified time coordinate corresponds to the inverse temperature. The temporal diffeomorphism invariance is naturally satisfied in the standard formulation of general relativity but might be not true for massive gravity where diffeomorphism invariance is explicitly broken. However, as we will see in Section 3, the massive gravity model which we consider is invariant under diffeomorphism in the t − z directions and therefore the relations (2.17) still hold.
From (2.17) we have the following relations among the corresponding functional deriva- where the partial derivatives with respect to the sources a (0) x and h (0) tx are to be taken keeping to zero the source upon which one does not differentiate. We underline that the sources a (0) x and h (0) tx are independent quantities. Stated this, in order to compute the explicit expressions of the transport coefficients in terms of the background quantities and the near-boundary fluctuations, we start taking double functional derivatives of the on-shell renormalized and quadratic action (2.14). Namely, and (2.22) We observe that the equation for the fluctuations of the gauge field is independent of h tx and that, because of equation (2.8), h (1) tx is completely determined in terms of a (0) x and the parameters of the background, Eventually, we have that the entries of the transport matrix (2.16) are all expressible in terms of the background quantities and a unique electric conductivity [3]: It is interesting to consider the thermal conductivityκ for a neutral black hole, namely for µ = 0. The only surviving contribution is the imaginary pole whose residue is proportional to E. Relying on the Kramers-Kronig relations this corresponds to a delta function at zero frequency in the real part ofκ which encodes the lossless heat transport through a momentum conserving medium induced by a thermal gradient.
To conclude this brief review we plot in Figure 1 the static limit of the electric conductivity σ DC = lim ω→0 σ(ω) as a function of the scale invariant temperatureT = T /µ. Usually in the gauge/gravity literature one mostly discusses the spectral properties of the electric conductivity without focusing on the temperature dependence. Instead, since we are concerned with the thermo-electric properties, we find it interesting to study the temperature dependence of the static limits of all the transport coefficients, i.e. also the Seebeck and the thermal conductivity. In fact, the experimental and real condensed matter investigations of the thermo-electric and thermal coefficients are usually more focused on temperature dependence rather than the spectral behavior. We have plotted the static electric conductivity for γ = 1 and µ = 1, however its behavior for different values of the parameter γ is the same since, as noted before, γ can be reabsorbed through a field redefinition, namely the system is invariant under the scaling µ → aµ and γ → γ/a.
It is important to note that, since there is a δ(ω) in the real part of the conductivity, the static conductivity is defined as the limit for ω → 0 of the spectral conductivity disregarding the delta function. From the numerical point of view, this coincides with computing the spectral conductivity at a value of ω much smaller than all the other scales in the system. From Figure 1 we observe that the behavior of σ DC presents two regimes; the "cross-over" region corresponds roughly with the energy scale set by the chemical potential µ. The two above-mentioned regimes consist in the following two behaviors: for T 1 the static conductivity goes to zero quadratically while forT 1 it saturates to 1/q 2 .

Thermo-electric transport in massive gravity
In this Section, after discussing the basic properties of the massive gravity model which we consider, we will explain how to compute the thermo-electric transport coefficients for this system; the detailed analysis of the numerical results that we obtained is postponed to Section 4. Unless specified otherwise, we refer to Section 2 for conventions and definitions.

The massive gravity model
The idea underlying the application of massive gravity in holography consists in breaking the diffeomorphism invariance in the bulk by introducing a mass term for the graviton in such a way that one has momentum dissipation in the boundary dual field theory. Actually, several ways to give a mass to the graviton had been studied, but, following [11], we work here with the formulation of the massive gravity presented for the first time in [13]. The action of the model is: where β is an arbitrary parameter having the dimension of a mass squared and the small square brackets denote a trace operation. Notice that the action (3.1) contains already the Gibbons-Hawking term necessary to have a well-defined bulk variational problem. The matrix (K 2 ) µ ν is defined in terms of the dynamical metric g µν and a fiducial fixed metric f µν in the following way 10 Along the lines of [11], we consider the following form for f µν : Considering this particular form for the fiducial metric means that the action is still invariant under diffeomorphism in the (z, t) plane, but not in the (x, y) plane. At the dual level this implies that the theory has conserved energy but no conserved momentum. At this point some comments are in order. In [14] it was proved that, in the limit of small momentum dissipation, namely when the temperature is greater than the characteristic momentum relaxation rate of the system, some observables computed in massive gravity are consistent with a hydrodynamical model which respects the modified conservation laws given in (1.1). The τ appearing in the modified conservation relations is the characteristic momentum relaxation time of the system. Relations (1.1) coincide exactly with the conservation laws proposed in [16] for a relativistic hydrodynamic model which includes impurity scattering in the limit of spatially isotropic perturbations.

Background and thermodynamic
The equations of motion descending from the action (3.1) are: where γ ≡ κ 4 q and (3.5) 10 Within this formulation of massive gravity, it is possible to consider also a linear term in the trace of K; namely an α[K] term in the Lagrangian density where α is a numerical coefficient. However in this paper we always consider the case α = 0. The reason for doing so is twofold: first a rigorous proof of the absence of ghosts in the model exists only in this α = 0 case [11]; secondly, as noted in [14], with α = 0 logarithmic terms appear in the near-boundary expansion of the bulk fields. The latter fact introduces non-standard divergences in the on-shell 2 + 1-dimensional action.
We want to study the system in the presence of a chemical potential, we then consider the same background ansatz as in (2.3). In the massive case the black-brane solution is: (3.6) In the limit β → 0 the emblackening factor f (z) reduces to that corresponding to the standard Reissner-Nordström solution. The black hole temperature is computed in the usual way leading to The full set of thermodynamical quantities were derived in [15]. For the sake of later need, we report here the explicit expressions for the entropy density S, the energy density E and the pressure P , Notice that the dual theory of a massive gravity has in general E = 2P . The equation of state E = 2P is expected for a 2 + 1 dimensional conformal theory but, as it happens with the conservation laws of the stress-energy tensor, the massive gravity set-up introduces modifications that are proportional to the mass parameter β.

Scales and scalings
As we have just noted observing the thermodynamic quantities, the massive parameter β introduces a new scale in the model. This new scale affects the scaling symmetries of the bulk fields. In fact, if we rescale the radial coordinate z as z → az, we find that the other quantities of the model must scale as in order for this scaling to be a symmetry of the action. In particular, if we consider the scale invariant temperatureT ≡ T /µ we find from (3.7) that this is a function of the scale invariant quantities β/µ 2 and µz h : Moving the temperature while keeping fixed both the chemical potential µ and the mass parameter β (which, as we will see, is related to the momentum dissipation rate in the dual field theory) corresponds to varying the horizon radius z h . Finally, we note that, as in the massless case, the constant γ can be rescaled away from the action (3.1) by means of a redefinition of the gauge field. In fact the system is invariant under the scaling γ → aγ , µ → µ/a , (3.11) namely the same scaling symmetry found in the Reissner-Nordström AdS black hole. This scaling affects in particular the transport coefficients and consequently to compute the transport coefficients at different values of γ is equivalent to compute the same quantities at the corresponding rescaled values of the chemical potential.

Linearized equations and asymptotic expansions
In order to obtain the transport coefficients, we need to expand the action (3.1) at the second order in the fluctuation fields. As in the massless bulk gravity case, we work in the zero momentum limit 11 . However, as opposed to the massless case, here the equations for h tx and h zx are independent and then we have to turn on both the fluctuations to be consistent. Hence we consider the following set of fluctuations Expanding the equations of motion (3.4) to the linear order in the fluctuations (3.12) we obtain: There are no derivatives of h zx in the first equation of motion which therefore can be algebraically solved to obtain h zx . We then substitute the solution inside the second equation. Finally we are left with two coupled equations for a x and h tx : (3.14) In the β → 0 limit the first equation in (3.14) reduces to (2.9) obtained in standard massless gravity. This, however, cannot be simply interpreted as the fact that the fluctuation dynamics in the limit β → 0 coincides with that arising in the massless gravity on the Reissner-Nordström black hole. Indeed, the second equation in (3.14) shows that the limits β → 0 and ω → 0 do not commute. Since we are interested in computing DC observables we always consider the ω → 0 first.

IR expansion
As usual, in order to compute the retarded correlators, we have to numerically solve the equations (3.14) imposing the in-going wave boundary conditions at the horizon z = z h , namely

(3.15)
It is important to note that, unlike the case of fluctuations on pure Reissner-Nordström black hole, it is impossible to combine the two equations (3.14) in a unique equation for a x . The dynamics of electric and thermal fluctuations is consequently more intimately mixed. From the bulk standpoint, it is possible to rescale to 1 only one of the two coefficients a 0 and b 0 . Said otherwise, the physics of the model is sensitive to the ratio η = a 0 /b 0 . In the computations aimed at getting the transport coefficients, in order to isolate the purely electric response of the system, we have to tune the coefficient η so that the thermal source vanishes. Symmetrically, to compute the pure thermal contribution, we must fix η so that the electric field source is zero 12 .

UV expansion
Near the boundary z = 0 the expansion of the fluctuations in powers of z is: (3.16) 12 In the context of mixed spin-electric transport a technically analogous situation arises in the unbalanced holographic superconductor [30].

The coefficients of the higher orders in the z expansions can be determined in terms of the background parameters and the integration constants h
x , a (1) x . Since we are concerned with solutions of a system of second-order differential equations, these integration constants remain arbitrary in the UV analysis. As usual, once one imposes the above-mentioned IR boundary conditions at the horizon they are determined and can be read from the full bulk solution. According to the standard holographic dictionary, we interpret h (0) tx and a (0) x as the sources of the dual operators whose vacuum expectation values are given by h (1) tx and a (1) x .

On-shell action and renormalization
The action (3.1) diverges if evaluated on-shell at the quadratic order in the fluctuations. The counter-term which is necessary to make the quadratic action finite is, as in the massless case, However, the reduced amount of symmetry in massive gravity allows one to introduce additional finite counter-terms which are forbidden in the massless case. More specifically, the larger freedom corresponds to the possibility of having terms that do not respect the spatial diffeomorphisms which are already broken by the graviton mass. Of course we still consider finite counter-terms which respect the power-counting (i.e. terms that do not require the introduction of further dimensionful coefficients), the (reduced) boundary symmetries and which lead to a well-defined bulk variational problem.
In accordance with the above-mentioned requirements, we are allowed to add only the following tower of finite counter-terms 13 for all values of n. Here N is a normalization constant that depends on the dimensional parameters of the bulk theory 14 . It is important to notice that for n = 0 the counter-terms (3.18) introduce polynomial contributions to the imaginary part of the T tx T tx correlator and that such contributions diverge at large frequency. We exclude this behavior on the basis of field theoretical arguments on the high-ω behavior of physical correlators and therefore we retain only the n = 0 case, namely 13 We remind the reader that the case under consideration has zero spatial momentum k; hence terms with spatial derivatives are automatically null. In such circumstances, terms involving the boundary Ricci scalar R[γ] are vanishing as well.
14 Not to be confuse with the N → ∞ rank of the boundary theory gauge group.
where a is a dimensionless parameter on which the finite counter-term depends. The freedom associated to the choice of a specific value for a appears as a renormalization ambiguity of the model or, said otherwise, to a renormalization scheme dependence. However, in order to eliminate an unphysical delta function at ω = 0 in the thermal conductivity, we must choose a = − 1 2 . We will comment further on this important aspect in the following Sections; here we anticipate the remark to underline that the physical model at hand is eventually not affected by renormalization ambiguities. Terms similar to (3.18) but containing h zx do not respect spatial translation invariance 15 .
The total on-shell action reduces to a purely boundary term. Fourier transforming the fields and substituting h zx by means of the second equation in (3.13) we obtain (3.20) where the prime denote the derivative with respect to the radial variable z, the arguments of the first and second fluctuation in each pair are respectively (−ω, z) and (ω, z) and V represents the volume of the spatial manifold. The boundary action (3.20) evaluated on the boundary expansions (3.16) allows us to compute the transport coefficients, (for details on the computation of the transport coefficients see Appendix A).

Definition of the transport coefficients
The computation of the transport coefficients is analogous to that illustrated for the massless case, but with two important differences. The first one is that, since we are dealing with two coupled differential equations, relations (2.24) are not valid and we have to keep into account that: 15 One can recover spatial translations considering a spatial diffeomorphism where the coordinate variation ξ is a constant. The component h zx has a non-vanishing variation contributed by the non-trivial Christoffel symbols involving the coordinate z. Of course, interpreting the massive gravity model as an effective way to account for spatial inhomogeneities, one would drop the spatial translation invariance requirement. In such circumstances it is possible that wider classes of counter-terms could be considered. This analysis is however beyond the purpose of the present treatment.
The second is that, on the computational level, in the massive case the IR parameter η = a 0 /b 0 coming from the boundary conditions at the horizon (3.15) has a physical relevance and cannot be simply rescaled to 1. Indeed we have to tune η depending on which source we need to set to zero in performing the functional derivatives. We resort to a numerical shooting method to the purpose of finding the value of η corresponding to the desired UV source set-up.
Finally, the explicit expressions of the electric conductivity σ, the thermal conductivitȳ κ and the thermo-electric conductivity s (obtained in Appendix A) are . (3.24) As anticipated, the thermal conductivityκ depends explicitly on the parameter a introduced by the finite counter-term (3.18) and, as explained in the next paragraph, we fix the value of a according to physical requirements.

Fixing the finite counter-term
As it is evident from (3.23), only the imaginary part of the the thermal conductivity depends on the value of the parameter a. This parameter, which corresponds to the normalization of the finite counter-term (3.18), has a key role in allowing us to get a sensible physical picture. For instance let us note that if we just set a = 0 we find as the result of the numerical computations that the imaginary part of the thermal conductivity has a pole at ω = 0. The Kramers-Kronig relations map such a pole to a delta function δ(ω) in the real part ofκ. A delta function in the thermal conductivity describes a perfectly efficient (lossless) transport of heat which is unphysical given that we are concerned with a system that dissipates momentum.
The apparent inconsistency can be completely fixed by setting a = −1/2. Observe that the divergence in the imaginary part ofκ is evidently contributed by the first term in (3.23) which diverges as i(1 − a)E/ωT . Actually also the the second term in (3.23) yields an analogous contribution which, however, needs to be uncovered and treated numerically.
Indeed an attentive numerical analysis shows that where the numerical factor in front of E does not depend (according to our numerical precision) on the particular value of the other parameters of the model. Notice that the numerical result (3.25) seems to provide an analytical insight. This noticeable conclusion is not only based on an accurate numerical treatment but on a theoretical expectation as well. Recall that in the massless gravity set-up the lossless thermal transport of a neutral black hole is proportional to the energy density E. This feature can be regarded as a generic characteristic independent of the details of the particular holographic model one considers. Also in massive gravity, where the lossless thermal transport would lead to unphysical consequences, we can reliably expect that it can be reabsorbed by means of tuning the coefficient with which E appears in the thermal conductivity. This argument supports us in distilling an analytical conclusion from the numerical data.
Let us rely on the same point looking the details of the formulae. For small ω the imaginary part of the thermal conductivity behaves as If we set a = 0 we find the same divergence as for massless gravity on the neutral black hole solution (see (2.26) with ρ = 0). However, in the massive case the symmetries of the model allow us to consider a = 0, and in particular if we set a = − 1 2 we find that the imaginary part of the thermal conductivity goes to zero as ω → 0. as expected for the imaginary part of a physical transport coefficient in the presence of momentum dissipation and in the DC limit (see Figure 2). scattering rate τ −1 ) was developed in [16]. Being the impurity scattering an elastic mechanism, [16] supposes that weak disorder affects only the momentum conservation while preserving the energy conservation as encoded in (1.1). As a general result of this hydrodynamic approach, once the scattering rate τ −1 and the thermodynamical quantities 16 of the system are provided, all the transport coefficients take the following form:

Transport coefficients analysis 4.1 The dissipation rate and the hydrodynamic regime
where σ Q has to be determined in terms of a constitutive description of the system. In [14,15] it was proven that massive gravity has a hydrodynamic regime when the scattering rate τ −1 is small (with respect to the temperature and the chemical potential) and therefore the momentum conservation violation is small as well. This regime is captured by the general hydrodynamic treatment described in [16]. In particular in [14], by analyzing the poles of the correlators in such hydrodynamic limit and determining the scattering rate as , (4.4) it was demonstrated that massive gravity is well described by the modified conservation law (1.1). More precisely, [15] provides an analytical expression for the static electric conductivity σ DC for every value of the temperature T in the massive gravity model at hand, which agrees with (4.1) when the scattering rate τ −1 is given by (4.4) and σ Q = 1/q 2 . It is important to underline again that formulae (4.1)-(4.3) and the argument followed in [14] are expected to be valid only when the scattering rate is the smallest energy scale in the system, namely τ −1 T and τ −1 µ. The reason being that the modified conservation law (1.1), which is the starting point of [14] and [16], is accurate only if the dissipation rate τ −1 and the frequency of the fluctuations ω are small compared to the characteristic scales of the system [1].
To have a complete picture of the behavior of the system at hand also beyond its hydrodynamical regime we need to study the whole range of the scale invariant temperaturẽ T = T /µ 17 . Keeping into account the expressions for the thermodynamical quantities given in (3.8) and expressing the scattering rate (4.4) as a functionT we obtain the following limiting behaviors forT 1 andT 1 We report them in Figure 3. According to (4.4) and Figure 3, the scattering time approaches a constant atT → 0 and decreases asT −1 whenT → ∞. This qualitative behavior is the same for every allowed values of the parameters of the model 18 . Hence, there is always a region at lowT where τ −1 is greater than the temperature and in such region the hydrodynamic approximation is no longer accurate. As already noted, the conductivity (4.5) obtained in [15] is valid for every value of the temperature T . The passages performed in [15] leading from the expression of the conductivity (4.5) to that of the scattering rate (4.4) are performed within the hydrodynamical regime. Our purpose is at first to cross-check the validity of the hydrodynamical approximation by comparing our numerical results regarding the other transport coefficients with 17 As explained already in Section 3, we recall that the correct way to vary the temperature corresponds to move the horizon radius r h keeping fixed the chemical potential µ. This because there are more scales in the system in addition to T and µ. As a consequence, to obtain the scattering rate as a function of the temperature, we have substituted z h (T ) in (4.4). 18 We remind the reader that, as explained in [11,14], β must be negative in order for the scattering rate to be positive.  For the moment being we stick to the hydrodynamical regime. As regards the electric conductivity, keeping into account that for the holographic model at hand the charge density is ρ = µ q 2 z h , it is evident that σ DC does not depend on the horizon radius z h and then on the temperature 19 . We have verified the correctness of our numerical computations by comparing the results for σ DC against the analytic formula (4.5).
The comparison between the thermal conductivity (3.23) and the Seebeck coefficient (3.24) computed numerically (blue solid lines) and the corresponding hydrodynamic formulae (4.2), (4.3) (red dashed lines) are plotted in Figures 4 and 5. All the numerical computations whose results are shown in the plots are obtained taking µ = 1 and for a particular choice of the parameter β, L, q and κ 4 . Nevertheless, it is essential to mention that the various behaviors plotted are qualitatively the same for all the allowed values one could choose for this quantities 20 .
From Figure 4 emerges that, in the highT region the transport coefficients computed numerically match exactly the hydrodynamic expectation (4.1)- (4.3). This confirms that, as proven in [14], the massive gravity model under study has a hydrodynamic regime that is well described by means of the modified conservation law (1.1). On the other hand, in the low-T region (magnified in Figure 5) the hydrodynamic condition τ −1 ≤ T is not satisfied and we find a disagreement between the hydrodynamic description and our numerical results. In particular note that the hydrodynamical plots coming from both (4.2) and (4.3) diverge asT → 0; this clearly indicates the intrinsic limit of the hydrodynamic description at lowT 21 . As we will further comment in Subsection 4.2, the Seebeck coefficient computed numerically approaches instead a constant value and the numerical thermal conductivity goes linearly to 0.

Beyond the hydrodynamic regime
In the region where T /µ 1 the hydrodynamic approximation ceases to be valid. Since the thermal conductivity κ DC =κ DC − s 2 T /σ goes linearly to zero and the electric conductivity σ DC is constant, we note that the Wiedemann-Franz ratio W = κ DC /(σ DC T ) is approximatively independent on the temperature. Said otherwise, the Wiedemann-Franz law holds and it is therefore reasonable to argue that, at least to a first approximation, a quasi-particle-like description of our system in this regime could be accurate.
To check and make the quasi-particle hypothesis more substantial, we compare our numerical results concerning the transport coefficients with the analytical results obtained with a standard Boltzmann approach in [1]. There one considers the thermoelectric transport properties of a fluid of interacting Dirac fermions (through Coulomb interactions) and a dilute density of charged impurities. The purpose being to study the graphene within and beyond its quantum-critical regime. The result obtained in [1] shows that, in the hydrodynamic regime where the fermion-fermion interaction is the dominant phenomenon, the transport coefficients agree with the hydrodynamical prediction (4.1)-(4.3) at the zero order in the weak disorder expansion. On the other hand, in the region where the elastic scattering due to impurities dominates on the fermion-fermion interactions and the hydrodynamic approximation is no longer valid, the Seebeck coefficient and the thermal conductivity assume the following form: where the first terms of (4.7) and (4.8) are completely determined in terms of the impurity scattering rate τ −1 and of the thermodynamical quantities of the system. Instead, s and α depend on the fermion-fermion interactions but can be neglected at large doping, namely when µ T (see [1]). We want to compare the phenomenology arising from the gauge/gravity model at hand with the picture of [1] that we have just briefly reviewed. In the region µ T where the hydrodynamic description is inaccurate 22 , our results can be compared with formulae (4.7) and (4.8) neglecting s andκ . Indeed, as we have already commented, in this regime the elastic scattering rate τ −1 dominates on the typical energy scale of the interactions of the charged plasma. In other terms we are in the region dominated by impurity scattering. We recall that in the impurity dominated region the Wiedemann-Franz ratio W is constant and it suggests that the Boltzmann equation, which is valid only in the presence of well-defined and sharp quasi-particle excitations, leads to quantitative and qualitative results that match accurately the transport properties of the holographic system at hand. The comparisons between our numerical results (blue solid line) and those arising from formulae (4.7) and (4.8) (red dashed line) are plotted in Figure 6. Form Figure 6, at least within our numerical precision, emerges that the numerical results match exactly with (4.7) and (4.8). This suggest strongly that a quasi-particle description might be accurate also for our strongly coupled holographic model for T µ. Of course to make this claim conclusive one requires a systematic and careful analysis of the quasi-normal modes of the model, which we postpone to future work [38].
It is noteworthy that the temperature T * at which the Wiedemann-Franz ratio W deviates from the constant value attained in the impurity-dominated regime, is approximatively the same temperature at which the elastic dissipation rate τ −1 becomes comparable with the temperature T of the system 23 . This fact allows us to define a specific quantitative limit of validity of the hydrodynamic approximation as applied to our model. All in all, by analyzing formulae (4.7) and (4.8), we note that the low-T regime of our model presents many features in common with the behaviour of the Fermi-liquid in the disorder-dominated regime. Namely the thermal conductivityκ DC goes linearly to zero with the temperature and is proportional to the heat capacity C = T ∂S ∂T , Furthermore, the electric conductivity σ DC is independent of the temperature, which constitutes another feature of the Fermi-liquid disorder-dominated regime.
The comparison of our model at low-T with the disorder-dominated Fermi-liquid appears however to be not complete. In particular, the Mott law describing the Fermi-liquid thermo-electric response is not satisfied even qualitatively. In fact equation (4.11) yields a Seebeck coefficient which goes linearly to zero as T → 0. On the contrary, as we have already noted, in the system at hand, s DC approaches a constant at T = 0 and then grows linearly with the temperature: This is due to the fact that the entropy S is non-zero at T = 0 (see (3.8) and (4.7)) . As a final remark, we note that in the region µ T it is possible to distinguish two different regimes, namely the regime β > µ 2 , where the elastic scattering rate τ −1 is greater that the chemical potential µ 24 , and the opposite situation β < µ 2 . In particular we find that when β > µ 2 the electric conductivity approaches its minimum value 1/q 2 while the Seebeck coefficient goes to zero.

Conclusion and future prospect
We have throughly studied and characterized the thermoelectric transport of a simple holographic model featuring momentum dissipation in the boundary theory. We regard the results obtained as interesting both from a purely theoretical perspective and from a phenomenological standpoint. Regarding the former, we demonstrated the possibility of obtaining a physically sound picture for the thermo-electric response of a gauge/gravity model possessing massive gravitons in the bulk. This feature leads to a breaking of some diffeomorphism in the gravity model which therefore has a lower amount of symmetry. Therefore, performing the holographic renormalization of a massive gravity model, one must consider a larger set of possible counter-terms. The additional freedom proves crucial in obtaining a consistent phenomenological picture because the appropriate choice of finite counter-terms allows one to prevent the appearance of an unphysical dissipation-less heat transport mode at null frequency.
From a more phenomenologically-oriented viewpoint, it is tantalizing to observe the closeness between the transport properties of the model at hand and the physics of the crossover between the quantum-critical to Fermi-liquid regimes discussed for the graphene. The behavior of the model at hand in the limiting high and low temperature regions respectively is in agreement with the non-holographic expectation of a hydrodynamic and quasi-particle regimes. On top of that, the holographic model allows one to study also intermediate regimes and offers the opportunity of having a complete setup interpolating the asymptotic regions.
A noteworthy fact is the possibility of the emergence, in our model, of a quasi-particlelike regime in the low-temperature region which is usually based on a standard Boltzmann description of quasi-particle degrees of freedom. Such description is not immediately connected with a microscopic detail of the model; indeed this quasi-particle regime arises in the deep IR (actually ω = 0). At any rate, it is interesting to observe that a Fermiliquid-like physical picture can arise from a strongly coupled, momentum dissipating gauge/gravity model. Of course, more investigation is needed in this respect. Both the assessment of this Fermi-liquid behavior and its detailed dynamics call for further exploration (e.g. the study of probe fermions on the massive gravity charged black hole).
In the writing of this paper we became aware of related studies about the thermoelectric transport in holographic systems with momentum dissipation [40]. In [40] the momentum dissipation is realized by means of additional scalar fields within the Q-lattice framework. Remarkably, the analytic formulae for the thermo-electric transport coefficients which they found are compatible with those found by us in the contest of massive gravity. It would be interesting to further investigate the relation between these two results.
One natural extension of the present analysis consists in studying the quasi-normal modes of the system in the whole temperature range. In other words, the extension of the study presented in [14] to the ballistic and intermediate regimes as well. Although possibly technically demanding, such an analysis could shed light on the intimate nature of the holographic plasma and some statements regarding the quasi-particle nature of the low-temperature physics could obtain conclusive evidence.
Another very promising direction for further work is represented by the inclusion of a magnetic field 25 . This not only allows one to study the mixed magnetic, electric and thermal transport, but could offer the possibility of studying other features which are based on experimental expectations. In particular, the presence of cyclotron modes which are intrinsically related to a collective nature of the quantum critical response. where the arguments of the first and second fluctuation field in each term are respectively −ω and ω. In order to simplify the notation, we introduce gothic letters to indicate the coefficients in the quadratic action:

Acknowledgements
where the correspondence between gothic letters and coefficients is easily understood by comparing (A.2) with (A.1).
The relation between the derivatives with respect to the physical quantities and those with respect to the sources of the bulk fields is given in (2.18). We remind the reader that the sources h (0) tx and a (0) x are independent and the derivative with respect to one of them is taken putting the other to zero (this fact is understood throughout our formulae). The off-diagonal term in the transport matrix is due to the mixed second order derivative. Exploiting linearity we obtain Since we are dealing with a system preserving time-reversal symmetry, due to Onsager's argument the transport matrix must be symmetric. To check the symmetrical character of the transport matrix offers a useful check of the correctness of the computations (which is slightly delicate due to the non-trivial relation between the physical and the bulk fields). On a technical ground, we need to verify that the functional derivatives commute, namely We have the right commutation between the derivatives and taking stock of the preceding computations, we have Repeating the same steps for the diagonal entries of the transport matrix (2.16), we obtain: