Bounds on charge and heat diffusivities in momentum dissipating holography

Inspired by a recently conjectured universal bound for thermo-electric diffusion constants in quantum critical, strongly coupled systems and relying on holographic analytical computations, we investigate the possibility of formulating Planckian bounds in different holographic models featuring momentum dissipation. For a simple massive gravity dilaton model at zero charge density we find robust linear in temperature resistivity and entropy density alongside a constant electric susceptibility. In addition we explicitly find that the sum of the thermo-electric diffusion constants is bounded.

Introduction.-The description of universal behaviors of strongly correlated systems despite the differences in their microscopic structure is challenging and of high physical relevance. One of the best known examples is the linear in temperature (T ) resistivity regime shared by many apparently very different materials like cuprates, heavy fermions, pnictides, ruthenates and fullurenes [1]. In recent times, a great effort has been put into characterizing the universal behavior of these materials by determining bounds on appropriate physical quantities. The reasons why this approach might be successful are twofold: first, a bound can be formulated independently of the microscopic detail of the system and then can aspire to be universal; secondly, in an incoherent quantum critical regime, the system equilibrates on a time scale independent of any microscopic energy scale and just determined by the temperature. Therefore, as a consequence of the insensitivity to energy scales other than the temperature, strongly correlated systems tend to saturate general temperaturedependent bounds.
Strongly correlated models amenable to quantitative theoretical control are extremely rare. In this respect, the gauge/gravity duality is very useful since it provides a dual description for some strongly interacting quantum field theories where equilibrium and transport properties can be computed. Actually, one of the most celebrated results obtained through holography is the evidence of a universal bound on the momentum diffusion rate which is approximatively saturated by strongly interacting systems. This bound was originally conjectured in [2] in terms of the ratio of the shear viscosity to the entropy density, namely η/s ≥h/(4πk B ). The approximate saturation of the viscosity bound has been subsequently observed in the quark gluon plasma created at the heavy ion colliders, as well as in the unitary fermionic cold atom gas [3]. Moreover, recent experimental ARPES measurements on an optimally doped cuprate show that, also in this case, the ratio η/s is close to saturate the afore mentioned bound [4], strongly suggesting that the transport properties of the strange metals due to the strongly correlated electron behavior could be described in the holographic framework.
The physical origin of the existence of the η/s bound can be explained with the concept of "Planckian" dissipation [5,6]: the fact that, in the finite temperature quantum critical state, physical quantities relax extremely rapidly and the equilibration rate is substantially determined by the uncertainty principle, namely τ eq ∼h/(k B T ). Recent experimental observations showing that a wide range of strange metals in the linear in temperature resistivity regime present an equilibration rate of this kind [1] strongly supports the idea that a unified framework might capture the properties of these strongly interacting materials.
Relying on a wider applicability of similar arguments, it is tempting to find whether other physical observables in strange metals and, more generally, in strongly correlated materials have to saturate a bound due to Planckian dissipation. However, the identification of the correct kinematic framework where one can formulate universal bounds is in general not easy. The reason being that in metallic materials the mechanisms for momentum relaxation, such as the lattice or scattering from disorder, which are necessary to make the conductivities finite, are extrinsic to the electron dynamics. Hence it is difficult to formulate intrinsic and universal bounds.
A recent proposal advanced in [7] suggested that, in analogy to the bound on η/s, the quantities that are universally bounded in strange metals are the thermo-electric diffusion constants D + and D − . More specifically in [7] it was argued that strongly correlated metals in the incoherent regime (where momentum is quickly degraded) have charge and heat diffusion constants that saturate the bound where C is an undetermined constant andv is a characteristic velocity of the system which, in genuinely relativistic arXiv:1411.6631v1 [hep-th] 24 Nov 2014 invariant systems, corresponds to to the speed of light c while in systems like the strange metals it can be identified with the Fermi velocity. Indeed, in strongly correlated metals the Fermi velocity appears in the linear dispersion relation for the low-energy excitations of the Fermi surface. The diffusion constants D + and D − are related to the transport coefficients via the Einstein relations, namely: where σ, α and κ are respectively the electric, the thermoelectric and the thermal conductivities (see [7] for a derivation); c ρ is the specific heat at fixed charge density, ζ is the thermo-electric susceptibility and χ is the compressibility. The bounds (1) can therefore be translated in terms of the thermo-electric transport coefficients of the model. In particular, relying on qualitative arguments, in [7] it was suggested that incoherent metals approximatively saturating the bounds (1) have a linear in temperature resistivity controlled precisely by the equilibration time-scale τ eq ∼h/(k B T ).
In this letter we explicitly compute within the framework of holography the charge and heat diffusion constants in strongly correlated systems featuring momentum dissipation. Particular attention is paid to distinguish intrinsic and extrinsic contributions to the diffusion constants. We find that in the critical regime the intrinsic contributions to the diffusion constants D ± saturate a bound like (1). Restricting to the intrinsic contribution, we also compute the value of the numerical factors C in front of the bound. The independence of this result on the specific parameters of the model and the comparison of different holographic models allow us to speculate about its possible universal relevance. Nevertheless, turning the attention back to the complete (namely, both extrinsic and intrinsic) diffusion constants, the picture is more involved. Appropriately defining a concomitantly critical and incoherent regime (where localization phenomena are irrelevant and no slowly-relaxing quantities are present) we are able to bound the complete diffusion constants according to the conjecture advanced in [7]. In this case, however, the coefficients are not universal showing actually sensitivity to the detail of the extrinsic scattering mechanisms. Eventually, in the particle-hole symmetric circumstance (i.e. µ = 0) in which the thermo-electric transport matrix is diagonal (namely the thermo-electric conductivity and susceptibility vanish), one is able to identify the diffusion constants D + and D − as the heat and charge diffusion constants respectively.
The holographic model.-Our computations rely on the phenomenologically motivated holographic model [9] which features massive gravitons in the dual gravitational 3 + 1 dimensional bulk. The UV derivation and ultimate consistency of massive gravity are still open questions which, to the present purposes, can be set aside as long as the phenomenology of the boundary theory is itself consistent. This was proven to be the case both in relation to the thermodynamics [11] and on the level of the thermoelectric transport properties [13].
The gravitational action is where represents the graviton mass term expressed in terms of the matrix K whose definition is K ≡ ( β is a parameter constrained to be negative by the stability of the model [9]. The matrix f is a non-dynamical auxiliary metric whose explicit form is f µν = diag(0, 0, 1, 1). The action (4) contains also the Maxwell-Einstein term (with F = dA) and a boundary term S c.t (see [13] for the proper definition of the latter) necessary to have a well-defined variational problem and a finite on-shell action.
Following the standard holographic dictionary, it is known that (4) is the gravitational dual of a strongly correlated system with charged degrees of freedom exhibiting an extrinsic elastic mechanism of momentum dissipation such as that generated by the presence of quenched disorder [9,[11][12][13].
As it is well discussed in the literature [9,[11][12][13], the model (4) admits the following black brane solution: where z h is the horizon radius defined by f (z h ) = 0. As derived in [11], the temperature T , the charge density ρ and the entropy density S can be expressed in terms of z h and the other parameters of the model in the following way: The parameter β controls the strength of the momentum dissipation mechanism, namely the momentum is dissipated faster as |β| increases. In particular in the coherent regime an extrinsic dissipation rate τ −1 ext proportional to |β| can be defined [12]: In order to obtain the diffusion constants from the system of equations (2) and (3), we need at first the explicit expressions of the DC transport coefficients. An analytical computation (see [14]) leads to the following expressions: secondly, we have to compute c ρ , χ and ζ for the model at hand. This can be easily achieved using their thermodynamic definitions, namely and keeping into account the specific thermodynamic relations (6). Diffusion constants at large temperature.-We have all the entries appearing in the right hand sides of equations (2) and (3), hence we can solve for the diffusion constants. The general expressions are rather complicated and not of particular interest by themselves; hence we do not report them here. However, in order to have an idea of the general behavior of D ± , we plot in Figure 1 the diffusion constants as functions of the temperature for a specific choice of the parameters of the model. As it is evident from Figure 1 low-T regime, one may expect localization phenomena to be relevant and consequently a lower bound like (1) may naturally be violated. From the viewpoint of the bounds, the high-T regime is more interesting; there, in fact, we expect localization phenomena to be irrelevant. Considering the large temperature limit of the diffusion constants we obtain The diffusion constant D − saturates the bound (1) with C = 3/(4π) while the D + (even though respecting the same bound) does not saturate (1) and it even diverges in the T → ∞ limit. The analytical control which led to the diffusion constants (10) and (11) allows us to distinguish the intrinsic and the extrinsic contributions; in fact, we simply rely on whether each term contains or not a dependence on the extrinsic parameter β. Isolating the intrinsic contributions we have We observe therefore that the intrinsic contributions to the diffusion constants at criticality are dominated by a 1/T behavior and actually saturate a bound like (1). Incoherent regime.-The simple high-temperature limit however does not respect the incoherence requirement τ −1 ext > ∼ k B T . Increasing the temperature without modifying the momentum dissipation rate τ −1 ext we are led to a coherent circumstance where τ −1 ext k B T and momentum is an almost-conserved quantity. To remain in an incoherent regime one need to consider the limit where b is an undetermined positive coefficient. The scaling of β is naturally connected to its dimensionality when we remind ourselves that it appears in the bulk model as a squared mass for the gravitons. Moreover, naively substituting the functional expression (14) in the formula for the dissipation rate τ −1 ext (7) we find τ −1 ext ∼ k b T , which is the scale at which a transition between the coherent and incoherent regime is expected [17]. In other words, the limit (14) does not introduce any "external" hierarchy to the model (τ −1 ext ∼ τ −1 int ) and respects the incoherence re- where f ± (b) are functions encoding the dependence of the coefficients in (15) and (16) on the detail of the incoherent limit (14). Thermo-electric decoupling.-In a system with vanishing thermo-electric mixed response (i.e. ζ = α = 0), from the Einstein relations (2), (3) we have that the diffusion constants are respectively purely electric, σ/χ, and purely thermal, κ/c ρ . In our holographic system, we can attain this thermo-electric decoupling condition considering the µ = 0 circumstance. When the thermal and electrical responses are decoupled, one can interpret the diffusivity constants and bounds thereof directly in terms of charge and heat diffusion respectively.
Observe that, in the µ = 0 case, the bounds on the intrinsic contributions (12) and (13) which is in line with the fundamental result obtained for the neutral N = 4 critical plasma [16] (where no extrinsic physics was included). Notice, however, that the individual numerical factors in (12) and (13) differ from those in [16].
Comparison with other models.-To corroborate the relevance of the results on the diffusion constants in a critical and incoherent regime, we repeated the same analysis in different holographic models exhibiting momentum dissipation. The simple "axion" model described in [10] has the same thermodynamics and DC transport coefficients of (4), and consequently it is completely equivalent to massive gravity regarding the diffusion constants. We also analyzed the model specified by the action . (17) bound only when the system is considered in a critical and incoherent regime which translates in an appropriate concomitant limit of high temperature and high momentumdissipation scattering rate. The picture here could however be not universal meaning that, although a bound is satisfied in form, the numerical coefficients are still sensitive on the extrinsic detail of the model. We analyzed all the currently known holographic and momentum-dissipating models possessing an analytical description. In addition to what described in the body of the letter, we also found that the "axion" model introduced in [10] leads precisely to the same thermodynamics and DC transport as massive gravity and, consequently, to the same conclusions about the diffusion constants. Moreover, considering axions instead of massive gravity in the action (17) led to the same identical coefficients for the asymptotics of the diffusion constants.
The general relevance of the present results and the fact that massive gravity was previously observed to account for "weak" holographic (explicit) lattices [20], suggests a possible universal meaning of massive gravity as the correct general framework to account for the leading effects of translation breaking in holography.
As a final comment, the results regarding the critical behavior of diffusion constants could be even more general that the massive gravity framework just mentioned. Indeed, also in solutions where deformations (e.g. modifications of the dilaton of (17)) could be relevant in the UV sense, the concomitant modification of the asymptotic behavior of thermodynamic and transport properties could conspire and lead to the same critical/incoherent structure for the diffusion constants.
The investigation about these last suggestions constitutes the main future prospect of this letter.