Abstract
We study the presence of universal bounds on transport in homogeneous holographic models with broken translations. We verify numerically that, in holographic systems with momentum dissipation, the viscosity to entropy bound might be violated but the shear diffusion constant remains bounded by below. This confirms the idea that \eta/sη/s loses its privileged role in non-relativistic systems and that, in order to find more universal bounds, one should rather look at diffusion constants. We strengthen this idea by showing that, in presence of spontaneously broken translations, the Goldstone diffusion constant satisfies a universal lower bound in terms of the Planckian relaxation time and the butterfly velocity. Additionally, all the diffusive processes in the model satisfy an upper bound, imposed by causality, which is given in terms of the thermalization time – the imaginary part of the first non-hydrodynamic mode in the spectrum – and the speed of longitudinal sound. Finally, we discuss the existence of a bound on the speed of sound in holographic conformal solids and we show that the conformal value acts as a lower (and not upper) bound on the speed of longitudinal phonons. Nevertheless, we show that the stiffness \partial p/\partial \epsilon∂p/∂ϵ is still bounded by above by its conformal value. This suggests that the bounds conjectured in the past have to be considered on the stiffness of the system, related to its equation of state, and not on the propagation speed of sound.
Highlights
The diffusion constant which reduces to charge diffusion at zero coupling violates the standard bound. (V) All the diffusive processes in the holographic models with spontaneously broken translations obey an upper bound on diffusion, where the lightcone speed is the speed of longitudinal sound and the thermalization time is extracted as the imaginary part of the first non-hydrodynamic mode
Taking into account all these points, here, we study the dynamics of the η/s ratio and the momentum diffusion constant DT in holographic systems with explicitly broken translations
We have shown numerically that the violation of the KSS bound in holographic systems with momentum dissipation can be saved by looking at the momentum diffusivity, as a more natural quantity to bound
Summary
(V) All the diffusive processes in the holographic models with spontaneously broken translations obey an upper bound on diffusion, where the lightcone speed is the speed of longitudinal sound and the thermalization time is extracted as the imaginary part of the first non-hydrodynamic mode Until this point, we have focused our discussion on the diffusive dynamics, which is typical of incoherent systems (e.g. systems where momentum is dissipated very fast) or systems with conserved quantities (e.g. charge conservation, energy conservation, etc). We will only consider holographic models in the large N , strong coupling, limit in which the gravitational dual is described by weakly-coupled classical gravity It is well known (see [19] for a review) that 1/N corrections can induce a violation of the KSS bound [8] and push the ratio η/s below its “universal” value of 1/4π. As a result of that, the violations of the KSS bound induced by these effects are not parametric (like in the case of broken spacetime symmetries which we consider in this work) but they just modify the O(1) constant # appearing in η/s ≥ #ħh/kB, which does not have any fundamental meaning
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