# Universal Bounds on Transport in Holographic Systems with Broken Translations

### Submission summary

 As Contributors: matteo Baggioli Preprint link: scipost_202005_00007v1 Date submitted: 2020-05-19 Submitted by: Baggioli, matteo Submitted to: SciPost Physics Discipline: Physics Subject area: High-Energy Physics - Theory Approach: Theoretical

### Abstract

We study the presence of universal bounds on transport in homogeneous holographic models with broken translations. We prove 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 $\eta/s$ is not a meaningful quantity 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 prove that the stiffness $\partial p/\partial \epsilon$ 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.

###### Current status:
Editor-in-charge assigned

### Submission & Refereeing History

Submission scipost_202005_00007v1 on 19 May 2020

## Reports on this Submission

### Report

The authors study the behaviour of black hole solutions in a specific class of models corresponding to massive gravity theories in order to test the validity of several conjectured (upper and lower) bounds on certain observables in the dual field theory. Such models have been used in the literature for modelling spontaneous and explicit breaking of translation invariance in holographic matter.

Identifying and understanding universal bounds on observables in Nature is certainly of great importance. Here, particular emphasis is give to the conjectured KSS bound for the shear viscosity to entropy density ratio, the lower bound on diffusion constants introduced by Hartnoll and extended by Blake, the upper bound on diffusion constants by Hartman et al as well as more recent bounds related to the speed of sound by Hohler et al. The authors carry out a set of numerical computations in order to test these bounds in field theories with (explicit or spontaneous) breaking of translations. These tests are new and, to the best of my understanding, correct. No new bounds have been introduced in this work.

At several points in the paper the authors claim that they have numerical proved" the conjectured bounds. The authors are reminded that numerically one can only disprove a conjecture by finding a counter-example. If the bound is not violated numerically, it simply means that in the specific model studied and for the specific values used, e.g. ($N,\mu, T,\dots$), the bound was respected. Having this in mind, the claims of the authors are too strong and should thus be toned down, in order to actually reflect the computations done. As the paper stands, the claims are too strong.

Let me now turn to more specific points (following the order of the paper):
1. It is well known that finite-N effects violate the KSS bound (but not finite $\lambda$ corrections). This itself suggests that this bound can not be a universal, regardless of the status of translation invariance. See 1108.0677 for a review.
2. $\tau_{eq}$ is the \textit{local} equilibration time and it's related to the applicability of hydrodynamics. This is different from the time needed to reach \textit{global} equilibrium, so please adjust the text. Furthermore, by now it's understood that the scale that sets $\tau_{eq}$ is not the distance to the first gapped quasinormal mode, but instead the location of the critical point (see e.g. 1904.12862) which can be much larger. How does this observation affect your results related to the upper bound?
3. For a system with both energy and charge density one needs to be more careful when defining the speed of sound, see e.g. 1205.5040
4. In the last sentence of Section 1, you say you will discuss the implications of the absence of a UV cutoff, but I can not find this discussion. Please indicate where this is done.
5. Please add a small appendix with the details of the numerical calculations that you carried out. This will add clarity and completeness to the paper and allow readers to follow easier. Also, add a few comments in the main text where you discuss the difference between explicit and spontaneous breaking in connection with the value of $N$ -- currently this is only done for the explicit case.
6. Figure 2, right panel: remove the y-axis label (since you display 2 different quantities on the same plot)
7. Page 7, point V, first 2 sentences: Essentially all the points on page 6 and 7 indicate that it's not clear what is the right definition for the shear viscosity when translations are broken. Given that, any conclusion for the violation or not of the KSS bound in this context can be attributed to simply using the wrong definition and thus not to carry any physical importance. This comment also applies to the orange line in Figure 2. I understand and I agree with the rest of the argument V regarding the mass of the shear mode.
8. Page 21: the fact that the stiffness becomes negative signals a dynamic instability.
9. It would be nice to add a table in your discussion section where you summaries the bounds and whether or not they are violated for spontaneous or explicit breaking of translations in your model.
10. The Hawking-Page transition has been associated to the confinement transition by Witten more than 10 years ago. How does this fit with the concept of the melting temperature?
11. Please check the text again for typographic errors. Furthermore, in several points the authors summarise the lack of deeper understanding of certain concepts in the form of questions, one after the other. This clearly shows their excitement for the topic, which is highly appreciated and valued, but it can be disruptive to the reader -- please consider formulating in a different way.

Given the above, I recommend this paper for publication in Sci.Post, after the authors have addressed the points raised here.

• validity: good
• significance: ok
• originality: good
• clarity: good
• formatting: good
• grammar: good