Planckian dissipation, minimal viscosity and the transport in cuprate strange metals

This is ironic. This report is helpful, since I had overlooked the work by Sean Hartnoll et al, arXiv:1601.02757 but unfortunately the report of the second referee rests on an awkard miscomprehension of this work.

It looked at first sight detrimental for my story, albeit for other reasons than quoted by this referee. Write $\eta/s = A_\eta \hbar/k_B$. The suggestion from the Hartnoll paper is that in the presence of a strong UV potential $A_{\eta} << 1/(4\pi)$. In the dimensional analysis associated with the linear resistivity on page 28 this would have the consequence that the disorder length $l_K$ would become much smaller than the lattice constant - a fatal inconsistency. Fortunately Sean is next door and he pointed out immediately that the standard way to compute the viscosity holographically (Eq. 12 in the revised manuscript) is actually misleading in the present context. This was elucidated in a follow up paper (new ref. 62): as I explain in the revised version one has to inspect the full shear propagator to find out that the viscosity associated with the emergent hydrodynamical regime is the usual one with $A_\eta = 1/ 4\pi$! Surely, the referee missed this development and therefore his comments 1-3 are irrelevant anyhow.

Frankly, I am at a loss with regard to his train of thought in these remarks. As spelled out in Hartnoll's paper "their" $\eta$ has no a-priori relationship to transport: when for whatever reason Galilean invariance is broken in the deep IR there is no such thing as a hydrodynamical viscosity. More alarming, it seems that he did not pay any attention to the phenomenology of the cuprates as discussed at length in my paper. Dealing with nature itself one cannot pick at will from the repertoire of possible outcomes of a theory. One first looks at the constraints set by experimental information. There are independent empirical reasons to assert that $z \rightarrow \infty$. It is just fact that the optical conductivity of the strange metal reveals sharp Drude peaks at temperatures below 400 K or so. There is no experimental evidence whatsoever for relevancy of translational symmetry breaking (charge order) in the strange metal regime. We do know from experiment that the strong periodic lattice potentials disappear completely in the deep IR, while there are good reasons to assume that the quenched disorder is quite weak. Under these perturbative circumstances it is perfectly save to employ a minimal bulk set up such as massive gravity and the bottomline is that hydro behavior will come to an end at the length scale Eq. (27).

With regard to his comment 4: of course the butterfly velocity is temperature dependent according to finite density holography. Hartnoll's case is that he claims this to be unreasonable based on bounds for operator spreading. After much debate the two of us reached the conclusion that this issue cannot be decided on theoretical grounds alone, reason for me to include it in the "caveats" section. This comment just highlights that both referee's did not take the time to familiarize themselves with Hartnoll's reasoning.

I am sceptical as well whether the minimal viscosity is at work. However, this is condensed matter physics and the role of the theorist is no more than to ask unusual questions to experiment, based on the capacity of equations to teach us to think differently. It may well be that the experimental answers are yet elsewhere but this represents usually progress. I added the original section 4.4 (related to the comment 4 of the referee) for the mere purpose to spell out to the experimental readership that holography is as any other theoretical physics activity in condensed matter physics to be looked at with distrust. But in my mind the "large N caveat" is even superseded by another one: it is very well understood that for the cuprate electrons to become interesting very large periodic lattice potentials are needed (Hubbard models, etc.). How can it then be that near perfect Galilean invariance physics emerges in the not so deep IR? I did not pay explicit attention to this obvious problem in the original paper for length reasons. I am grateful to the second referee for pointing out the Hartnoll viscosity paper (arXiv:1601.02757) because it represents a concise way to explain how it works in holography. Assuming that the linear axions have anything to do with the strong lattice potentials, the computations show that these are ${\em strongly}$ irrelevant. Upon departing from the UV, these dive down rapidly under renormalization, forming a quite large regime at low temperatures where \eta/s is constant signaling unambiguously the hydrodynamical regime. The lattice is more irrelevant than in e.g. a Fermi liquid where Umklapp switches on algebraically in the flow from the IR fixed point -- the $T^2$ resistivity.

For this reason I decided to expand the "caveat" section 4.4 in the revised manuscript. I added an overall introduction for rhetorical reasons. I text edited the "large N caveat" part (the original section 4.4), wiring in the temperature dependence issue of the butterfly velocity explicitly to help holographic readers like the present referee's to appreciate Hartnoll's point of view. This has now become subsection 4.4.2. I added a new subsection (4.4.1) that is relatively long for the reason that various aspects of the paper arXiv:1601.02757 (new reference 61) are quite informative in the present context. It also serves the purpose of advertising the insightful (new) ref. 62 that is apparently not yet widely disseminated in the holographic community (e.g., referee 2). I discuss the relation of $\eta/s$ to entropy production in the non-hydrodynamical setting, explain the presence of a large effectively Galilean invariant IR regime, to then elucidate the results of (new) ref. 62 showing that the emerging IR hydrodynamical viscosity is the minimal one. I conclude that a similar miracle should be at work in cuprates in order for the mechanism underlying Sections 5,6 to be possible altogether.