Bad Metals from Fluctuating Density Waves

The manuscript makes several interesting points about applying the hydrodynamics of phase fluctuating density waves to describe the conductivity of bad metals.
By extracting the location of the displaced peak and its width from data for a variety of bad metals, the paper identifies a simple dependence of both quantities on the Planckian timescale, a characteristic signature of quantum critical systems.
This elegant result provides evidence for the importance of including phase relaxation in the hydrodynamics analysis, as the authors stress. They also do a good job highlighting the crucial roles played by different scales in the system, and in particular the interplay between momentum dissipation and phase relaxation.
Finally, the paper makes concrete predictions for experimental signatures of fluctuating density wave order and clearly motivates searching for them in the bad metal region of the phase diagram of many materials.

I have no objection to the physics, but I think the results would be easier to appreciate and understand if the paper was structured slightly differently, with more material from the supplemental sections included in the main text, as a minimal change.
As is, some of the discussion comes across as being rather scattered, and it is hard for the reader to easily see a coherent picture and appreciate the significance of all the points made by the authors.
The discussion of the way in which the location and width of the peaks were extracted, as well as the extended Drude formalism, could easily be moved to the main text. This is a minimal change which would improve the manuscript.

This is a very interesting paper which discusses the temperature dependence of the optical conductivity for several families of bad metals, focusing on understanding the shift of the Drude peak away from zero frequency with increasing temperature. The goal is to identify collective low energy dynamics and provide a unified description of transport in these systems. The analysis relies on the optical conductivity results obtained by the authors in their earlier paper [40], in which they examined the hydrodynamics of charge density waves, taking into account the effects of broken translational invariance as well as phase relaxation. Inspecting the available conductivity data for a variety of bad metals, the authors extract the location of the displaced Drude peak and its width (see Fig. 2 and Eq. 3). The dependence of both quantities on the Planckian timescale, apparent from Eq.3, is characteristic of a quantum critical system. One of the crucial points of the analysis is to emphasize that the hydrodynamics of phase fluctuating (charge) density waves can describe the features seen in the transport properties of bad metals, and to stress in particular the need to include phase relaxation. The authors also identify concrete experimental signatures of fluctuating density wave order, such for example peaks in the frequency dependence of the scattering rate. The paper makes several valuable points about the effectiveness of the hydrodynamic description of density wave order - and the role of phase relaxation in addition to momentum dissipation - to understand the behavior of bad metals.

To improve the clarity of the discussion, I suggest that the authors incorporate the contents of Sections B and C into the main text.
I leave it up to them to do so.

This work represents a fresh eye on an old subject. What to expect for the conductivity of a fluid subjected to relatively weak breaking of Galilean invariance, that is however characterized by strong commensurate thermally fluctuating density-density correlations? This was already hotly pursued in the seventies (Fukuyama-Lee-Rice-Anderson) and it revived in the 1990's as related to fluctuating stripes. The hard work was actually done by the present authors in ref. [40]: in a characteristic style, completely resting on hydrodynamical principles, they derived eq. 1 for the optical conductivity. The surprise is that for a growing pinning energy a peak develops at finite frequency in the optical conductivity (Fig. 1). In hindsight this makes much sense but I was caught by surprise when I saw it for the first time, and it seems that in this regard I was not the only one. An evolution of the kind shown in Fig 1 as function of increasing temperature is measured in an variety of metallic oxides and organics (see Fig. 2) and the authors make the case that these electron fluids could be generically characterised by such strong crystalline correlations. This is a highly provocative idea; I am quite sceptical but its merit is that it may generate attempts to test it experimentally. Given the recent advances in experimental techniques it should become possible to try to falsify in a near future. When it turns out to be correct it would completely change the basic outlook on bad metal physics. In summary, this type of thinking out of the box is highly desired in this branch of physics.

There are many weaknesses: I am actually very sceptical. But I can't prove the authors to be wrong and it is just bad taste to use such guts feelings to censor papers. I repeat, this will likely provoke new experimental activity and I would cherish it when it turns out that this paper is on a right track.
Some of the reasons: (1) I wonder to what extent these authors realise the established principe that "transport is the first thing that one measures and the last thing that is explained." The issue is that there is not much information in (optical) conductivities while it may be sensitive to many aspects of the complex physics in oxides etc. As a veteran of condensed matter phenomenology I would be myself most hesitant to build a case as the present solely resting on transport information. (2) At the least one should attempt to arrive at high quality fits of the data itself with the Eq. 1 in the text. Dealing with real data it is just a bit very hand waiving to not get beyond the observation that "there is some peak in the data". (3) Also in this regard, transport would be the last alley to look for firm evidence for strong charge order correlations in the fluid. This can be measured directly using RIXS and EELS. At least in the cuprates, although clear signals are now picked up in the under doped- and overdoped regime the latest data indicate that up to room temperature optimally doped cuprates (showing the linear-in-T resistivity) are devoid of any correlations of the kind. To my eyeballs, the theory of the authors gets the transport just quite wrong in the regime where charge correlations have been established (underdoped, also the CDW organics). (4) A very interesting result by itself is the scaling of the peak energies and width, determined from experimental data (Fig. 3). The intriguing finding is that both increase linearly in temperature, being even set quantitatively by the "Planckian dissipation" principle. The authors express the hope that this will be eventually explained in terms of quantum critical states involving charge order, whatever. To the best of my understanding this is impossible for very simple reasons. The "first law" governing pinning energies insists that these are proportional to the strength of the charge-order correlations. This is canonical; in the conventional CDW cases one finds that the pinning energy goes to zero when the (mean-field) order parameter disappears at Tc; typically one may find a transition from the pinned (commensurate) phase to a truly incommensurate state when temperature is raised. In order to see the increase of \omega_0 with temperature it has to be that the charge correlations are {\em growing} when temperature increases. Regardless whether quantum fluctuations are in one or the other way in the game, this defeats general statistical physics principle: charge correlations represent order and when temperature increases entropy will invariably win when temperature gets high enough. (5) Specifically for the cuprates, it is just a myth that their secrets reside in bad metal behaviour. It is correct that above \sim 400 K the strange metals are nominally bad metals but at lower temperature they turn into good metals and at very low temperature they appear to become {\em excellent} metals (absence of residual resistivity, microns mean free path in quantum oscillations). What needs to be explained is that there is no sign of cross-overs between these different regimes: the resistivity is perfectly linear. It is odd to address this in terms of theories that are geared up to explain only {\em high} resistivities.

See the "strengths" and "weaknesses". Whatever happens, at the least Fig.2 will be remembered: even ignoring the interpretations put forward by the authors this is a very intriguing, purely phenomenological observation that has not been noticed before.

I leave it to the authors to use some of my observations to improve the text. When they have the energy they may attempt to actually use eq. 1 to fit some real data to see whether there is more to the story than just a peak.