Hydrodynamics of a relativistic charged fluid in the presence of a periodically modulated chemical potential

The authors initiate the study of relativistic hydrodynamics around a state with a periodic chemical potential defined on a lattice. This is a novel construction that has potential relevance for electronic transport in solids. This work satisfies the journal's acceptance criteria of opening a new pathway in an existing research direction and so I think is suitable to be published subject to the comments below.

The paper is well written and the authors describe in careful detail the properties of the hydrodynamic theory they construct and study. They also provide a consistency check by quantitatively comparing the predictions of this theory to exact results arising from microscopic holographic models where this type of hydrodynamics is expected to emerge in the infrared.

The one area of the paper that I think could be improved is the conclusions. It would be helpful here for the authors to provide a more physical explanation of any new phenomena arising in their theory. I think it is fair to say that the one physical feature highlighted in the conclusion – the relaxation of momentum and the multiple contributions to it – has previously been realised in more basic theories, while the other statements (having to include Brioullin zone copies etc) are a bit removed from direct physical implications. For example, the authors briefly mention that the condensed matter physics relevance was analyzed in a companion paper – I think it would be beneficial to the readers of this paper to briefly mention any key results of this type here, even if they have to look elsewhere for the details.

1 - The developed hydrodynamic framework is novel and the subsequent derived results are backed up by holographic examples.

2 - The relation of the developed framework to other techniques in the literature (e.g. the memory matrix formalism) is outlined.

3 - Novel effects in the dispersion relations of the hydrodynamic modes are discovered and clearly explained.

4 - The work opens up a new pathway in an existing research direction with clear potential for follow up work.

1 - The paper sometimes uses terminology or makes assumptions that are not always clearly described, but are familiar to those that work in the field of applied holography.

The paper is written in a clear and intelligible manner - and while generally free of unnecessary jargon would benefit from fixing the few places where this is not the case (as discussed in the requested changes). Similarly, with the requested changes, the abstract and introduction clearly explain the context and results of the work.

The appendices describe well how certain results in the paper are produced, and would enable an expert in the field to repeat the numerical analysis used to produce the figures. Similarly, there is sufficient detail in the bulk for a qualified expert to reproduce the presented calculations. Previous works are clearly cited and represent a complete record of the relevant literature.

In my opinion, the analysis is new, interesting and complementary to other approaches in the literature. Subject to satisfying the requested changes I would recommend publication at SciPost.

I would request that the following points are addressed prior to publication:

1 - In general I would qualify more carefully how a spatially varying chemical potential mimics, but is not equivalent to, an ionic lattice – in particular, one should be careful with “i.e. an ionic lattice”.

2 - On page 2, “Sound waves, however, are hydrodynamic fluctuations…” - the authors have not yet defined the term “hydrodynamic” and it may be worthwhile to do so prior to this point as readers familiar with material physics may not understand what relation sound waves in a solid have to those in a fluid.

3 - On page 3, in the text around equations (3) and (4), it would be worthwhile to briefly discuss a little more on the formalism of Bloch waves and in particular the role of “n” for those unfamiliar with this terminology.

4- On page 4, eqs. (6), are those for a fluid which respects microscopic spatial parity invariance. This should be noted.

5 - On page 4, in reference to the sentence “This is the reference frame where the fluid velocities vanish”. Initially, I misread this in the context of the previous statement - “The above constitutive relations also hold in a static equilibrium background. ” As I am sure the authors are aware, in the absence of an external field we can boost the fluid to have non-zero spatial velocity and still be in global thermodynamic equilibrium. They should elaborate briefly on why one can or cannot have a non-zero spatial velocity in the presence of a background field.

6- Pages 8 and 9 – in particular Eqs. (32) and (34) should not depend on time. Only time independent quantities are related by the susceptibility as in (32). See footnote (5) on pg. 19 of https://arxiv.org/pdf/1205.5040.pdf . This affects the discussion on both pages and should be remedied.

7 - Page 17, penultimate paragraph – the longitudinal optical conductivity as written receives contributions from all modes of the theory, as it is just a scaling of the current-current correlator. As a consequence the following sentence beginning “Generically this current will receive… all hydrodynamic fluctuations” is confusing as it could suggest to a casual reader that only the hydrodynamic poles make contributions to this quantity. The authors should clarify what they mean here.

8 - To fully satisfy the criteria of the journal, I feel the conclusion of the paper needs to offer a discussion of the limitations and reach of the results, which is currently missing.