Finding universal structures in quantum many-body dynamics via persistent homology

1) Nice introduction to topological data analysis techniques

2) Clear discussion of the performed numerical simulations

3) Potential importance for future studies of experimental data in ultracold atoms systems with topological data analysis

1) Not enough discussed how the method can be used to treat general, strongly interacting QFTs

2) Not enough motivated the claim that "a refined classification of nonequilibrium universal phenomena" is (or can be) obtained.

The paper introduces topological data analysis techniques, introducing the so-called "persistent homology observables". After such an introduction, the authors considered data from a classical-statistical simulation of a 2D Bose gas far from equilibrium. Birth and death radii distributions are introduced and numerical results discussed.

The idea of using topological data analysis is certainly interesting, and motivates me to give an overall positive evaluation. Also appreciated by me has been the clarity with which the numerical simulations are presented and discussed.

The point which at variance leaves me skeptical is the particular system studied. To reach and motivate the ambitious goal the authors set, I feel they should/could have discussed (before their example or even instead of) a simpler, and in my opinion, clearer example: a system slightly perturbed from the critical point, and studying the critical dynamics with their methods. I would have expected to see this in a mean-field model, to see what their machinery would have produced. In that case the authors could have used a description based on Gross-Pitaevskii or time-dependent Ginzburg-Landau, and compared with known results, including how the obtained scaling exponents (of birth and death radii distributions) would have been related to the known dynamical critical exponent z, at least for some of the Hohenberg-Halperin classes. That could have be applied to a Bose-Hubbard model near the superfluid-Mott transition, or even to a Bose-Einstein condensate (without optical lattice) near the critical point. How/what the "scaling exponents" introduced in the paper would have been? It is possible that I may have missed literature doing topological data analysis on these simplified models, but in case such results already exist (or maybe the authors have already worked out) I would have appreciated very much an introductive discussion on this point.

The authors instead decided to go for the 2D Bose gas, and there one expects strong deviations from mean-field. It is not even clear to me whether with their approach they can reproduce the equation of state [Phys. Rev. Lett. 87, 270402 (2001)] and the highly non-perturbative parameter in it. In the regime in which they put themselves, the classical-statistical approximation, expectation values of quantum observables are computed as ensemble-averages of classical field configurations: so, one may raise the question of how the approach would work when classical-statistical approximation fails. And, similarly, what should be the procedure to compare with data extracted from the experiment when classical-statistical approximation fails? One answer to this objection could be that one should start form the simplest case before, but then we are back to my previous question on why simpler setups (such a weakly interacting Bose gas near its critical temperature) have been not discussed before. From my point of view even the 1D Bose gas would have better, but this is - I admit - certainly matter of taste.

Finally, I am bit surprised by the emphasis the authors give on the "scaling exponents". Indeed, analyzing Monte Carlo snapshots of Ising models (or percolation) one could define critical exponents from peculiar non-trivial subset one can construct from them, such as the Fortuin-Kasteleyn representation. Also there one can define appropriate critical exponents, the real question being how they are related to the other critical exponents.

In conclusion, I liked the idea of the paper. I feel that a more physical introduction to the problem, and in particular a discussion on mean-field-like examples [or near T_BEC, where experimental data could be worked out] would have been important and would have considerably improved both the clarity of the paper and the significance of the presented method.

In the report I mentioned several points which in my opinion could be discussed better. I do understand that too much material cannot be added (or changed) from the present version, but my suggestion is to consider the previous comments and accordingly improve the discussion, possibly including of what the topological data analysis machinery would produce near a critical point (at least in mean-field). Even if this discussion would be very simple with respect to the rich phenomelogy of the 2D Bose gas, I think it would help the clarity of the paper.

1) visionary
2) cross-disciplinary
3) bold

1) hard to read
2) full of jargon
3) weak interpretation

By reviewing the manuscript by Spitz et al. I developed a number of contrasting feelings about their work.

Their starting motivation seem to extend topological data methods to the analysis of quantum experiments. This is certainly original and they end up delivering a manuscript which is very much trans-disciplinary and certainly highly original.

The first issue is that such originality is perhaps too far reaching. By admission of the authors, their explanation on the continuous scaling is highly conjectural (mid of page 14) and motivation for future studies or even their original intention is hard to find (again, the authors are honest about it — see final paragraphs of the conclusions).

Combined with a strongly jargon language and some cryptic passages (see below), all this risks to prevent the interested or curious reader from benefiting of their results and potentially expanding such research line.
So I believe it should be in the own interest of the authors to significantly improve the manuscript.

I now report a number of instances mostly taken from the first ten pages, which carry on to the entire article:

— the intro is too technical; I understand they have a summary of homologies later on, but it is quite discouraging (the reader is bombarded with technical words from algebraic topology and would tend to skip the first two pages)

— sec. 2.1: what is the classical-statistical approximation? (I am aware they reference) why Q is used several lines before properly being introduced? the factor ’50’ in eq (1) appears weird — why not using a tunable parameter? this section is overall fine but this additional imprecisions increase the sense of discouragement in keep going with the manuscript

— hard/cryptic paragraphs persist even in section 2.2 which should help the reader: for instance at page 6 the paragraph that starts with ‘The construction of persistent homology groups etc etc’

— these birth and death radii should be connected with something physical, otherwise the entire original motivation is lost

— I am not sure how much the reader can follow on non-thermal fixed points and turbulence if he/she is disconnected from previous results (I am aware the authors work on this since several years)

It really seems sometimes that the paper has not been proof-read before submission. The five examples above could serve as a guideline to massively improve the presentation.

Two concerns about the main result:

— it is really destablising to talk about scaling exponents that depend on non-universal parameters like the filtration and Qt_min — can the authors mitigate this in the revised version of the manuscript?

— is there anything special about the model they consider? does this scaling of the homologies depend only on dimensionality, symmetries, and other few relevant quantities as in conventional scaling theory in stat mech?
Conjectural thoughts would already help to put all this in proper context

In conclusion: the work might definitely help the community to take a new angle in solving the many-body problem with science data/advanced math, however, at the moment the presentation is borderline with being not comprehsible and it would be a pity (in primary instance, for the authors themselves).

1) review the presentation following the criteria in the report
2) comment on universality of the results