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Finding selfsimilar behavior in quantum manybody dynamics via persistent homology
by Daniel Spitz, Jürgen Berges, Markus Oberthaler, Anna Wienhard
This Submission thread is now published as
Submission summary
As Contributors:  Daniel Spitz · Anna Wienhard 
Preprint link:  scipost_202102_00007v2 
Date accepted:  20210831 
Date submitted:  20210622 10:48 
Submitted by:  Spitz, Daniel 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
Inspired by topological data analysis techniques, we introduce persistent homology observables and apply them in a geometric analysis of the dynamics of quantum field theories. As a prototype application, we consider data from a classicalstatistical simulation of a twodimensional Bose gas far from equilibrium. We discover a continuous spectrum of dynamical scaling exponents, which provides a refined classification of nonequilibrium selfsimilar phenomena. A possible explanation of the underlying processes is provided in terms of mixing strong wave turbulence and anomalous vortex kinetics components in point clouds. We find that the persistent homology scaling exponents are inherently linked to the geometry of the system, as the derivation of a packing relation reveals. The approach opens new ways of analyzing quantum manybody dynamics in terms of robust topological structures beyond standard field theoretic techniques.
Published as SciPost Phys. 11, 060 (2021)
Author comments upon resubmission
1) “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).”
a) It is known that nonthermal fixed points and the related scaling behavior can encompass very different theories. For instance, the scaling properties in the infrared momentum regime of relativistic and nonrelativistic scalar field theories agree [Phys. Rev. D 92 (2015)]. These apparent similarities need to be tested against refined classification schemes. As mentioned in the introduction of our manuscript, it is a key motivation for our work to search for suitable quantities which both capture relevant properties of nonthermal fixed points and provide novel degrees of freedom to study them. Particularly, in light of the found scaling exponent spectrum, our results constitute a potentially enriching step in that new direction, the filtration function providing a new degree of freedom in order to discriminate between fieldtheoretic information under study.
2) “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.”
a) In the revised manuscript we included an updated introduction which contains less mathematical terms and should thus be better accessible to a readership educated in physics. We hope that the included updates of the section which introduces persistent homology to the reader can further support the readability.
3) “[T]he 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)”
a) We updated the introduction, accordingly, making it more accessible to nonmathematicians and only including mathematical jargon wherever crucial.
4) “[S]ec. 2.1: what is the classicalstatistical 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”
a) In the revised version, the classicalstatistical approximation is in brevity described in the section on selfsimilarity in occupation numbers. Valid equivalently, in the revised manuscript we change $Qt=0$ to $t=0$, not employing the scale $Q$ before introducing it later. Furthermore, in the new manuscript version we introduced the chosen initial conditions more carefully, mentioning the chosen initial momentum space box in occupation numbers, keeping the amplitude $f_0$ more general and subsequently only restricting to a value of $f_0=50/(2mGQ)$.
5) “[H]ard/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’”
a) We updated the paragraph on the robustness of the employed computational approach to include less mathematical terms which have not been introduced properly before (revised manuscript, Sec. 3.1). Admittedly, the remaining text passages of this section should already be amenable to a broad readership with physics background, given that we introduce the notion of filtration functions with care and discuss in detail the physical degrees of freedom included in point clouds of different filtration parameters.
6) “[T]hese birth and death radii should be connected with something physical, otherwise the entire original motivation is lost”
a) Persistent homology provides a means to study connectivity structures present in point clouds in a robust fashion which is insensitive to noise in the data due to the topological constructions undertaken. It is a crucial feature of persistent homology to be able to discriminate between topological structures of different physical length scales involved. The construction of point clouds as sublevel sets of field amplitudes allows for a discrimination between vortices and bulk excitations – a distinction known to discriminate between crucial dynamical degrees of freedom in the twodimensional Bose gas [New J. Phys. 19, 093014 (2017)]. Our topological data analysis pipeline with distributions of birth and death radii as the central objects under investigation features wellinterpretable geometric objects – loops of different dimensions, shapes, and sizes – contained in easily interpretable data which is even experimentally accessible as we elaborate on in a paragraph on the applicability of our methods to optical density images.
7) “I am not sure how much the reader can follow on nonthermal fixed points and turbulence if he/she is disconnected from previous results (I am aware the authors work on this since several years)”
a) In the revised manuscript we included a section which discusses nonthermal fixed points and the scaling of twopoint functions in the twodimensional Bose gas, moving the content of Appendix F to the main text. This should help the reader not familiar with nonthermal fixed points understand the core messages of our work.
8) “[I]t is really destab[i]lising to talk about scaling exponents that depend on nonuniversal parameters like the filtration and Qt_min — can the authors mitigate this in the revised version of the manuscript?”
a) The timedependent scaling exponent analysis carried out to obtain Fig. 7 in our manuscript reveals two key findings: A plateau for $\bar{\nu} < 0.5$ which is constant in time and the presence of a peak which comparably slowly shifts gradually towards higher filtration parameters $\bar{\nu}$. Both these findings persist in time and as such are to be regarded independent of the chosen value of $Qt_\min$. Solely the precise position of the peak changes in the course of time and is to be studied via timedependent scaling exponents. The entire phenomenological discussion of the scaling exponent spectrum in our work is based on these two findings and not on the precise peak position. As such, we regard the timedependency of the scaling exponents as a mere technicality.
b) The dependence on the filtration function is from our point of view not a mere technicality but instead a feature of our analysis. Persistent homology is robust in the sense that slightly varying the filtration function, the obtained results are nearly unaltered. However, one may choose completely different filtration functions such as phases computed from the complexvalued field configurations instead of amplitudes. We cannot expect these very different filtration functions to give rise to the same selfsimilar behavior of persistent homology observables. Crucially, via the choice of a filtration function one can study lengthscale resolved connectivity structures from selected fieldtheoretic information.
c) Given that our data is still well described by a selfsimilar scaling ansatz, we carry on regarding the corresponding exponents as scaling exponents. Nonetheless, in the revised manuscript at multiple positions in the text and in the title, we have been more careful about connecting the found scaling exponents with universality far from equilibrium. The latter certainly demands more attention beyond the scope of the present manuscript, such as investigating the dependence on initial conditions.
9) “[I]s 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.”
a) Fieldtheoretically, the singlecomponent nonrelativistic Bose gas in two spatial dimensions is the simplest system which can be both analysed reasonably by persistent homology techniques and shows nonthermal fixed points [New J. Phys. 19, 093014 (2017)]. For instance, to find nonequilibrium selfsimilar dynamics in a single spatial dimension, internal degrees of freedom are necessary [Phys. Rev. A 99, 033611 (2019)].
b) The plateau in observed scaling exponents at low filtration parameters with exponent values of approximately 0.2 can be attributed to anomalous vortex dynamics [New J. Phys. 19, 093014 (2017), SciPost Phys. 8, 039 (2020)]. Being topological defects of field configurations in position space, vortices, and their dynamics in general strongly depend on dimensionality. Defectdominated nonthermal fixed points are thus expected to depend on dimensionality, symmetries, and the fieldcontent under study. On the other hand, nonthermal fixed points can show similar scaling behavior across dimensions and symmetry groups due to waves interacting nonlinearly [Phys. Rev. D 92, 025041 (2015), Phys. Rev. D 101, 091902 (2020)]. The concrete role of topological defect and nonlinear wave dynamics in nonthermal fixed point behavior is still to be untangled, conclusively. Our work shows that persistent homology observables can provide novel degrees of freedom which can be of use here.
Detailed replies to comments made by the second referee
Comments on the denoted weaknesses:
1) “Not enough discussed how the method can be used to treat general, strongly interacting QFTs”
a) See the answer to point 4 below.
2) “Not enough motivated the claim that "a refined classification of nonequilibrium universal phenomena" is (or can be) obtained.”
a) With the novel Sec. 2 of the revised manuscript we gladly incorporate a more careful description of the selfsimilar scaling via the wellestablished occupation number spectrum, in particular showing that a single pair $(\alpha,\beta)$ of scaling exponents allows for a successful rescaling of the latter. Clearly, the obtained persistent homology scaling exponent spectrum contains more structure, as such providing from our point of view a sound basis for the above claim. In particular, given the freedom to choose a filtration function, the computational pipeline provides further novel quantities to study selfsimilar phenomena far from equilibrium, possibly allowing for untangling relevant degrees of freedom. We interpret the presence of a peak in the deduced scaling exponent spectrum as a signature in favor of the suitability of persistent homology observables for that purpose.
Detailed answers to points included in the report:
1) “The point which at variance leaves me sceptical 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 meanfield model, to see what their machinery would have produced. In that case the authors could have used a description based on GrossPitaevskii or timedependent GinzburgLandau, 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 HohenbergHalperin classes. That could have be applied to a BoseHubbard model near the superfluidMott transition, or even to a BoseEinstein condensate (without optical lattice) near the critical point.”
a) The critical behavior under investigation is to be distinguished from critical behavior in equilibrium. In our work the selfsimilar scaling of relevant observables unfolds in the course of time and not in the immediate vicinity of a critical temperature. In addition, to reach the vicinity of nonthermal fixed points no finetuning of (order) parameters is necessary. Nonthermal fixed points even form nonequilibrium attractors of timeevolutions of quantum manybody systems encompassing different types of initial conditions [Phys. Rev. D 89, 114007 (2014), Nature 563, 217220 (2018)]. Nonthermal fixed points being comparably well studied with clear phenomenological signatures (selfsimilar scaling), they provide an ideal testing ground for persistent homology observables in dynamical quantum manybody systems.
b) Fieldtheoretically, the singlecomponent nonrelativistic Bose gas in two spatial dimensions is the simplest system which can be both analysed reasonably by persistent homology techniques and shows nonthermal fixed points [New J. Phys. 19, 093014 (2017)]. For instance, to find nonequilibrium selfsimilar dynamics in a single spatial dimension, internal degrees of freedom are necessary [Phys. Rev. A 99, 033611 (2019)].
c) Taking different approaches, in recent years persistent homology has also been applied to critical behavior in equilibrium [Phys. Rev. E 93, 052138 (2016), Phys. Rev. Research 2, 043308 (2020)], with the main goal of topologically identifying phase transitions and providing novel degrees of freedom to capture order in physical systems with more complicated degrees of freedom. We included a description of the main available papers describing equilibrium models in the introduction. Our approach should be easily adaptable to the thermal equilibrium case, which, however, lays beyond the scope of the present paper.
2) “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.”
a) Far from equilibrium our study is the first one investigating scaling behavior of persistent homology observables in quantum manybody systems. Even in equilibrium there has been to the best of our knowledge no study which analysed persistent homology observables with regard to scaling phenomena. As such, we expect no such discussion to be present in the literature already. Solely, we can again refer to studies in equilibrium which qualitatively investigated phase transitions [Phys. Rev. E 93, 052138 (2016), Phys. Rev. Research 2, 043308 (2020)].
3) “The authors instead decided to go for the 2D Bose gas, and there one expects strong deviations from meanfield. 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 nonperturbative parameter in it.”
a) As mentioned already, the twodimensional singlecomponent Bose gas is among the simplest models to both reveal selfsimilar dynamics far from equilibrium and give rise to at least onedimensional persistent homology classes with nonzero birth radii. The persistence diagram of zerodimensional homology classes would just amount to a descriptor of interpoint distances in point clouds. One and higherdimensional persistent homology classes instead contain topological information on loops and voids, etc.
b) The classicalstatistical approximation is valid for large occupation numbers and small couplings [Phys. Rev. Lett. 88, 041603 (2002), Phys. Rev. A 76, 033604 (2007)]. Quantum thermal equilibrium is not amenable to a classicalstatistical description. It is due to this fact that our approach will not be able to reproduce the equation of state referenced by the referee. Still, via the classicalstatistical description with the GrossPitaevskii equation at its heart the dynamical evolution away from and towards the BKT phase transition can be studied [Phys. Rev. A 86, 013624 (2012)].
4) “In the regime in which they put themselves, the classicalstatistical approximation, expectation values of quantum observables are computed as ensembleaverages of classical field configurations: so, one may raise the question of how the approach would work when classicalstatistical approximation fails. And, similarly, what should be the procedure to compare with data extracted from the experiment when classicalstatistical approximation fails?”
a) The classicalstatistical approximation covers a wide variety of physically interesting scenarios and phenomena [Phys. Rev. Lett. 88, 041603 (2002), Phys. Rev. A 76, 033604 (2007)]. As such, our topological data analysis pipeline is already quite generally applicable to classical systems, as well as quantum systems in the classicalstatistical regime. Furthermore, in the section on point cloud phenomenology we speculate about possible applications to experimental data. Optical density images providing absolute squares of nonrelativistic field values, our approach can be employed directly as it is. In principle, our approach can be applied as well to field configurations obtained from importancebased sampling techniques such as Monte Carlo simulations and as such extends widely beyond the classicalstatistical regime.
5) “From my point of view even the 1D Bose gas would have better, but this is  I admit  certainly matter of taste.”
a) Nonequilibrium selfsimilar dynamics in a single spatial dimension requires internal degrees of freedom, else, kinetic arguments prohibit necessary scattering processes for the redistribution of particles from taking place. This would complicate the entire topological analysis.
6) “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 nontrivial subset one can construct from them, such as the FortuinKasteleyn representation. Also there one can define appropriate critical exponents, the real question being how they are related to the other critical exponents.”
a) Given the robust observation of selfsimilar scaling in persistent homology observables, it is natural to talk about scaling exponents as the exponents appearing in the selfsimilar scaling ansatz to the asymptotic persistence pair distribution. Discussing scaling exponents, we a priori do not assume them to be universal, i.e., including different physical systems, dimensions, quantities and initial conditions. In the revised manuscript we have been more careful in distinguishing the mere observation of scaling with the involved scaling exponents from possible universal aspects of it.
7) “[M]y 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 meanfield). Even if this discussion would be very simple with respect to the rich phenomenology of the 2D Bose gas, I think it would help the clarity of the paper.”
a) We gladly included in the introduction a brief discussion of equilibrium works present in the literature and moved the discussion of scaling of twopoint functions in the system from the Appendix to a new section directly after the introduction. There we also clarify nonthermal fixed point characteristics, providing a more pedagogical introduction to selfsimilar scaling behavior in the vicinity of nonthermal fixed points and strengthening the motivation for persistent homology observables.
List of changes
 Updated title, focussing more on selfsimilarity instead of universality
 Updated abstract slightly
 Changed beginning and later paragraphs of the introduction, such that it becomes more accessible to nonmathematicians, in addition focussing as well more on selfsimilarity instead of universality. Information on available papers studying equilibrium phases with persistent homology methods has been added.
 Introduced a novel Sec. 2, which discusses nonthermal fixed points in the wellestablished occupation number spectrum. Therein, the chosen initial conditions are introduced more carefully than before.
 Revised manuscript, Sec. 3.1, includes less mathematical terms not introduced properly before.
 Tiny changes throughout the work in order to discriminate more clearly between the observation of selfsimilarity and possible universal aspects of it.
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 2 on 2021826 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202102_00007v2, delivered 20210826, doi: 10.21468/SciPost.Report.3454
Report
The authors took care of many of the comments and points raised by the referees. I have to say that, despite this effort, I continue to think that the method could have better explained in relation to critical phenomena (and/or to systems "simpler" than the ones studied). Put in another way, I feel that many other interesting things may follow from the method presented in this paper. However, the method presented is interesting and the clarity considerably improved, so I am favour of the publication of the paper.
Anonymous Report 1 on 2021622 (Invited Report)
Report
The authors have taken into account large part of the comments in the previous round of Referral and they have done an effort in improving the quality of their presentation, which was one of the main critical aspects. I am glad to accept their work for publication in SciPost.