Análisis de la Dinámica Cuántica de Muchos Cuerpos mediante Homología Persistente

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Finding self-similar behavior in quantum many-body dynamics via persistent homology Daniel Spitz and Jürgen Berges Institut für theoretische Physik, Ruprecht-Karls-Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany Markus K.

Oberthaler Kirchhoff-Institut für Physik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany Anna Wienhard Mathematisches Institut, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany and HITS gGmbH, Heidelberg Institute for Theoretical Studies, Schloss-Wolfsbrunnenweg 35, 69118 Heidelberg, Germany 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 classical-statistical simulation of a two-dimensional Bose gas far from equilibrium.

We discover a continuous spectrum of dynamical scaling exponents, which provides a refined classification of nonequilibrium self-similar 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 many-body dynamics in terms of robust topological structures beyond standard field theoretic techniques. I.

INTRODUCTION Over the past two decades the mathematical field of topological data analysis (TDA) has gained considerable attention, accompanied by far-reaching theoretical and computational developments [1, 2]. Prominently, with the notion of persistent homology the TDA toolbox of- fers a versatile and numerically fairly simply applicable tool to study topological features contained in data, such as connected components, loops or voids [3–5]. In par- ticular, persistent homology associates length scales to such topological features, allowing for a numerical dis- crimination of dominant features and possible noise in data.

To accomplish this, simplicial complexes such as so-called Čech complexes, Vietoris-Rips complexes or al- pha shapes [6, 7] are employed. Besides the mathemat- ical investigations on persistent homology, very fruitful applications to physical systems include studies in as- trophysics and cosmology [8–11], physical chemistry [12], amorphous materials [13], quantum algorithms [14–18] and the theory of quantum phase space [19]. In partic- ular, persistent homology has been successfully applied to the detection of equilibrium phase transitions in sta- tistical mechanics [20] as well as to the identification of phases in lattice spin models [21].

In this work, we propose persistent homology observ- ables for the analysis of the dynamics of quantum many- body systems. As a prototype application, we consider a Bose gas far from equilibrium. While there are many dif- ferent ways of driving a Bose gas away from equilibrium, Electronic address: spitz@thphys.uni-heidelberg.de it has recently been demonstrated experimentally that the subsequent relaxation dynamics can exhibit universal properties that are insensitive to the details of the initial conditions and system parameters [22–24].

Theoretical results based on field correlation functions indicate that vastly different systems far from equilibrium may share very similar universal scaling properties, ranging from post-inflationary dynamics in the early universe [25, 26], and ultra-relativistic collision experiments with heavy nuclei [27–29], to ultra-cold quantum gases in the labora- tory [30, 31]. In particular, quantum as well as classical statistical field theories appear to belong to the same non- thermal universality class [32]. These similarities have to be tested against refined analysis and classification schemes.

We will exploit the multi-scale topological in- formation encoded in a family of alpha complexes and in associated persistent homology groups in order to analyze self-similar scaling dynamics in position space variables. More precisely, serving as a numerical testbed, we apply TDA techniques to the dynamics of the single- component nonrelativistic Bose gas in two spatial dimen- sions, described by the time-dependent Gross-Pitaevskii equation with quantum initial conditions. The latter ex- hibits a rich phenomenology far from equilibrium, in- cluding various nonthermal fixed points associated to regimes of weak and strong wave turbulence [33–35].

Fo- cussing on the nonperturbative strong wave turbulence regime, a vertex-resummed two particle-irreducible ex- pansion scheme has been successfully employed to ob- tain analytical predictions for relevant scaling exponents [32, 36]. The existence of corresponding nonthermal fixed points has been confirmed by means of numerical lattice simulations [37]. In addition, the infrared nonthermal arXiv:2001.02616v3 [cond-mat.quant-gas] 22 Sep 2021 2 fixed point can be dominated by vorticial excitations in- teracting anomalously with each other via 3-vortex in- teractions [37, 38], that is, altering the universal scaling behavior.

It has been conjectured that this anomalous vortex kinetics is associated to the formation of Onsager vortex clusters out of equilibrium via evaporative heat- ing [39, 40]. Recently, experimental evidence for scale- invariant dynamics and Onsager's model has been re- ported [41, 42]. Guided by numerical results for the two-dimensional Bose gas, we reveal that at late times far from equi- librium persistent homology observables can show self- similar scaling characteristic to a nonthermal fixed point.

We discover a continuous spectrum of dynamical scal- ing exponents, depending on a filtration parameter to construct point clouds, which provides a refined classi- fication of nonequilibrium self-similar phenomena. The existence of such a scaling exponent spectrum seems to indicate scaling species mixing, in our case between the strong wave turbulence and the anomalous vortex kinet- ics nonthermal fixed points present in the infrared of the particular Bose gas. The analysis is supplemented by a thorough investigation of accompanying subtleties of the chosen persistent homology approach such as amplitude redistribution-induced exponent shifts.

On the theoretical side, we define persistent homology observables. We introduce the notion of a persistence pair distribution and its statistical asymptotics in order to infer self-similar behavior of the latter. We reveal that the appearing scaling exponents probe the geometry at hand, as indicated by a packing relation heuristically de- rived in this study. This publication is structured as follows. We first de- scribe the lattice simulations and discuss self-similar scal- ing for the occupation number spectrum in Sec. II. With the Bose gas simulations at hand, we introduce and study point clouds and persistent homology groups in Sec. III.

Rediscovering self-similarity, this exploration culminates in the existence of a scaling exponent spectrum. In Sec. IV we carry out the construction of persistent homology observables in the classical-statistical framework, intro- duce the asymptotic persistence pair distribution and re- lated geometric quantities and investigate a correspond- ing self-similar scaling ansatz. We discuss amplitude redistribution-induced exponent shifts, persistences and Betti number distributions in Sec. V. Finally, in Sec. VI we summarize, draw conclusions and issue an outlook. II.

SELF-SIMILARITY IN OCCUPATION NUMBERS Laying the foundations for the introduction of persis- tent homology observables, we first discuss self-similar scaling in the two-dimensional Bose gas for the well- established occupation number spectrum. The two- dimensional Bose gas is among the simplest systems to give rise to different nonthermal fixed points and to allow for the fast and reasonable[43] computation of persistent homology observables. We start this section by introduc- ing the lattice simulations. A.

Simulation prerequisites The nonrelativistic Bose gas can be described by com- plex scalar fields ψ(t, x) depending on time and space, in numerical simulations restricted to a spatial lattice and time-evolved in discrete time-steps. We focus on the overoccupied regime, in which the classical-statistical approximation is suitable [32]. Accordingly, at initial time t = 0 a number k of classical field configura- tions is sampled from a Gaussian ensemble, computing their individual subsequent dynamics according to the time-dependent Gross-Pitaevskii equation as described in Appendix E.

In the classical-statistical approxima- tion expectation values of an observable are computed as ensemble-averages of the observable evaluated for in- dividual field configurations. Given a field configuration ψ(t, x), we define the sta- tistical two-point correlation function F(t, t0 , x x0 ) = 1 2 hψ(t, x)ψ (t0 , x0 ) + ψ(t0 , x0 )ψ (t, x)i, (1) h·i indicating evaluating the expectation value in the classical-statistical ensemble. Subsequently, with mo- mentum denoted by p we define the occupation number spectrum f(t, p) via f(t, p) + (2π)3 δ(3) (p) "ψ0"2 (t) Z d3 x e ipx F(t, t, x).

(2) Due to spatial isotropy of expectation values in the sys- tem, the distribution function only depends on the mod- ulus of momenta, f(t, p) f(t, "p"). The term "ψ0"2 (t) represents a condensate occurring in the system. We choose the initial occupation number spectrum to describe overoccupation up to a characteristic momen- tum scale Q. To this end, initial field configurations are defined as f(0, p) = f0Θ(Q "p"), (3) with f0 = 50/(2mgQ) in the simulations.

Unlike a sys- tem in thermal equilibrium, where the typical occupancy is of order unity at a characteristic temperature scale T, here we consider a nonequilibrium system where the oc- cupancy at a given characteristic scale Q is much higher than unity. Any dimensionful physical quantity will be given in units of Q. We set the mass m/Q = 8 and coupling Qg = 0.0625 throughout this work. Outside the box, no 'quantum-half' is taken into account and no initial condensate is specified. Spatial coordinates are re- stricted to a square lattice, Λ, consisting of a regular grid of N2 points within a volume L2 with periodic boundary conditions.

Throughout this work, the lattice spacing reads Qa = 0.0625, the number of lattice sites N = 1536, such that Λ = {(an1, an2) " n1, n2 {0, . . . , N 1}}. (4) 3 If not stated differently, we average over k = 72 classical- statistical realizations to compute classical-statistical ex- pectation values. For further details on the numerical simulations we refer to Appendix E. B. Self-similarity in the occupation number spectrum After a relatively short time interval with a quick re- distribution of the initial mode occupancies, the dynam- ics slows down and begins to indicate the vicinity of a nonthermal fixed point by means of self-similarity.

Self- similar scaling of the occupation number spectrum f(t, p) is described by a scaling ansatz including two scaling ex- ponents, α and β, f(t, p) = (t/t0 )α f(t0 , (t/t0 )β p). (5) In the infrared regime, a thorough numerical analysis as described in Ref. [32] yields the following scaling expo- nents, β = 0.189 ± 0.011, α = 0.395 ± 0.025, (6) choosing reference time Qt0 = 1250, fitting momenta be- tween p/Q = 0.07 and p/Q = 0.7 and times between Qt = 1875 and Qt = 37500. Thus, α/β = 2.09 ± 0.18. In Fig. 1 occupation number spectra are displayed in the in- frared regime. By means of the residuals the correctness of the extracted scaling exponents can be easily verified.

The results confirm the findings for box initial condi- tions in Ref. [38], in which the infrared dynamics of a two-dimensional relativistic scalar field theory has been mapped to that of nonrelativistic complex scalar fields. The extracted scaling exponent β is in very good agree- ment with the prediction for the anomalous vortex ki- netics nonthermal fixed point in a nonrelativistic single- component Bose gas, attributed to the specific dynam- ics of vortex defects and related vortex interactions [37]. Additionally, α/β 2 indicates the transport of particle numbers to lower momenta [32]. III.

PERSISTENT HOMOLOGY IN A BOSE GAS Given the lattice simulations of the nonrelativistic Bose gas described in the previous section, we intro- duce a simple approach to construct point clouds from field configurations, namely as sublevel sets of field am- plitudes. A rather intuitive sketch of the construction of alpha complexes and persistent homology groups from such point clouds is provided. In corresponding far-from- equilibrium simulations we discover growing geometric structures and self-similar scaling at large length scales. In particular, the existence of a scaling exponent spec- trum is revealed.

By means of the mixing of scaling dy- namics species we offer a possible route to explain this finding. Figure 1. Occupation number distributions in the infrared. In black: The initial unrescaled occupation number distribution. Figure 2. Amplitudes (left) and phases (right) of an example field configuration at time Qt = 3750. A. Phenomenology of point clouds Given a classical-statistical field realization ψ(t, x), an immense freedom of choice exists in constructing point clouds, which are, generally speaking, finite sets of points in an arbitrary Euclidean space. We define a filtration function f to be a map from C to R used to generate point clouds as subsets of the lattice Λ.

We may construct point clouds as sublevel sets of f(ψ(t, ·)), that is, at time 4 Figure 3. Amplitudes of an example field configuration and corresponding point clouds. First column from the left: Spatially- resolved field amplitudes, "ψ(t, x, y)". Second to fourth column: Point clouds Xν (t) for the different ν̄-values indicated. First row: Qt = 3750. Second row: Qt = 11250. Qt define them as {x Λ " f(ψ(t, x)) ( , ν]} for a filtration parameter ν. In this work, point clouds are generated as sublevel sets of the field amplitude, thus defining Xν(t) := {x Λ " "ψ(t, x)" ν}.

(7) By means of this definition, the ensemble of classical- statistical field realizations translates for each time Qt into an ensemble of point clouds. Numerically, we specify the filtration parameter ν by means of the dimensionless variant ν̄, ν̄ := ν/h"ψ(t = 0)"ivol, (8) with the volume-averaged initial field amplitude h"ψ(t = 0)"ivol = 1 N2 X x Λ "ψ(t = 0, x)". (9) We want to emphasize that in experiments with cold atoms optical density images as given by the square of the amplitudes displayed in Fig. 2 and used in the filtration protocol, Eq. (7), form a typical observational quantity and can be easily accessed via absorption images.

Vary- ing the filtration parameter ν̄ amounts to measurements up to the square root of corresponding condensate den- sities, highlighting the physical significance of the em- ployed point cloud construction via Eq. (7). Simulating on a spatial square lattice with constant lattice spacing, we want to stress that to obtain finite point clouds by means of Eq. (7) the finiteness of the lattice is crucial. Else, Xν(t) might consist of infinitely many points. The subsequent construction of persistent homology observables, described in detail in Sec. III B, is robust against perturbations of the lattice points[44].

This renders the microscopic form of the lattice irrele- vant for later numerical persistent homology results. The constant lattice spacing and finite lattice volume solely amount to a smallest and a largest length scale amenable to the investigated real-time dynamics. In Fig. 2 amplitudes and phases of a single classical- statistical field realization are displayed. One may first note from the amplitudes on the left that in position space the system comprises two major components: fluc- tuations in the bulk around a mean amplitude value larger than zero and distinct minima with minimum val- ues near to zero.

While phases differ locally only slightly in regions where minima are absent, around each mini- mum phase windings with shifts of ±2π occur. Thus, the minima can be identified with elementary vortex nuclei. In Fig. 3 at two different times we show spatially- resolved amplitudes and a variety of point clouds com- puted from a single classical-statistical field realization. In point clouds Xν(t) as defined by Eq. (7), at both times visualized we find clear manifestations of the aforemen- tioned two components appearing in amplitudes. Having approximately zero amplitude at the center of their nu- clei, vortices dominate the point clouds Xν(t) for small filtration parameters such as ν̄ = 0.2.

In the limit of ν̄ 0 point clouds actually comprise mostly vortex po- sitions themselves, although the presence of points origi- nating from bulk density fluctuations cannot be excluded. Described by point vortex models, for this reason the low- ν̄ limit can be associated to the incompressible limit of the theory. Increasing ν̄, in point clouds points first ac- cumulate around vortex nuclei but at moderately high values such as ν̄ = 0.6 also occur in the bulk. The higher ν̄ gets, the denser point clouds become, reducing the av- erage distance between points. Hence, studying point clouds at different ν̄-values effectively probes the system on different length scales.

Comparing the two times displayed, we note that the number of vortices decreases with time, or, equivalently, 5 the average inter-vortex distance increases. In Fig. 3 point clouds at ν̄ = 0.2 reflect this behavior, becoming sparser in the course of time. Similarly, at higher values of ν̄ the density of points in point clouds decreases in regions where vortices are absent. All this indicates that in the temporal regime of the displayed times geomet- ric structures in point clouds continuously grow at large length scales. Yet, one may notice that for ν̄ = 0.6 and ν̄ = 0.7 the number of points in the bulk decreases faster compared to the decline in vortex numbers.

This provides a first hint at the presence of different components, whose dynamics differ in terms of "speed". B. An introduction to persistent homology To obtain a robust quantitative means of the topolog- ical structure present in a point cloud Xν(t), persistent homology can be employed. Aiming at an intuitive treat- ment, with a point cloud Xν(t) at hand as it appears in the Bose gas simulations we introduce relevant notions from computational topology. From given input data we first define the Delaunay complex and a notion of the size of a simplex.

The so-called Delaunay radius func- tion can then be used to construct a nested sequence of subcomplexes, called alpha complexes, whose persistent homology groups form our objects of interest and even- tually provide multi-scale information on the topological structure of the input point cloud. While we carry out constructions in two spatial dimensions here, they gener- alize easily to higher dimensions. In Appendix A we rigorously introduce relevant funda- mental algebraic topology notions and discuss the math- ematical construction of persistent homology groups. For a general introduction to algebraic topology we refer to Ref.

[45]; for a thorough introduction to computational topology the interested reader may consult Refs. [2, 5], for instance. 1. Alpha complexes Let Xν(t) be a point cloud as defined by Eq. (7). We construct persistent homology groups from a nested family of simplicial complexes. A simplicial complex S on Xν(t) comprises the set Xν(t) together with a collec- tion S of subsets of Xν(t). The defining property of a simplicial complex is that for all points x Xν(t), the vertex {x} S, and if τ σ S, then τ S, i.e. S is closed under taking subsets. The elements of S are called its simplices.

Combinatorially, this structure allows for the computation of various descriptors of its topology, in particular the homology groups of S. We deliver details in Appendix A 1. Let us construct the particular type of simplicial com- plexes employed in this work: alpha complexes. Clearly, for any three points in Xν(t) that do not lie on a sin- gle straight line, a unique circumsphere passing through Figure 4. Alpha complexes of various radii Qr of the point cloud Xν (Qt = 3750) for ν̄ = 0.6 as displayed in Fig. 3. Panel (a): Qr = 1.0. Panel (b): Qr = 3.0. Panel (c): Qr = 7.0. Panel (d): Qr = 20.0. the points exists.

Any two points can be trivially iden- tified with a zero-dimensional circumsphere. We shall assume that the points in Xν(t) are in general position. This excludes, for example, the possibility that three or more points are collinear or that four or more points lie on a single circle[46]. Then, any two or three points in Xν(t) have a unique zero- or one-dimensional circum- sphere passing through these points, respectively[47]. We call a circumsphere empty, if all points of Xν(t) lie on or outside the sphere. The Delaunay complex, Del(Xν(t)), can be defined to consist of all points in Xν(t) as well as those edges and triangles whose circumspheres are empty [48].

Speaking about terminology, a point is a zero-dimensional simplex, an edge between two points is a one-dimensional simplex and a triangle is a two-dimensional simplex. As described in Ref. [49], for point clouds in general position this pro- cedure yields that the corresponding Delaunay complex is a simplicial complex, allowing for the construction of homology groups as described intuitively below. The Delaunay radius function Rad : Del(X) [0, ) is defined to map every simplex to the smallest radius of all its empty circumspheres. Intuitively, it provides a measure for the size of a simplex. In Fig.

4d the Delau- nay complex of an example point cloud Xν(t) as it ap- pears in the Bose gas simulations is displayed for ν̄ = 0.6. Note that simplices of different Delaunay radii are visu- ally of distinct dominance, typically. Smaller simplices appear foremost around local accumulations of points, while simplices of larger radii mainly make up the large- scale structure between them. 6 Figure 5. Persistence diagram of one-dimensional homology classes for the sequence of alpha complexes partially displayed in Fig. 4, Dgm1(Xν (t)). Let Qr [0, ) be some length scale.

Capturing ap- pearing structures of particular sizes, from the Delaunay radius function we finally construct alpha complexes[50] as its sublevel sets, αr(Xν(t)) := {σ Del(Xν(t)) " Rad(σ) Qr}. (10) For all 0 r s we find αr(Xν(t)) αs(Xν(t)). To this end, we obtain what is called a filtration of the Delaunay complex Del(Xν(t)), that is, a nested sequence of alpha complexes little by little filling out all Del(Xν(t)), αr1 (X) · · · αrκ (X) = Del(X), (11) with ri rj for all i < j. Again referring to the example point cloud Xν(t), in Fig. 4 corresponding alpha complexes of different radii Qr are displayed.

Note that at a small radius such as Qr = 1.0 the alpha complex mainly reflects the local ac- cumulations of points in Xν(t). Topological structures such as holes are of tiny size and each connected com- ponent loosely corresponds to a local accumulation of points. Besides seemingly random connected structures, at intermediate radii comparably large-scale holes appear in the alpha complexes, such as visible in the Qr = 7.0 alpha complex displayed in Fig. 4c. At even larger radii, the full Delaunay complex is recovered, in accordance with Eq. (11).

Leading to the notion of persistent ho- mology, it is a crucial insight that independent connected components disappear at a certain radius, merging with other components, and that holes only appear in alpha complexes of a certain radius and disappear again at a higher radius. 2. Persistent homology and the persistence diagram This intuitive picture can be turned into a mathemat- ical concept: persistent homology. In Appendix A 2, we provide a more rigorous introduction to it, while here we focus on capturing its intuitive essence. Alpha complexes of zero radius only consist of the ver- tices, that is, all points contained in the point cloud Xν(t).

Certainly, the number of connected components in the alpha complex of zero radius equals the cardinal- ity of Xν(t). Increasing the radius, at a certain value a first edge between two vertices appears in the alpha complex. A previously independent connected compo- nent dies. We call the minimum radius at which it is not present anymore in the corresponding alpha complex its death radius. The radius rising further, more and more connected components die, merging into a larger and larger complex. From a certain radius onwards, only one connected component is present in the corresponding alpha complexes. In Fig.

4 the process of connected com- ponents merging one by one into larger complexes can be observed as the sequence of alpha complexes is traversed towards larger radii. With radii increasing, in the sequence of alpha com- plexes holes begin to appear as is clearly visible in Figs. 4b and 4c. The minimum radius at which an independent hole first appears in the sequence of alpha complexes is called its birth radius. We say that it is born at its birth radius. Successively, a given hole is filled out with tri- angles in alpha complexes of rising radii, until from its death radius onwards the hole vanishes, being fully filled.

In fact, in simplicial homology independent connected components are described by zero-dimensional homology classes and independent holes by one-dimensional ho- mology classes. If the point clouds of interest lived in a higher-dimensional Euclidean space, one could continue analogously to describe the birth and death of higher- dimensional homology classes. This includes, for in- stance, independent enclosed voids represented by two- dimensional homology classes. Homology classes of di- mension `, appearing and disappearing again as the se- quence of alpha complexes is traversed, are collected in groups, the `-th persistent homology groups, cf. Appendix A 2.

Summarizing the structure of `-th persistent homology groups, the `-th persistence diagram Dgm`(Xν(t)), is de- fined to contain all birth radius-death radius pairs (rb, rd) of `-dimensional homology classes appearing in the se- quence of alpha complexes of Xν(t), taking respective multiplicities into account for coinciding such pairs[51]. In Fig. 5 the persistence diagram of one-dimensional ho- mology classes is displayed for the sequence of alpha com- plexes partially shown in Fig. 4. Certainly, in a persis- tence diagram all points lie above the diagonal rb = rd, since the death of any homology class happens at a higher radius than its birth.

We find that in the bottom-left of the diagram an accumulation of pairs is present, corre- sponding to comparably small one-dimensional homology classes (holes). The partly vertical alignment of points can be attributed to the homogeneity of the square lat- tice, on which Xν(t) resides. In addition, we find a sec- ond accumulation of pairs in the top-right of the dia- gram, corresponding to larger-size one-dimensional ho- 7 mology classes in corresponding alpha complexes. On these length scales birth and death radii are approxi- mately independent from the microscopic lattice geome- try. 3.

Statistical measures: birth and death radii distributions To obtain expectation values in the classical-statistical framework, ensemble-averages of quantities describing persistence diagrams of individual classical-statistical re- alizations are required. Persistence diagrams themselves are difficult objects to study statistically. Without mod- ifications not even a statistical average can be defined unambiguously. Nevertheless, there exist multifarious quantities suitable for a statistical treatment [52]. We introduce two of these here, postponing the general de- scription to Sec. IV A. We explicitly construct classical- statistical ensemble-averages.

To this end, let X (i) ν (t), i N, be an ensemble of point clouds, all constructed from individual field realizations according to Eq. (7). Denote by D (i) ` (t) := Dgm`(X (i) ν (t)) the `-th persistence diagram of the i-th such point cloud. Let σ > 0 be a constant. We define the expectation values of the `- th distribution of birth radii and the `-th distribution of death radii as hB`i(t, rb) = lim k 1 k k X i=1 X (r0 b,r0 d) D (i) ` (t) 1 2πσ2 exp (rb r0 b)2 2σ2 , (12a) hD`i(t, rd) = lim k 1 k k X i=1 X (r0 b,r0 d) D (i) ` (t) 1 2πσ2 exp (rd r0 d)2 2σ2 , (12b) respectively. Note that these distributions are statisti- cally well-behaved, such that averages and the denoted limits exist [53].

The parameter σ is chosen sufficiently small, such that numerical outcomes are independent from its particular value. C. Growing geometric structures in persistent homology Using a computational topology pipeline as described in Appendix B, we can numerically investigate birth and death radii distributions for different filtration parame- ters ν̄ in the aforementioned Bose gas simulations. For large length scales, in Fig. 6 death radii distributions of zero-dimensional homology classes and birth radii dis- tributions of one-dimensional homology classes are dis- played at times between Qt = 3750 and Qt = 35625.

Zero-dimensional persistent homology classes are always born at radius Qrb = 0, turning the distribution of birth radii of zero-dimensional homology classes trivial. The occurring oscillations in distributions are due to statisti- cal uncertainties, being computed from only a finite num- ber of classical-statistical samples. We first discuss unrescaled variants of the displayed distributions. It is important to note that in any of the distributions the maximum number of counts in birth and death radii distributions decreases with time. Simultane- ously, the steep decline at largest radii in birth and death distributions constantly shifts to higher radii.

Clearly, these are manifestations of geometric structures in the system growing at large length scales as conjectured in Sec. III A from the point clouds themselves. Beyond this, the approximately constant form of the distributions al- ready provides a first hint at self-similar dynamics. In first death radii distributions a clear peak is visible, in particular for ν̄ = 0.2 as displayed in Fig. 6, panel (a3). Point clouds for small ν̄-values being dominated by accumulations of points around vortex nuclei, we ex- pect this distinguished length scale to provide a measure for the average inter-vortex distance.

At higher ν̄-values such as ν̄ = 0.6 the peak is blurred by means of bulk points entering corresponding point clouds. D. Unveiling a spectrum of scaling exponents Motivated by the approximately constant form of the distributions displayed in Fig. 6, we examine whether they can be consistently described by a self-similar scal- ing ansatz. We say that birth and death radii distribu- tions scale self-similarly, if exponents η1, η0 1 and η2 exist, such that for all times t, t0 , hB`i(t, rb) = (t/t0 )η0 1 η2 hB`i(t0 , (t/t0 ) η1 rb), (13a) hD`i(t, rd) = (t/t0 )η1 η2 hD`i(t0 , (t/t0 ) η0 1 rd). (13b) In Sec.

IV C we deduce this particular form of scaling be- havior from a scaling ansatz to a more general quantity that describes persistent homology groups, the asymp- totic persistence pair distribution. Notice that in this scaling ansatz a possible dependence on the dimension ` of homology classes is neglected, supported by numerics. Using the numerical protocol described in Appendix G, scaling exponents are extracted from birth and death 8 Figure 6. Birth and death radii distributions in the infrared. Columns 1 and 2: Death radii of of zero-dimensional homology classes. Columns 3 and 4: Birth radii of one-dimensional homology classes.

Individual columns show data for the indicated filtration parameter, ν̄. Row 1: unrescaled distributions. Row 2: rescaled distributions. The employed time-dependent scaling exponents are displayed in Fig. 8. radii distributions of one-dimensional homology classes. Given a time Qtmin, birth and death radii distributions at times Qtmin, Qtmin + 625 and Qtmin + 1250 are fit- ted simultaneously against distributions at reference time Qt0 = 3750. A measure for the quality of a self-similar description of the investigated distributions is provided by means of residuals.

For instance, for the distribution of birth radii residuals at time Qt are computed as Res.(hB`i)(t, rb) := (t/t0 )η0 1 η2 hB`i(t0 , (t/t0 ) η1 rb) hB`i(t, rb) 1. (14) Indeed, distributions can be consistently rescaled by means of the scaling ansatz described in Eqs. (13a) and (13b). This can be deduced from Fig. 6 with resid- uals of rescaled distributions scattering approximately evenly around zero. Note that distributions of both zero- and one-dimensional homology classes can be con- sistently rescaled with the same triple of exponents, vali- dating that in the scaling ansatz we neglected a possible `-dependence.

However, filtration parameter- and time- dependent scaling exponents are necessary for a success- ful rescaling. In Fig. 7 we show the scaling exponents for a single minimum fitting time Qtmin, highlighting the size of er- ror bars. Errors origin from a finite number of classical- statistical samples taken into account and from fitting uncertainties. For values of ν̄ . 0.4 the displayed expo- nent values approximately lie around 0.2. A rise in values takes place for ν̄ & 0.5, up to a maximum value of ap- proximately 0.8. Thus, we make the crucial observation that a continuous spectrum of scaling exponents exists, depending on the filtration parameter ν̄.

Within error bars η1 equals η0 1 at all ν̄-values inves- Figure 7. Persistent homology scaling exponents at Qtmin = 18750. tigated here. This provides numerical evidence for that birth and death radii show the same dynamics at large length scales. In addition, for all ν̄-values analyzed η2/η1 = 4 within the indicated error bars. This relation results from the bounded packing of homology classes of a given size into the constant lattice volume, as shown in Sec. IV C 2. Comprehensively, results are summarized in Fig. 8, in which exponents are displayed in the full (ν̄, Qtmin)- plane.

The gradual shift of the peak in scaling exponents to higher ν̄-values with increasing fitting time Qtmin is a result of the redistribution of amplitude values with time, discussed in Sec. V A. The scattering of exponent values at larger ν̄-values is due to statistical uncertainties. 9 Figure 8. Persistent homology scaling exponents for different filtration parameters ν̄ and minimum fitting times Qtmin. E. Scaling species and exponents mixing conjecture An observation such as the existence of a whole spec- trum of scaling exponents at large length scales requires an explanation.

We conjecture that its appearance is linked to different dynamical scaling species occurring in the infrared of the two-dimensional Bose gas. First, note that momenta in the infrared regime corre- spond to large length scales. Hence, if infrared dynamics is visible in quantities describing the persistent homology of alpha complexes, it will show at correspondingly large birth and death radii. Vice versa, if ultraviolet physics is visible in persistent homology, it will show up at compa- rably small birth and death radii. To this end, we identify the regime of large birth and death radii in their distribu- tions with the infrared regime of the system.

This offers the possibility of linking aforementioned results to known momentum space dynamics of physical quantities. In addition, for positive scaling exponents η1 = η0 1 and η2 the scaling ansatz described by Eqs. (13a) and (13b) corresponds to a blow-up of length scales as a power-law with exponent η1, as we detail in Sec. IV C. Hence, a comparison of the exponent η1 with scaling exponents appearing in power-laws of further physical length scales is reasonable. We restrict the following discussion to η1. For ν̄ .

0.4, the exponent η1 meets the value of 1/5 associated to the anomalous vortex kinetics nonthermal fixed point [37, 42] and confirmed by the self-similar dynamics of occupation number spectra in the given simulations, Eq. (6). Point clouds, alpha complexes as well as birth and death radii distributions reflect the occurring vortex dynamics for small ν̄, correspondingly. This is in accordance with the observation made in Sec. III A that for ν̄ . 0.4 point clouds mainly comprise accumulations of points around vortex nuclei. The exponent η1 increases with ν̄ up to maximum val- ues of between 0.7 and 0.9 depending on Qtmin, cf. Fig. 8 — a value which is significantly different from 1/5.

We take a small detour to provide a physical interpretation for this phenomenon. Collectively, the vortices show anomalous kinetics and dominate point clouds at low ν̄-values: η1(ν̄ = 0.05) 0.2. It is well-known, however, that the two-dimensional nonrelativistic Bose gas not only exhibits the anomalous vortex kinetics nonthermal fixed point with β = 0.2, but also incorporates strong wave turbulence characterized by β = 0.5 [32, 37, 42, 54]. If the vortices were absent or coupled strongly to sound excitations in the bulk, only self-similar scaling with β = 0.5 would be visible, as ar- gued for in Ref. [37].

Motivated by this, we infer that in the configurations investigated it is sound excitations in the bulk that reflect strong wave turbulence. Corre- spondingly, if bulk points enter point clouds, then birth and death radii distributions might show scaling behavior deviating from η1 = 0.2. As can be seen in Figs. 3, 7 and 8 this is the case for growing ν̄-values and explains the increase of η1. With this admittedly loose association of bulk points to strong wave turbulence and vortex nuclei points to anomalous vortex kinetics in mind, we refer to the underlying phenomenon as scaling species mixing in point clouds. Yet, the maximum value of η1(ν) exceeds 0.5 signifi- cantly for all Qtmin.

A heuristic geometric explanation proceeds as follows. Restrict to the dynamics of a single classical-statistical field configuration and corresponding point clouds Xν(t). Let Yν(t) Xν(t) be associated to anomalous vortex kinetics and Zν(t) Xν(t) asso- ciated to strong wave turbulence in the bulk, such that Xν(t) = Yν(t) Zν(t). The alpha complexes of Xν(t), αr(Xν(t)), however, do not simply decay into αr(Yν(t)) and αr(Zν(t)). Instead, depending on the precise ar- rangements of points in Yν(t) and Zν(t), there may be a lot of simplices contained in αr(Xν(t)) which incorporate points of both Yν(t) and Zν(t).

In addition, simplices that only consist of points in Yν(t) or Zν(t) can be very different from the ones in αr(Yν(t)) and αr(Zν(t)). The construction of alpha complexes from Yν(t) and Zν(t) is a highly nonlinear process. Birth and death radii distri- butions can reflect this behavior. IV. PERSISTENT HOMOLOGY OBSERVABLES AND SELF-SIMILARITY In this section we embed alpha complexes and per- sistent homology descriptors into the classical-statistical regime of quantum field theory (QFT). By means of func- tional summaries of persistence diagrams, this leads to the definition of persistent homology observables.

In quite a few examples of these the same integral kernel appears, which we call the asymptotic persistence pair 10 distribution. This paves the way to a self-similar scaling approach for the asymptotic persistence pair distribution, whose outgrowths for birth and death radii distributions are given by Eqs. (13a) and (13b). In Sec. III D this particular scaling behavior has been shown to describe simulation outcomes well. A. Persistent homology observables via functional summaries Naturally, studying persistent homology in QFT re- quires a statistical treatment. Persistence diagrams themselves, however, do not admit a clear notion of aver- ages [52].

Instead, we propose to focus on so-called func- tional summaries, providing general statistically well- behaved descriptors of persistence diagrams. In Sec. IV B we reveal that the investigated birth and death radii distributions given by Eqs. (12a) and (12b) are corre- sponding examples. Let D be the space of persistence diagrams, that is, the space of finite multisets of points within {(rb, rd) [0, )2 " rd rb}. Let F be a collection of functions, f : R for all f F, being a compact space. Following Ref. [53], a functional summary is in full gen- erality any map from the space of persistence diagrams to a collection of functions, F : D F.

Upon the classical-statistical approximation, expec- tation values of quantum observables are computed as ensemble-averages of classical field configurations, which are time-evolved via the corresponding classical equa- tion of motion starting from fluctuating initial conditions. The range of validity of this approximation is typically restricted to high occupation numbers [32]. We propose to proceed analogously for functional summaries of per- sistence diagrams. To this end, any such summary F may be evaluated on the level of individual field config- urations and its expectation value hFi computed as the ensemble-average.

We assume that the range of validity of this approach coincides with the well-known classical- statistical regime. Certainly, for any functional summary F this proposal requires the existence of a corresponding linear operator F, such that in the classical-statistical regime for any s , tr(ρ(t) F)(s) = hFi(t, s), (15) ρ(t) being the time-dependent density operator of inter- est, the trace taken over the corresponding quantum the- ory Hilbert space and the right-hand side being computed via the aforementioned evaluation scheme. However, the existence of such an operator F is a priori not clear and will be discussed in a future work.

We need to assure that in the limit of averaging in- finitely many individual functional summaries of field configurations the statistical mean of the functional sum- mary is recovered. This is guaranteed for by a mathemat- ical statement on the pointwise convergence of so-called equicontinuous and uniformly bounded functional sum- maries, the details of which can be found in Proposition 1 of Ref. [53]. For the sake of this statement we restrict our proposal to functional summaries of persistence dia- grams with these two fairly general conditions. By means of the described classical-statistical evaluation scheme we refer to such functional summaries as persistent homology observables.

We want to stress that this proposal is neither re- stricted to the computation of persistent homology from equal-time alpha complexes, that is, alpha complexes computed from point clouds constructed at individual instances of time as done in this work, nor to alpha com- plexes themselves. B. The asymptotic persistence pair distribution and geometric quantities Let F : D F be a functional summary in the above sense. We say that F is additive, if F(D + E) = F(D) + F(E) for any two persistence diagrams D, E D. Here, D +E denotes the sum of multisets, that is, the union of D and E with multiplicities of elements in both D and E added.

Let D(t) D be a persistence diagram computed at time t as specified in Sec. III B 2 and F an additive func- tional summary. We then find for all s , F(D(t))(s) = X (rb,rd) D(t) F({(rb, rd)})(s) = Z 0 dr0 b Z 0 dr0 d F({(r0 b, r0 d)})(s) P(t, r0 b, r0 d), (16) with the persistence pair distribution P(t, r0 b, r0 d) := X (rb,rd) D(t) δ(r0 b rb) δ(r0 d rd), (17) δ denoting the Dirac delta function. Let (D (i) ` (t))i N D be a classical-statistical ensemble of persistence diagrams describing `-dimensional persis- tent homology classes at time t.

We denote the persis- tence pair distribution of D (i) ` (t) by P (i) ` (t) and define the asymptotic persistence pair distribution, hP`i, at any time t implicitly, requiring that for any equicontinuous and uniformly bounded functional summary F as in the above proposal, Z 0 dr0 b Z 0 dr0 d F({(r0 b, r0 d)})(s) hP`i(t, r0 b, r0 d) := lim k 1 k k X i=1 Z 0 dr0 b Z 0 dr0 d F({(r0 b, r0 d)})(s) P (i) ` (t, r0 b, r0 d), (18) 11 for arbitrary s . Functional summaries of relevance in this work include the distribution of birth and death radii that have been defined in Eqs. (12a) and (12b), respectively.

With an obstacle to be described below, both can be computed as marginal distributions of hP`i, hB`i(t, rb) = Z 0 drd hP`i(t, rb, rd), (19a) hD`i(t, rd) = Z 0 drb hP`i(t, rb, rd). (19b) In addition, we define the persistence distribution, that is, the distribution of rd rb, hP`i(t, r) = Z 0 drd hP`i(t, rd r, rd). (20) Natural quantities to study are the `-th Betti numbers hβ`i(t, r). Intuitively, the zeroth Betti number hβ0i(t, r) specifies the number of connected components minus one[55] present in the alpha complex of radius Qr and the first Betti number hβ1i(t, r) specifies the correspond- ing number of holes.

Being zero in the present work, higher Betti numbers count how many nontrivial higher- dimensional homology classes are present in correspond- ing complexes. Betti numbers can be computed from the asymptotic persistence pair distribution via hβ`i(t, r) = Z r 0 drb Z r drd hP`i(t, rb, rd). (21) A mathematical obstacle appears with regard to defi- nitions such as Eqs. (19a) and (19b). A priori, the sets of functions hB`i(t, rb), of hD`i(t, rd), of hP`i(t, r) and of hβ`i(t, r) are not equicontinuous. However, only func- tional summaries which have this property are persistent homology observables in the sense of Sec. IV A. For all positive σ we define ζσ(s) := 1 2πσ2 exp s2 2σ2 .

(22) By convolution with it at each time individually, sets of functions such as hB`i(t, rb) can be rendered equicontinuous[56]. In fact, this way Eqs. (12a) and (12b) for birth and death radii distributions arise from Eqs. (19a) and (19b). In everything that follows we omit the convolution procedure in notations. As mentioned previously, the convolution procedure is numerically ir- relevant. In computations, convergence of persistent ho- mology observables is numerically verified, cf. Appendix F. The average number of persistent homology classes is encoded in hP`i, too, hn`i(t) = Z 0 drb Z 0 drd hP`i(t, rb, rd). (23) Various length scales may be constructed from hP`i.

An interesting length scale is the average maximum death radius hrd,`,maxi(t), which can be computed from the asymptotic persistence pair distribution via[57] hrd,`,maxi(t) = lim p Z 0 drb Z 0 drd rp d hP`i(t, rb, rd) 1/p . (24) Analogously, the average maximum birth radius can be computed. The average number of persistent homol- ogy classes and the average maximum death (birth) ra- dius constitute persistent homology observables as con- structed above. C. Self-similar scaling approach By means of the scaling behavior visible in birth and death radii distributions, in Sec.

III D we have already begun the study of self-similarity in persistent homology observables in the vicinity of a nonthermal fixed. Here, we introduce a more general scaling ansatz for the asymp- totic persistence pair distribution. We provide a heuristic packing argument relating the appearing scaling expo- nents. In Appendix D we provide a brief discussion on the relation between the self-similar scaling ansatz described here and known notions of self-similar scaling appearing across the literature. 1. Scaling ansatz to the asymptotic persistence pair distribution Let hP`i(t, rb, rd) be a time-dependent asymptotic per- sistence pair distribution as it appears in Eq. (18).

We say that hP`i(t, rb, rd) scales self-similarly, if exponents η1, η0 1 and η2 exist, such that for all times t, t0 , hP`i t, rb, rd = (t/t0 ) η2 hP`i t0 , (t/t0 ) η1 rb, (t/t0 ) η0 1 rd . (25) Due to the time-dependence of hP`i derived geomet- ric quantities become time-dependent, too. Immediately, 12 from Eq. (25) for birth and death radii distributions the scaling behavior described by Eqs. (13a) and (13b) follows. Assuming η1 = η0 1, the persistence distribution scales as hP`i(t, r) = (t/t0 )η1 η2 hP`i(t0 , (t/t0 ) η1 r).

(26) The total number of persistence pairs scales as hn`i(t) = (t/t0 )η1+η0 1 η2 hn`i(t0 ) (27) and the average maximum death radius as hrd,`,maxi(t) = (t/t0 )η1 hrd,`,maxi(t0 ). (28) Though not explicitly given here, the average maximum birth radius scales the same way. This provides evidence for the geometric intuition of persistence length scales blowing up or shrinking in the course of time upon self- similar scaling. Provided that η1 = η0 1, the `-th Betti numbers scale as hβ`i(t, r) = (t/t0 )2η1 η2 hβ`(t0 , (t/t0 ) η1 r). (29) 2. A heuristic packing relation We assume that η1 = η0 1 and consider a general spatial dimension d here.

A fairly general heuristic argument leads to the packing relation η2 = (2 + d)η1. Intuitively, the argument encodes that only a finite number of persis- tent homology classes of a given size can be packed into a constant volume V . Let point clouds be dominated by a time-dependent length scale L(t). The d-dimensional volume V in which the point clouds reside is kept constant. Heuristically, a number hnd 1i(t) of (d 1)-dimensional persistent ho- mology classes fits into V , with this number scaling as hnd 1i(t) V L(t)d , (30) since the volume that each (d 1)-dimensional persistent homology class occupies generically may scale as L(t)d .

Inferring the scaling of length scales as described by Eq. (28), that is, L(t) tη1 , we find hnd 1i(t) t dη1 . (31) On the other hand, from Eq. (27) we obtain hnd 1i(t) t2η1 η2 . (32) Hence, η2 = (2 + d)η1, (33) which shows that persistent homology observables repre- sent in a direct fashion the geometry at hand. Of course, the assignment of occupied volumes to (d 1)-dimensional homology classes is highly heuristic, bearing in mind that a homology class is an equivalence Figure 9. Distribution of amplitude-values at different times, averages taken across classical-statistical sampling runs. class of many cycles within a simplicial complex, render- ing any such mapping ambiguous.

However, one may use elements of the proof of the Wasserstein stability theorem for persistence diagrams, carried out in Ref. [58], to de- duce Eq. (33) more rigorously from physically reasonable assumptions. In Appendix C we sketch the correspond- ing derivation, provided in detail in Ref. [59]. V. EXPONENT SHIFTS, PERSISTENCES AND BETTI NUMBER DISTRIBUTIONS In this section the due explanation of temporal shifts of the scaling exponent spectrum observed in Sec. III D is given as well as numerical outcomes for persistence distributions and Betti numbers.

The latter provide fur- ther evidence for the suitability of the self-similar scaling ansatz for the asymptotic persistence pair distribution, as given by Eq. (25). A. Amplitude redistribution-induced exponents shifts The scaling exponents displayed in Fig. 8 change in time for ν̄ & 0.5. To discuss the origins of this effect, in Fig. 9 amplitude distributions are displayed for different times between Qt = 3750 and Qt = 37500. As is clearly visible, amplitudes redistribute with growing times to- wards the peak at around "ψ(t)"/h"ψ(t = 0)"ivol 1.05. As indicated in Fig. 10, point clouds Xν(t) with ν̄ .

1.0 become sparser with time, that is, for a fixed ν̄ the car- dinality of point clouds decreases. As deduced earlier, at low ν̄-values point clouds are dominated by accumulations of points around vortex nu- clei, while for ν̄ & 0.4 points in the bulk enter point clouds. With point clouds getting sparser in the course of time it is first bulk points to disappear from point clouds. Accumulations of points around vortex nuclei remain, as can be seen from Fig. 11, in which point 13 Figure 10. The average cardinality of point clouds varying with ν̄ at different times, averages taken across classical- statistical sampling runs. Figure 11.

Example point clouds Xν (t) for different ν̄-values as indicated. Row (a): time Qt = 3750. Row (b): Qt = 7500. Row (c): Qt = 11250. clouds are displayed for different filtration parameters and times. Given the example point cloud for ν̄ = 0.5 at time Qt = 3750, we observe that it is made up from accu- mulations of points (around vertices) mixed with random points in between, while at time Qt = 11250 the point cloud consists of nothing but the accumulations. The be- havior of point clouds at ν̄ = 0.6 is similar, although the point cloud at Qt = 11250 still contains random points associated to sound excitations between accumulations.

Point clouds at ν̄ = 0.70 only get sparser but still contain many bulk points. The average maximum death radius of 1-dimensional 103 104 Qt 101 4 × 100 6 × 100 2 × 101 Qhr d,1, max i(t) ν̄ = 0.05 ν̄ = 0.20 ν̄ = 0.30 ν̄ = 0.40 ν̄ = 0.50 ν̄ = 0.60 ν̄ = 0.70 ν̄ = 0.80 Figure 12. The average maximum death radius of 1- dimensional persistent homology classes varying with time, displayed for ν̄-values as indicated. persistent homology classes, hrd,1,maxi(t), is displayed for different ν̄-values in Fig. 12.

Comparably large fluc- tuations and outliners occur, since hrd,1,maxi(t) is very sensitive to particular geometric arrangements of points in point clouds of individual classical-statistical samples. According to Eq. (28), if the system's asymptotic per- sistence pair distribution scales self-similarly in time and η1 = η0 1, then hrd,1,maxi(t) tη1 . Indeed, hrd,1,maxi(t) shows power-law behavior within individual periods of time and confirms the shifts in scaling exponents as in- dicated by the results displayed in Fig. 8, which have been deduced from birth and death radii distributions. For instance, for ν̄ = 0.6 a shift occurs between times Qt 9000 and Qt 13000.

Recently, the phenomenon of prescaling has been dis- covered, that is, the rapid establishment of a universal scaling form of distributions long before the universal values of corresponding scaling exponents are realized [60, 61]. Although we also study time-dependent scal- ing exponents of constant-form distributions, we want to stress that in our case this is not a manifestation of prescaling. Instead, it is an artifact of the sharp cutoff at the filtration parameter to generate point clouds, render- ing point clouds themselves and their persistent homol- ogy groups sensitive to amplitude redistribution effects. B. Persistence distributions In Fig.

13 persistence distributions for different fil- tration parameters are displayed. Again, fluctuations are due to statistical uncertainties. Distributions can be rescaled using time-dependent scaling exponents as given in Fig. 8. To this end, we attribute the observed behavior to the physics at large length scales. We want to emphasize that the persistence distributions at a low filtration parameter such as ν̄ = 0.2 show distinctly a power-law behavior at all times. A power-law fit of the rescaled distributions for ν̄ = 0.2 reveals a scaling with 14 Figure 13. Persistence distributions. Each column shows data for the indicated filtration parameter, ν̄.

The employed time- dependent scaling exponents are displayed in Fig. 8. Insets show corresponding residuals. persistence as (rd rb) ζ with[62] ζ = 1.468 ± 0.021. (34) The relation of the exponent ζ to known signatures of for example strong wave turbulence is to date not clear to us. C. Betti numbers as a consistency check In Sec. IV C we derived that if the asymptotic per- sistence pair distribution scales self-similarly, then Betti number distributions do so as well, described by Eq. (29). Having extracted scaling exponents from birth and death radii distributions in Sec. III D, we investigate Betti number distributions as a consistency check. In Fig.

14 Betti number distributions for both zero- and one-dimensional homology classes are displayed at ν̄ = 0.2. For all times hβ0i(t, r) is a monotonically de- creasing function, since zero-dimensional persistent ho- mology classes are born at zero radius and hβ0i(t, r) cap- tures only their death. We find a peak in unrescaled hβ1i(t, r), which, again, decreases in magnitude and shifts to higher radii as an indication of growing geometric structures. Approximately, Betti numbers display self-similar scal- ing behavior. However, residuals of the rescaled hβ0i(t) Figure 14. Betti number distributions for ν̄ = 0.2 are shown for dimensions ` as indicated.

The employed time-dependent scaling exponents are displayed in Fig. 8, setting η0 1 := η1. Insets show corresponding residuals. increase at large radii and hβ1i(t) shows comparably large fluctuations. Nonetheless, rescaled Betti number distri- butions confirm previously extracted exponents. VI. CONCLUSIONS In the present study we proposed a novel class of observables, persistent homology observables, to study the dynamical behavior of quantum fields. Serving as a prototype application, we investigated the self-similar dynamics at nonthermal fixed points in the classical- statistical approximation.

Accompanied by mathemat- ical considerations that guarantee, for example, for the convergence of averages, we studied functional summaries of persistent homology groups. We found that the notion of an asymptotic persistence pair distribution is a suit- able probability measure for a self-similar scaling ansatz. By means of simulations of the two-dimensional nonrel- ativistic Bose gas we revealed that the self-similar scal- ing dynamics characterizing nonthermal fixed points is a phenomenon that also appears in persistent homology observables.

Crucially, this way we discovered a continu- ous spectrum of scaling exponents, depending on a filtra- tion parameter that appears in the construction of point clouds. We provided a possible explanation in terms of scaling species mixing associated to two different dynam- ical processes: Strong wave turbulence and anomalous vortex kinetics. For all times investigated we found a power-law in per- 15 sistence, possibly providing a direct indication in persis- tent homology observables for the presence of a turbulent cascade.

It is currently unclear to us how to relate the de- duced persistence power-law exponent to known power- law exponents appearing in occupation number spectra, typically signaling strong wave turbulence or hinting at topological defect structures [28, 37, 38]. Describing the wrapping of finite-size homology classes into a finite volume, by means of a packing relation we argued that self-similarity in persistent homology observ- ables reflects the geometry at hand. Further exploring the relation between such geometric effects and conserved quantities associated to transport processes at nonther- mal fixed points would be interesting, but lies outside the scope of this work.

Of particular relevance in the proposed persistent ho- mology ansatz is the filtration function to generate point clouds from individual field configurations. We showed that already a simple variant such as the amplitude of the complex-valued fields can give rise to interesting observa- tions. It is a feature of our analysis that the information on phase windings around vortex nuclei is not necessary in order to show the existence of further dynamical com- ponents beyond vortices. Nonetheless, we want to stress that at this point of the analysis scheme an immense free- dom of choice exists, rendering the persistent homology ansatz highly flexible.

Also without such a filtration procedure the proposed methods can be applied to for instance point vortex mod- els. Surpassing the present work, one does in princi- ple not need a lattice to construct persistent homology groups. Even for fields with an arbitrary smooth and triangulable manifold as their domain there exist mul- tifarious ways to construct persistent homology groups [5]. Myriad of interesting further applications of persis- tent homology within QFT exist. With regard to the recent experimental…