Quantitative and Interpretable Order Parameters for Phase Transitions from Persistent Homology
QQuantitative and Interpretable Order Parametersfor Phase Transitions from Persistent Homology
Alex Cole ∗ a , Gregory J. Loges † b and Gary Shiu ‡ ba GRAPPA and ITFA, Institute of Physics, University of Amsterdam,Science Park 904, 1098 XH Amsterdam, the Netherlands b Department of Physics, University of Wisconsin-Madison,1150 University Ave, Madison, WI 53706, USA
Abstract
We apply modern methods in computational topology to the task of discovering and char-acterizing phase transitions. As illustrations, we apply our method to four two-dimensionallattice spin models: the Ising, square ice, XY, and fully-frustrated XY models. In particular,we use persistent homology, which computes the births and deaths of individual topologicalfeatures as a coarse-graining scale or sublevel threshold is increased, to summarize multi-scale and high-point correlations in a spin configuration. We employ vector representationsof this information called persistence images to formulate and perform the statistical taskof distinguishing phases. For the models we consider, a simple logistic regression on theseimages is sufficient to identify the phase transition. Interpretable order parameters are thenread from the weights of the regression. This method suffices to identify magnetization,frustration, and vortex-antivortex structure as relevant features for phase transitions in ourmodels. We also define “persistence” critical exponents and study how they are related tothose critical exponents usually considered. ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p ontents Given an unknown condensed matter system sitting in front of you, the zeroth order questionyou may ask is: what is its phase structure? With sufficient technical ability, one may varythe various coupling constants, external temperature, etc, and measure its ensuing equilibriumconfigurations. One way to understand the phase structure is to carefully search through theentire parameter space, and deduce for which parameter regimes the system looks similar (i.e.,the system remains in the same phase). In doing so, one may occassionally encounter boundariesof the parameter space where some symmetry is broken or some specific heat diverges, indicatinga new phase. Having identified these phases, a natural next question is how to distinguish themin practice, i.e. what order parameters describe the various phase transitions. These questionsare naturally phrased in the language of machine learning (ML). Namely, the question “howmany phases are there” is an exercise in unsupervised learning , while the question “how aredifferent phases distinguished” is an exercise in supervised learning . Note that this is an exericisein distinguishing statistical ensembles, and incurs some amount of uncertainty.Recently, ML techniques have been applied to these very tasks. Unsupervised methods suchas Principal Component Analysis, clustering algorithms ,and autoencoders have been used to2dentify phase transitions (see, e.g., [1–5]). Support vector machines have been shown to be auseful tool in quantifying characteristics of the Ising phase transition [6]. Supervised learningwith neural networks has proven useful in this classification task (see, e.g., [7–11]), but oftenlacks the desired level of interpretability.In this paper we use persistent homology [12, 13] (see [14–16] for reviews) as a tool for de-tecting and characterizing phase transitions. As illustrations, we apply our method to studytwo-dimensional lattice spin systems. Persistent homology is a technique from Topological DataAnalysis (TDA) that identifies the births and deaths of topological features throughout a familyof discrete complexes. This family often corresponds to the data set at various coarse-grainingscales. By now, persistent homology has been fruitfully applied in a wide variety of fields,including sensor networks [17], image processing [18], genomics [19], protein structure [20, 21],neuroscience [22], cosmology [23, 24] and string theory [25, 26], to name only a few. In thecontext of spin systems, persistent homology encodes multiscale and high-order correlations ina data set. The main takeaway from our work is that this representation of a spin system con-figuration is not only sufficient to distinguish phases in spin systems, but additionally providesinterpretable order parameters for the phase transitions. For example, we find that persistenthomology identifies such varied phenomena as magnetization, frustration, and (anti)vorticesin spin systems. Additionally, as a multiscale technique, persistent homology can capture asystem’s approach towards scale-invariance, i.e. its critical behavior. We work with persistenceimages [27], which are vectorized representations of persistent homology information. Thisframework allows us to define quantitative order parameters and quantify the uncertainty thata particular spin configuration belongs to a particular phase.Persistent topological methods have been applied to statistical mechanics in a few cases, butso far these applications have been largely qualitative in terms of statistics. Ref. [28] studiedthe relationship between phase transitions and topology changes in configuration space. Morerecently, [29] studied the relaxation dynamics of a two-dimensional Bose gas with persistenthomology, and [30] performed unsupervised learning on persistence diagrams to visualize theirphase structure. However, the use of persistent topological methods in obtaining quantitative information about statistical mechanics systems is in our view an underdeveloped subject. Thepurpose of this paper is to provide an initial foray in this important direction.In this manuscript we use persistent homology to quantitatively characterize phase transi-tions in four different lattice spin systems. We consider discrete and continuous spin modelswith and without frustration in the ground state: see Figure 1. Each example contains a distinctlesson. We begin with an obligatory analysis of the two-dimensional Ising model (Is). We areable to easily identify the model’s phase transition relying only on training data far from thecritical temperature. The magnetization as order parameter is immediately extracted from theweights of the corresponding logistic regression. Additionally, we examine the multiscale natureof the information probed by persistence images. In particular, we define persistence criticalexponents that capture the model’s approach towards criticality, finding interesting connectionsto the critical exponents usually considered. We then turn to the square-ice model (SI), forwhich there is no local order parameter due to frustrated low-energy dynamics. We again finda successful classification and are able to identify an order parameter associated with the low-3sing(Section 3.2) Square-ice(Section 3.3)XY(Section 3.4) Fully-frustrated XY(Section 3.5)
Unfrustrated Frustrated D i s c r e t e C o n t i nu o u s Figure 1:
Overview of models.temperature phase’s “scale of frustration.” Our technique quickly picks up on the scale of thisfeature. We subsequently turn to continuous spin models where a configuration consists of amap f from the lattice Λ to the unit circle S . For these models, rather than a coarse-grainingscale we study the topologies of the sublevel sets f − ( − π, ν ] ⊆ Λ. We study the XY model,where a simple logistic regression on the persistence images discovers the Kosterlitz-Thouless(KT) phase transition and corresponding vortex-antivortex structure in the low-temperaturephase. Vortex-antivortex pairs are shown to give a distinctive signature in the persistence im-ages that the logistic regression discovers and decides to use on its own. Finally, we consider thefully-frustrated XY model (FFXY), where frustration prevents the formation of (anti)vortices inthe low-temperature phase. In this case, our method identifies small scale correlations betweennext-to-nearest neighbors that reflect the system’s attempt to satisfy competing constraints.An important feature of our analysis is the simplicity of our machine learning architecture.Once the relevant spin configurations are reduced to persistence images, the phase classificationand extraction of order parameters can be achieved via a simple logistic regression. This re-flects the fact that persistent homology condenses these data sets into their most relevant (andinterpretable) features.The organization of this manuscript is as follows. In Section 2 we give a brief introductionto persistent homology, persistence images, and our computational choices for applying thesetechniques to spin models. In Section 3 we apply our methods to spin models of increasingcomplexity. We conclude in Section 4.
Note:
While this manuscript was being prepared, [31] appeared, which considers the taskof understanding the phase structure of a lattice model using persistent homology. Our workdiffers not only in the models considered (we study models with discrete as well as continuousspins) but also in the methods and goals. [31] computes pairwise distances under a particularmetric on the persistence diagrams of spin configurations and visualizes the phase diagram via adimensional reduction. In this work, we consider the statistical task of distinguishing differentphases from data (including quantitative uncertainties), which leads us to compute logisticregressions of persistence images. This approach enables us to obtain quantitative information4f the statistical mechanical systems (such as the critical temperature and critical exponents),and provide physical interpretations of the order parameters.
We are interested in developing general interpretable order parameters for phase transitions inspin systems. Some inspiration can be drawn from the hallowed two-dimensional (ferromagnetic)Ising model. In this case, spontaneous magnetization in the low-temperature phase leads tolarge, continuous domains where all spins are aligned. As the temperature is decreased towards T = 0, these domains grow, so that at sufficiently low temperature, the entire system is aligned.On the other hand, in the high-temperature phase, spins receive enough thermal energy to berandomly oriented. In the language of ML, the (non)existence and scale of magnetic domainsmanifests as a pattern in the hierarchical clustering of aligned spins. In other words, if weconsider the set of aligned spins and perform successive coarse-graining transformations, wewould be able to distinguish these two phases by the number of domains at different coarse-graining scales. Note that this is a multiscale concept that probes high-order correlationsfunctions.In fact, clustering can be viewed as the most basic topological information about a data set,giving the total number of “connected components.” We may then consider the hierarchical (i.e.multiscale) topologies corresponding to higher-dimensional features as well: for example, loops.A unified description of topological features of all dimensions is given by algebraic topology,and the hierarchical or multiscale version of algebraic toplogy is persistent homology.We now give a brief description of simplicial homology, referring the reader to [14, 15] fordetails. We begin by embedding our data in a discrete complex. We use both simplicial andcubical complexes in this work. In a simplicial complex, points (0-simplices) may be connectedin pairs by edges (1-simplices), in triples by triangular faces (2-simplices), and so on. Simplicialcomplexes must be closed under taking faces: for example, if a 2-simplex is in the complexthen so too must be its three edges and three vertices. A cubical complex is similar, but itconsists of points (0-cubes), line segments (1-cubes), squares (2-cubes), and so on. Topologicalaspects of the simplicial or cubical complex are then captured by its homology groups. Thesegroups, denoted H p , consist of equivalence classes of p -cycles, where two p -cycles are in the sameequivalence class if they can be smoothly deformed into one another. H consists of connectedcomponents, H consists of noncontractible loops, and so on, with the Betti numbers b p givingthe number of inequivalent, nontrivial p -cycles.The core insight of persistent homology is that such a procedure can be significantly en-hanced in its stability and information context if instead of a single complex, a monotonicallygrowing family, called a filtration , is considered. Often the growing of the filtration correspondsto the increasing of a coarse-graining scale, so that multiscale information is captured. SeeFigure 2. As this coarse-graining scale increases, p -cycles are created (for example, loops form)and destroyed (for example, loops are “filled in”). The mathematics of persistent homologyallows us to track the births and deaths of individual topological features. This information is5 Figure 2: (Top) Four steps in the α -filtration for a grid of points, such as appears in ourdiscrete-spin models. The filtration parameter when a p -simplex is included is α , where α is the radius of the simplex’s circumsphere. The α -complexes are pictured in black/red, withthe most recently added p -simplexes being shown in red. (Bottom) Four steps in the sublevelfiltration for a function (represented by grayscale) defined on a 4 × ν , for the sublevel setsand the cubical complexes are pictured in red.usually summarized via a persistence diagram (see Figure 3), which is a scatter plot of thesebirths and deaths.While persistence diagrams are often suitable for visualization, they are not very well suitedfor statistical analysis. In the end, we are interested in the statistical task of quantifyingthe probability that a given spin configuration belongs to a particular phase of the system.Therefore, rather than scatter plots, we might prefer a summary statistic that lives in a vectorspace. These also aid us in quantitatively characterizing the change in the system’s persistenthomology as some parameter is varied. We therefore use persistence images for our analysis,which are formed by appropriately smoothing the persistence diagram and binning so as to havea low(er)-dimensional representation of the persistence data. A weight factor which vanishesat zero persistence is used in order to highlight those more important features which have highpersistence values. (See [27] for more details on the stability properties of persistence images.) The data we consider come from square-lattice spin models, some with discrete, ± , spins andothers with continuous angles. We now describe the filtrations we use for these two cases, andgive examples of how features of the spin configurations are captured by the persistence data.6 a) Features of size 5 × (b) Features of size 6 × Figure 3:
Random configurations with features of a characteristic size and their correspondingpersistence diagrams. Squares of spins are randomly flipped to be spin-up until 50% of spinsare aligned.
Filtrations for discrete spins
With discrete spins on a square lattice we choose to representour data via a point cloud, taking the locations of all spins aligned with a pre-determineddirection as the data. We choose to take all spins which are aligned with the total magnetization(no matter how small). After creating the point cloud from a given spin configuration, we thenuse an α -filtration to create the persistence diagram/image. The filtration corresponds to acoarse-graining of the point cloud, parameterized by the areas of balls enclosing the simplices:see 2 for a small example.As an example, Figure 3 shows two (fake) spin configurations and their corresponding per-sistence diagrams. The configurations consist of (possibly overlapping) squares of aligned spinsof a characteristic size. 1-cycles are born relatively early as adjacent, aligned spins in the latticeare connected by an edge. We see that there is a noticeable shift in the distribution of persis-tences as the characteristic size of the squares changes. Note that this manifests in features ofsize smaller than the squares as well, due to their overlapping nature. Additionally, note thatall 0-cycles are born at the beginning of the filtration. This is true in general for the α -filtration.As such, when considering discrete spins we will generally use the 1-cycles for our statisticaltests. To compute the persistence of α -complexes we use the GUDHI class
AlphaComplex [32].7 igure 4:
Random configuration with characteristic “stripe” pattern and its correspondingpersistence diagram. Along the top we see steps in the sublevel filtration.
Filtrations for continuous spins
We also consider models where the spins are continuous.In these cases, a spin configuration is a function f : Λ → S from the lattice, Λ, of N spinsites to their angles. We consider models with global O(2) symmetry so that we may placethe total magnetization (no matter how small) at angle θ = 0 and think of the function f as mapping Λ into ( − π, π ]. The sublevel sets with threshold ν ∈ S , f − ( − π, ν ], then give afiltration of (periodic) cubical complexes. These sublevel sets experience topology change whenthe threshold ν passes a critical point of f , as is familiar from Morse theory [33]. In this case0-cycles have nontrivial births, corresponding to spin values where f has a local minimum. Assuch we include both 0- and 1-cycles in the derived persistence images.Figure 4 shows an example continuous-spin configuration with a “stripe” pattern on top ofGaussian noise. There are a number of 0-cycles which are born early as the stripes appear in thefiltration: the high-persistence 0-cycles are the 10 blue stripes which remain disconnected untiljoined by red spins. The high-persistence 1-cycles born near ν = 0 correspond to the nontrivialloops around the blue stripes (there are periodic boundary conditions). The low-persistencecycles are a result of the noise; for example for ν = π there still remain a number of 1-cyclesaround those spins which just happen to have angles much larger than their neighbors. Tocompute the persistence of cubical complexes we use the GUDHI class
PeriodicCubicalComplex [34].
In this section we apply our methods to the task of phase classification in simple two-dimensionallattice spin models. We consider four such models: the Ising and square-ice models have discrete,8 a) Is, T = 2 . (b) SI, T = 1 . (c) XY, T = 0 . (d) FFXY, T = 0 . (e) Is, T = 3 . (f ) SI, T = 4 . (g) XY, T = 1 . (h) FFXY, T = 1 . Figure 5:
Sample spin configurations in the low- and high-temperature phases for each model. ± , spins and the XY and fully-frustrated XY models have continuously varying spins. Samplespin configurations for each model are generated at a number of temperatures using standardMonte-Carlo sampling techniques. Example spin configurations at low and high temperaturesfor each model are shown in Figure 5.For each model considered, classification into two phases is performed using only the per-sistence images. A subset of samples with extreme temperatures are used to train a logisticregression and then the accuracy of the regression is evaluated using the known temperatures ofall samples. We normalize our persistence images using the (cid:96) -norm, so they may be interpretedas probability densities for finding cycles with particular births/deaths for a given system. Un-normalized persistence images contain information about the total number of p -cycles and alsolead to a successful classification. In the following sections we will be classifying spin configurations based on their persistenceimages. Since the persistence images are information-rich, we are able to use perhaps thesimplest classification scheme, logistic regression, to great effect. Here we quickly recall theprocedure of logistic regression. One benefit of logistic regression is that it is easy to tell whataspects of the data are used by the classification algorithm. We will use these to extract orderparameters for the phase transitions under consideration.Persistence images x ∈ R n ( n ∼ O (400) in our examples) are vectors of positive num-bers representing the distribution of cycles in the birth-persistence plane. A logistic regression9epends on parameters λ and λ ∈ R n and the sigmoid function σ : R → (0 , σ ( z ) = 11 + e − z . (1)The sigmoid interpolates between σ ( −∞ ) = 0 and σ ( ∞ ) = 1. A persistence image is declared tobe in “category 0” if σ ( λ + λ · x ) < and in “category 1” if σ ( λ + λ · x ) > . In our examples“category 0” will correspond to a low-temperature phase and “category 1” will correspond toa high-temperature phase. The parameters λ i =0 ,...,n are learned by training on a subset of thedata, x ( k ) , which are labeled into the two categories (i.e. phases) with y ( k ) ∈ { , } . Trainingamounts to maximize the log-likelihood, (cid:88) k (cid:16) y ( k ) log (cid:2) σ ( λ + λ · x ( k ) ) (cid:3) + (1 − y ( k ) ) log (cid:2) − σ ( λ + λ · x ( k ) ) (cid:3)(cid:17) − C (cid:88) i λ i , (2)where the constant C controls the (cid:96) -regularization used to prevent overfitting. By trainingon extreme temperatures, we incur some inaccuracy due to nonlinerarities as criticality is ap-proached; these will not concern us too much, as we will find successful classification regardless.Upon training, the regression can be applied to the rest of the persistence images to givean “average classification” at each temperature. This can be interpreted as quantifying theregression’s “certainty” that a particular temperature belongs to a particular phase. In addition,the learned coefficients λ i may be investigated to learn which bins (i.e. regions) of the persistenceimages are most discerning when it comes to distinguishing the low- and high-temperature data.Bins where λ i (cid:29) λ (cid:28) The Ising model on a two-dimensional square lattice is very well understood, largely in partto Onsager’s exact solution [35]. Spins s i ∈ {− , } live at the vertices of the lattice withferromagnetic interactions governed by the local Hamiltonian H Is = − (cid:88) (cid:104) i,j (cid:105) s i s j , (3)where the sum is over nearest-neighbor pairs. In the thermodynamic limit there is a second-order phase transition at T Is = √ ≈ .
27. At low temperatures there is spontaneousmagnetization, while there is a disordered phase at high temperatures. While this model iswell-understood, it provides a good first application of our method. We are able to easilyextract the magnetization as order parameter from a simple logistic regression. We additionallystudy the relationship of new “persistence” critical exponents to those usually studied.
For temperatures T ∈ { . , . , . . . , . } we generate 1000 sample spin configurations for a50 ×
50 square lattice of 2500 spins. For each sample we construct the persistence image using10 a) Sample spin configuration, persistence diagram and persistence image for T = 1 .
90 (top)and T = 3 .
50 (bottom). (b)
Average classification of testing data and learned logistic regression coefficients for theIsing model. The training data have temperatures in the highlighted regions. In the regressioncoefficients, blue regions are more populated in the low-temperature phase and red regions aremore populated in the high-temperature phase.
Figure 6:
Ising model persistence data and phase classification.a weight log (1 + p ), as in Figure 6a. Training of a logistic regression on the persistence imagesis conducted only on a subset of samples with extreme temperatures, well within the expectedphases (see the left-hand side of Figure 6b). The classification extrapolates very well to theintermediate temperatures and gives an estimate of T ≈ .
37 for the critical temperature. Thediscrepancy from the known critical temperature may be attributed to finite-size effects.The coefficients of the trained logistic regression (see the right-hand side of Figure 6b) showthat the low-temperature configurations are identified by their having many small, short-livedcycles. These may be understood as arising both from 2 × igure 7: Ising model death distributions. The slight horizontal stripes in the figure on theright (e.g. at death = 25) are symptomatic of the underlying lattice.to very short-lived 1-cycles) as well as 1-cycles wrapping small groups of isolated spins whichare flipped relative to the large domains of aligned spins: the latter become more and moreimportant as the temperature is increased. In the high-temperature phase, spins are orientedrandomly, leading to a more uniform distribution of 1-cycle sizes. Using persistent homologywe are able to easily identify the magnetization as the order parameter, as is well known.
Since persistent homology contains multiscale information about a spin configuration it seemsreasonable that one should be able to probe a model’s approach towards scale-invariance viacritical exponents. Indeed, we are able to see aspects of scale-invariance appearing at criticalityby looking at statistics derived from the persistence diagram. One-dimensional statistics suchas the Betti numbers, births and deaths can be found by counting points in different regions ofthe persistence diagrams. In this way we may compute the 1-cycle death probability density,D T ( d ), at each temperature, which quantifies the distribution of feature sizes in the spins. InFigure 7 we see that deaths are exponentially distributed with a long tail forming at criticality,indicative of a diverging correlation length and the emergence of power-law behavior.To be more quantitative, we may fit each D T ( d ) to a function of the formD T ( d ) = A d − µ e − d/a death . (4)Here d is the filtration parameter at the death scale of a cycle, and A is a numerical constant.There are two critical exponents to be extracted: µ gives the power-law behavior at criticality,while the correlation area a death diverges at criticality according to a death ∼ | T − T c | − ν death . Weare limited by the IR cutoff of the model, namely the finite area of the lattice, but we maystill estimate these exponents. As a consistency check, we ask how these might be related topreviously studied critical exponents. Using scaling arguments, one can show that at criticalitythe proportion of clusters of k aligned spins goes asP(cluster of size k ) ∼ k − τ , (5)12 igure 8: µ and a death for Ising death distributions. The red dashed lines indicates thepreviously estimated critical temperature T ≈ .
37. Error estimates are derived from fittingmultiple simulations.where the critical exponent is τ ≈ .
032 [36, 37]. The function D T ( d ) is not directly measuringthe size of clusters, since the death of a 1-cycle around an island of spins is influenced nontriviallyby the shape and “nesting” of clusters. Nevertheless, it seems reasonable to expect that atcriticality the distribution of 1-cycle deaths should follow a similar power-law distribution.Recall that the value of the filtration parameter at the death of a 1-cycle is the area of the disksplaced on each point in the point cloud, and so roughly corresponds to the number, k , of spinsenclosed by the 1-cycle.The fit parameters µ and a death are shown in Figure 8, where we see clearly the divergingcorrelation area as criticality is approached from above. The value of µ at our previouslyestimated critical temperature, T ≈ .
37, is consistent with µ ≈ τ = 2 .
032 as anticipated above,although a more detailed study would be needed to determine the value of µ more exactly.We see also the linear behavior of a − with temperature, indicating ν death ≈
1. That thisis the same degree of divergence as the correlation length of the spin-spin correlation function (cid:104) s (0) s ( r ) (cid:105) ∼ e − r/ξ , ξ ∼ | T − T c | − can be understood by the following rough argument.The death of a 1-cycle in the α -filtration roughly corresponds to the area of the clusterof spins that it encloses and D T ( d ) roughly corresponds to the probability that a contiguousregion of spins with area d is aligned. Consider for simplicity looking to estimate the probabilitythat a disk of spins with radius R are all aligned. At infinite temperature where the spins arerandomly aligned, this probability would simply be 2 − πR /(cid:96) , where (cid:96) is the lattice spacing. If wesuppose that P disk ( R ) ∼ e − R /a for some “correlation area” a even at finite temperature, thenhow is a related to ξ as defined by (cid:104) s ( R ) s (0) (cid:105) ∼ e − R/ξ in the disordered phase? To estimateP disk ( R + (cid:96) ) ∼ e − ( R + (cid:96) ) /a ≈ e − R /a − (cid:96)R/a , imagine asking that a circle of ∼ R spins all bealigned with the disk of (aligned) spins of radius R that they encircle. For simplicity, we ignoreconditional aspects of the probability and subleading terms. This should then take the form e − R /a e − R/ξ , from which we conclude that a ∼ (cid:96)ξ : in particular, a ∼ ξ ∼ | T − T c | − ν with thesame critical exponent as the critical temperature is approached from above.It would be interesting to further understand the relationship between the persistence criticalexponents we defined and those typically studied.13 a) Sample spin configuration, persistence diagram and persistence image for T = 0 . T = 4 . (b) Average classification of testing data and learned logistic regression coefficients for thesquare-ice model. The training data have temperatures in the highlighted regions. In theregression coefficients, blue regions are more populated in the low-temperature phase and redregions are more populated in the high-temperature phase.
Figure 9:
Square-ice persistence data and phase classification.
The square-ice model places spins, s i ∈ {− , } , on the edges rather than vertices of a squarelattice and is governed by the local Hamiltonian H SI = (cid:88) v ∈ Λ (cid:16) (cid:88) i : v s i (cid:17) , (6)where i : v denotes those spins on edges adjacent to the vertex v . In contrast to the Isingmodel there is no spontaneous magnetization at low temperatures. Rather, the ground state14s highly degenerate: any configuration with exactly two up and two down spins adjacent toevery vertex has zero energy. This leads to frustration in the low-energy dynamics, as adjacentvertices v compete to minimize ( (cid:80) i : v s i ) . This competition takes place at small scales, so thatmany 1-cycles die very quickly in the filtration. Nevertheless, we are still able to identify a shiftin the distribution of p -cycle births and deaths and reliably classify samples into two phases.In this case, the frustration introduces a particular length scale to the topological features inthe low-temperature phase, while the distribution of sizes in the high-temperature phase is lessrestricted. We generate 1000 sample spin configurations for a 50 ×
50 lattice with 5000 spins at temperatures T ∈ { . , . , . . . , . } . Each sample gives a persistence image with a weight log (1 + p ), suchas that shown in Figure 9a. Again training a logistic regression only on those persistenceimages with extreme temperatures (Figure 9b), we find an estimate of T ≈ . The XY model is a continuous-spin generalization of the Ising model. At each site of the squarelattice spins take values in S and are governed by H XY = − (cid:88) (cid:104) i,j (cid:105) cos ( θ i − θ j ) . (7)There is a well-known KT phase transition at T XY ≈ .
892 (see [38, 39], among others). This isan infinite-order phase transition where at low temperatures there are bound vortex-antivortexpairs while at high temperatures free vortices proliferate and spins are randomly oriented.
With continuous spins each spin configuration implicitly contains much more information aboutthe underlying dynamics. For temperatures T ∈ { . , . , . . . , . } we generate 200 samplespin configurations on a 20 ×
20 lattice with 400 spins. Persistence images are created for eachsample, as in Figure 10a. The zeroth homology, in contrast to the α -complexes used for discretespins, is very rich for the cubical complexes and we include both H and H persistence datain the persistence images. There is always a single 0-cycle and two 1-cycles which never die:these correspond to the p -cycles of the torus on which the lattice lives. We distinguish theseimmortal p -cycles from those cycles with late deaths ( d ≈ π ) by giving the former a death of15 a) Sample spin configuration, persistence diagram and persistence image for T = 0 .
10 (top)and T = 1 .
40 (bottom). (b)
Average classification of testing data and learned logistic regression coefficients for theXY model. The training data have temperatures in the highlighted regions. In the regressioncoefficients, blue regions are more populated in the low-temperature phase while red regionsare more populated in the high-temperature phase.
Figure 10:
XY persistence data and phase classification. The dashed lines visually separatethe infinite persistence 0- and 1-cycles from those which have finite death. d = π by hand. Omitting these infinite persistence “torus cycles” results in a comparablephase classification.Performing a logistic regression of the concatenated H and H persistence images by train-ing on configurations with temperatures far away from the anticipated transition leads to theclassification of Figure 10b. The critical temperature is estimated as T XY ≈ .
9. We see thatthe low-temperature phase is characterized by p -cycles on the “boundary” of the persistenceimages. This we can understand in the following way. A (small enough) loop around an isolatedvortex has nontrivial winding number, which ensures that there are spins with angles close toboth − π and π if a vortex is present. This explains the strong blue regions in the corners ofthe logistic regression coefficients: for ν ≈ − π a number of 0-cycles are born very early for16 igure 11: Example spin configuration with three vortex-antivortex pairs, with seven sublevelsets shown. Each vortex-antivortex pair corresponds to a number of 0-cycles which are bornvery early and 1-cycles which die very late.
Figure 12:
XY 0- and 1-cycle death distributions.
Figure 13:
Average counts of 0- and 1-cycle deaths for fixed number of vortex-antivortex pairsat low temperatures ( T ≤ . d ≤ − π and 1-cycles with d ≥ π . We see that these features are correlated with the number of vortex-antivortex pairs in the configuration.each vortex and antivortex, giving the lower-left corner of the H coefficients. One of these0-cycle lives forever, giving the upper-left corner of the H coefficients. In addition, there are1-cycles which are born close to ν ≈ π , again corresponding to the extreme angles associatedwith the (anti)vortices. See Figure 11 for an example of this interpretation in practice. Whenvortex-antivortex pairs happen to not be present at low temperatures, then all of the spins arealigned close to θ = 0, giving the short-lived features centered around a birth of zero along thebottom edges of both the H and H coefficients.As before we may consider the distribution of p -cycle deaths as a function of temperature.17n Figure 12 we see that low-temperatures there are two “populations” of both 0- and 1-cycleswhich merge into one as we pass into the high-temperature phase. This again can be attributedto the presence of vortex-antivortex pairs in the following way. Using the raw spin configurationswe may count the number of (anti)vortices simply by looking for nontrivial winding in 2 × d ≤ − π ) and the number of 1-cycles with late death (e.g. d ≥ π ). Averaging over sampleswith temperatures below 0 .
20 where the number of vortex-antivortex pairs is reasonably smallon the 20 ×
20 lattice leads to Figure 13. There is a clear correlation between the number ofextreme-death p -cycles and the number of vortex-antivortex pairs as determined directly fromthe spins. This topological signature of vortex-antivortex pairs should exist rather generally.Previous investigations of the XY model and its KT phase transition using neural networksand PCA have faced difficulties in identifying vortices at low temperatures [40, 41]. It is worthemphasizing the relative ease with which persistent homology identifies (anti)vortices as animportant feature at low temperatures. A frustrated version of the XY model is obtained by changing some of the nearest-neighborinteractions to be antiferromagnetic. One such choice which is fully frustrated is H FFXY = − (cid:88) (cid:104) i,j (cid:105) J ij cos ( θ i − θ j ) , (8)where J ij = − J ij = +1 everywhere else. Thereare two phase transitions that occur at temperatures which are very close together: a phasetransition at T ≈ .
454 is associated with the loss of Z symmetry, and a phase transition at T ≈ .
446 is associated with the loss of the SO(2) rotational symmetry [42]. Because of theirproximity we are unable to identify both transitions without an extensive set of simulations.
We generate 200 sample spin configurations on a 20 ×
20 lattice with 400 spins for temperatures T ∈ { . , . , . . . . } . As before with the XY model, the zeroth homology is quite richand we include it in the persistence images, such as that in Figure 14a. Training the logisticregression leads to the classification in Figure 14b, where the critical temperature is estimatedas T FFXY ≈ .
39. A more accurate estimation can be achieved by using training data closer tothe phase transition. The learned coefficients show a strong tendency for both 0- and 1-cycles toshift to have persistence around π in the high-temperature phase. As in the square-ice model,our order parameter probes the small-scale structure of the frustration pattern. In particular,the low-temperature phase exhibits “pseudo-domains” where many next-to-nearest neighborstake similar spin values. The alternating structure induced by the antiferromagnetic bandstherefore leads to more isolated local minima (i.e. 0-cycles in the sublevel filtration) in thelow-temperature phase. In the high-temperature phase, most of the local minima are born at θ ∼ − π , while in the low-temperature phase there are local minima at higher θ protected by18 a) Sample spin configuration, persistence diagram and persistence image for T = 0 .
15 (top)and T = 1 .
00 (bottom). (b)
Average classification of testing data and learned logistic regression coefficients for theFFXY model. The training data have temperatures in the highlighted regions. In the re-gression coefficients, blue regions are more populated in the low-temperature phase and redregions are more populated in the high-temperature phase.
Figure 14:
FFXY persistence data and phase classification. The dashed lines visually separatethe infinite persistence 0- and 1-cycles from those which have finite death.these pseudo-domains. This explains the blue band at the bottom of the H logistic regressioncoefficients. The lack of vortices can be seen from the death distribution as a function oftemperature in Figure 15.In our discussion we have used a sublevel filtration with cubical complex to quantify thehomology of continuous-spin configurations. Another approach would be to construct pointclouds by taking the locations of spins in a (sub)levelset and using an α -filtration. By scanningthrough levelsets one can capture the topological features of f : Λ → S in a different way.For the fully-frustrated XY model this leads to a comparable classification and estimate for thecritical temperature. 19 igure 15: FFXY 0- and 1-cycle death distributions.
In this paper we have explored the use of persistent homology in quantitatively analyzing thephase structure and critical behavior of lattice spin models. While the models we consider can beunderstood via other means, the use of persistent homology provides an interesting perspectiveinto their statistical properties. Nonlocal features are naturally accounted for in this frameworkand could prove to be useful in more complex systems as well.Phase classification using persistence images alone is accomplished successfully for theIsing, square-ice, XY and fully-frustrated XY models, providing a mix of examples with dis-crete/continuous spins and some with frustration. The resulting trained logistic regressionsreveal those regions of the persistence image/diagram which are characteristic of low- and high-temperature phases. This allows for an easily interpretable classification, where (sometimesdrastic) shifts in the distributions of p -cycle births and deaths are associated with a phasetransition. In the case of the XY model, there is a clear correlation between the number ofearly-death 0-cycles, late-death 1-cycles and the number of bound vortex-antivortex pairs atlow temperatures.The persistence data also display features of critical phenomena. For the Ising model oneobserves the emergence of power-law behavior in the distribution of 1-cycle deaths as the criticaltemperature is approached. We are able to estimate two critical exponents associated with thisbehavior: the correlation area diverges as a death ∼ | T − T c | − ν death with ν death ≈ µ , introduced through D T ( d ) ∼ d − µ ,to be µ ≈
2, in agreement with expectations from the known power-law behavior of cluster sizesat criticality.We have demonstrated the quantitative statistical capabilities of persistent homology forrelatively simple 2D lattice spin systems. It would interesting to apply these ideas and techniquesto more complicated lattice spin models in higher dimensions or with no known order parameter.In more than two dimensions the higher homology groups may serve useful in quantifyingnonlocal structures. We leave such work for the future.20 cknowledgments
We thank Jeff Schmidt for useful discussions. We also thank the participants of the “TheoreticalPhysics for Machine Learning” and “Physics ∩ ML” workshops at the Aspen Center for Physicsand Microsoft Research, respectively, where partial results of this work were presented in early2019, for discussion. G.J.L. and G.S. are supported in part by the DOE grant DE-SC0017647and the Kellett Award of the University of Wisconsin. The code and data used in our analysisis provided in the following GitHub repository: gloges/TDA-Spin-Models . References [1] L. Wang,
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