🍩 Database of Original & NonTheoretical Uses of Topology
(found 20 matches in 0.005701s)


Unsupervised Topological Learning for Identification of Atomic Structures (2022)
Sébastien Becker, Emilie Devijver, Rémi Molinier, Noël JakseAbstract
We propose an unsupervised learning methodology with descriptors based on topological data analysis (TDA) concepts to describe the local structural properties of materials at the atomic scale. Based only on atomic positions and without a priori knowledge, our method allows for an autonomous identification of clusters of atomic structures through a Gaussian mixture model. We apply successfully this approach to the analysis of elemental Zr in the crystalline and liquid states as well as homogeneous nucleation events under deep undercooling conditions. This opens the way to deeper and autonomous study of complex phenomena in materials at the atomic scale. 
Quantitative Analysis of Phase Transitions in TwoDimensional XY Models Using Persistent Homology (2022)
Nicholas Sale, Jeffrey Giansiracusa, Biagio LuciniAbstract
We use persistent homology and persistence images as an observable of three different variants of the twodimensional XY model in order to identify and study their phase transitions. We examine models with the classical XY action, a topological lattice action, and an action with an additional nematic term. In particular, we introduce a new way of computing the persistent homology of lattice spin model configurations and, by considering the fluctuations in the output of logistic regression and knearest neighbours models trained on persistence images, we develop a methodology to extract estimates of the critical temperature and the critical exponent of the correlation length. We put particular emphasis on finitesize scaling behaviour and producing estimates with quantifiable error. For each model we successfully identify its phase transition(s) and are able to get an accurate determination of the critical temperatures and critical exponents of the correlation length. 
Unsupervised Topological Learning Approach of Crystal Nucleation (2022)
Sébastien Becker, Emilie Devijver, Rémi Molinier, Noël JakseAbstract
Nucleation phenomena commonly observed in our every day life are of fundamental, technological and societal importance in many areas, but some of their most intimate mechanisms remain however to be unravelled. Crystal nucleation, the early stages where the liquidtosolid transition occurs upon undercooling, initiates at the atomic level on nanometre length and subpicoseconds time scales and involves complex multidimensional mechanisms with local symmetry breaking that can hardly be observed experimentally in the very details. To reveal their structural features in simulations without a priori, an unsupervised learning approach founded on topological descriptors loaned from persistent homology concepts is proposed. Applied here to monatomic metals, it shows that both translational and orientational ordering always come into play simultaneously as a result of the strong bonding when homogeneous nucleation starts in regions with low fivefold symmetry. It also reveals the specificity of the nucleation pathways depending on the element considered, with features beyond the hypothesis of Classical Nucleation Theory. 
Unsupervised Topological Learning Approach of Crystal Nucleation in Pure Tantalum (2021)
Sébastien Becker, Emilie Devijver, Rémi Molinier, Noël JakseAbstract
Nucleation phenomena commonly observed in our every day life are of fundamental, technological and societal importance in many areas, but some of their most intimate mechanisms remain however to be unraveled. Crystal nucleation, the early stages where the liquidtosolid transition occurs upon undercooling, initiates at the atomic level on nanometer length and subpicoseconds time scales and involves complex multidimensional mechanisms with local symmetry breaking that can hardly be observed experimentally in the very details. To reveal their structural features in simulations without a priori, an unsupervised learning approach founded on topological descriptors loaned from persistent homology concepts is proposed. Applied here to a monatomic metal, namely Tantalum (Ta), it shows that both translational and orientational ordering always come into play simultaneously when homogeneous nucleation starts in regions with low fivefold symmetry. 
A Topological Perspective on Regimes in Dynamical Systems (2021)
Kristian Strommen, Matthew Chantry, Joshua Dorrington, Nina OtterAbstract
The existence and behaviour of socalled `regimes' has been extensively studied in dynamical systems ranging from simple toy models to the atmosphere itself, due to their potential of drastically simplifying complex and chaotic dynamics. Nevertheless, no agreedupon and clearcut definition of a `regime' or a `regime system' exists in the literature. We argue here for a definition which equates the existence of regimes in a system with the existence of nontrivial topological structure. We show, using persistent homology, a tool in topological data analysis, that this definition is both computationally tractable, practically informative, and accounts for a variety of different examples. We further show that alternative, more strict definitions based on clustering and/or temporal persistence criteria fail to account for one or more examples of dynamical systems typically thought of as having regimes. We finally discuss how our methodology can shed light on regime behaviour in the atmosphere, and discuss future prospects. 
Topological Electronic Structure and Weyl Points in Nonsymmorphic Hexagonal Materials (2020)
Rafael GonzálezHernández, Erick Tuiran, Bernardo UribeAbstract
Using topological band theory analysis we show that the nonsymmorphic symmetry operations in hexagonal lattices enforce Weyl points at the screwinvariant highsymmetry lines of the band structure. The corepresentation theory and connectivity group theory show that Weyl points are generated by band crossings in accordionlike and hourglasslike dispersion relations. These Weyl points are stable against weak perturbations and are protected by the screw rotation symmetry. Based on firstprinciples calculations we found a complete agreement between the topological predicted energy dispersion relations and real hexagonal materials. Topological charge (chirality) and Berry curvature calculations show the simultaneous formation of Weyl points and nodallines in 4d transitionmetal trifluorides such as AgF3 and AuF3. Furthermore, a large intrinsic spinHall conductivity was found due to the combined strong spinorbit coupling and multiple Weylpoint crossings in the electronic structure. These materials could be used to the spin/charge conversion in more energyefficient spintronic devices. 
Crystallographic Interacting Topological Phases and Equvariant Cohomology: To Assume or Not to Assume (2020)
Daniel Sheinbaum, Omar Antolín CamarenaAbstract
For symmorphic crystalline interacting gapped systems we derive a classification under adiabatic evolution. This classification is complete for nondegenerate ground states. For the degenerate case we discuss some invariants given by equivariant characteristic classes. We do not assume an emergent relativistic field theory nor that phases form a topological spectrum. We also do not assume shortrange entanglement nor the existence of quasiparticles as is done in SPT and SET classifications respectively. Using a slightly generalized Bloch decomposition and Grassmanians made out of ground state spaces, we show that the \$P\$equivariant cohomology of a \$d\$dimensional torus gives rise to different interacting phases. We compare our results to bosonic symmorphic crystallographic SPT phases and to noninteracting fermionic crystallographic phases in class A. Finally we discuss the relation of our assumptions to those made for crystallographic SPT and SET phases. 
Finding Universal Structures in Quantum ManyBody Dynamics via Persistent Homology (2020)
Daniel Spitz, Jürgen Berges, Markus K. Oberthaler, Anna WienhardAbstract
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 simulated data of a twodimensional Bose gas far from equilibrium. We discover a continuous spectrum of dynamical scaling exponents, which provides a refined classification of nonequilibrium universal phenomena. A possible explanation of the underlying processes is provided in terms of mixing wave turbulence and vortex kinetics components in point clouds. We find that the persistent homology scaling exponents are inherently linked to the geometry of the system, as the derivation of a packing relation reveals. The approach opens new ways of analyzing quantum manybody dynamics in terms of robust topological structures beyond standard field theoretic techniques. 
Topological Persistence Machine of Phase Transitions (2020)
Quoc Hoan Tran, Mark Chen, Yoshihiko HasegawaAbstract
The study of phase transitions from experimental data becomes challenging, especially when little prior knowledge of the system is available. Topological data analysis is an emerging framework for characterizing the shape of data and has recently achieved success in detecting structural transitions in material science such as glassliquid transition. However, data obtained from physical states may not have explicit shapes as structural materials. We propose a general framework called topological persistence machine to construct the shape of data from correlations in states; hence decipher phase transitions via the qualitative changes of the shape. Our framework enables an effective and unified approach in phase transition analysis. We demonstrate the impact in highly precise detection of BerezinskiiKosterlitzThouless phase transitions in the classical XY model, and quantum phase transition in the transverse Ising model and BoseHubbard model. Intriguingly, these phase transitions have proven to be notoriously difficult in traditional methods but can be characterized in our framework without requiring prior knowledge about phases. Our approach is thus expected applicable and brings a remarkable perspective for exploring phases of experimental physical systems. 
The Persistence of Large Scale Structures I: Primordial NonGaussianity (2020)
Matteo Biagetti, Alex Cole, Gary ShiuAbstract
We develop an analysis pipeline for characterizing the topology of large scale structure and extracting cosmological constraints based on persistent homology. Persistent homology is a technique from topological data analysis that quantifies the multiscale topology of a data set, in our context unifying the contributions of clusters, filament loops, and cosmic voids to cosmological constraints. We describe how this method captures the imprint of primordial local nonGaussianity on the latetime distribution of dark matter halos, using a set of Nbody simulations as a proxy for real data analysis. For our best single statistic, running the pipeline on several cubic volumes of size \$40~(\rm\Gpc/h\)\textasciicircum\3\\$, we detect \$f_\\rm NL\\textasciicircum\\rm loc\=10\$ at \$97.5\%\$ confidence on \$\sim 85\%\$ of the volumes. Additionally we test our ability to resolve degeneracies between the topological signature of \$f_\\rm NL\\textasciicircum\\rm loc\\$ and variation of \$\sigma_8\$ and argue that correctly identifying nonzero \$f_\\rm NL\\textasciicircum\\rm loc\\$ in this case is possible via an optimal template method. Our method relies on information living at \$\mathcal\O\(10)\$ Mpc/h, a complementary scale with respect to commonly used methods such as the scaledependent bias in the halo/galaxy power spectrum. Therefore, while still requiring a large volume, our method does not require sampling longwavelength modes to constrain primordial nonGaussianity. Moreover, our statistics are interpretable: we are able to reproduce previous results in certain limits and we make new predictions for unexplored observables, such as filament loops formed by dark matter halos in a simulation box. 
Revealing Key Structural Features Hidden in Liquids and Glasses (2019)
Hajime Tanaka, Hua Tong, Rui Shi, John RussoAbstract
A great success of solid state physics comes from the characterization of crystal structures in the reciprocal (wave vector) space. The power of structural characterization in Fourier space originates from the breakdown of translational and rotational symmetries. However, unlike crystals, liquids and amorphous solids possess continuous translational and rotational symmetries on a macroscopic scale, which makes Fourier space analysis much less effective. Lately, several studies have revealed local breakdown of translational and rotational symmetries even for liquids and glasses. Here, we review several mathematical methods used to characterize local structural features of apparently disordered liquids and glasses in real space. We distinguish two types of local ordering in liquids and glasses: energydriven and entropydriven. The former, which is favoured energetically by symmetryselective directional bonding, is responsible for anomalous behaviours commonly observed in watertype liquids such as water, silicon, germanium and silica. The latter, which is often favoured entropically, shows connections with the heterogeneous, slow dynamics found in hardspherelike glassforming liquids. We also discuss the relationship between such local ordering and crystalline structures and its impact on glassforming ability. 
Persistent Homology to Quantify the Quality of SurfaceSupported Covalent Networks (2019)
Abraham Gutierrez, Mickaël Buchet, Sylvain ClairAbstract
Covalent networks formed by onsurface synthesis usually suffer from the presence of a large number of defects. We report on a methodology to characterize such twodimensional networks from their experimental images obtained by scanning probe microscopy. The computation is based on a persistent homology approach and provides a quantitative score indicative of the network homogeneity. We compare our scoring method with results previously obtained using minimal spanning tree analyses and we apply it to some molecular systems appearing in the existing literature. 
PhaseField Investigation of the Coarsening of Porous Structures by Surface Diffusion (2019)
PierreAntoine Geslin, Mickaël Buchet, Takeshi Wada, Hidemi KatoAbstract
Nano and microporous connected structures have attracted increasing attention in the past decades due to their high surface area, presenting interesting properties for a number of applications. These structures generally coarsen by surface diffusion, leading to an enlargement of the structure characteristic length scale. We propose to study this coarsening behavior using a phasefield model for surface diffusion. In addition to reproducing the expected scaling law, our simulations enable to investigate precisely the evolution of the topological and morphological characteristics along the coarsening process. In particular, we show that after a transient regime, the coarsening is selfsimilar as exhibited by the evolution of both morphological and topological features. In addition, the influence of surface anisotropy is discussed and comparisons with experimental tomographic observations are presented. 
NonEmpirical Identification of Trigger Sites in Image Data Using Persistent Homology: Crack Formation During Heterogeneous Reduction of IronOre Sinters (2018)
M. Kimura, I. Obayashi, Y. Takeichi, R. Murao, Y. Hiraoka 
Homological Analysis of MultiQubit Entanglement (2018)
Alessandra di Pierro, Stefano Mancini, Laleh Memarzadeh, Riccardo MengoniAbstract
We propose the usage of persistent homologies to characterize multipartite entanglement. On a multiqubit data set we introduce metriclike measures defined in terms of bipartite entanglement and then we derive barcodes. We show that, depending on the distance, they are able to produce different classifications. In one case, it is possible to obtain the standard separability classes. In the other case, a new classification of entangled states of three and four qubits is provided. 
Topological Edge Modes by Smart Patterning (2018)
David J. Apigo, Kai Qian, Camelia Prodan, Emil ProdanAbstract
We study identical coupled mechanical resonators whose collective dynamics are fully determined by the patterns in which they are arranged. In this work, we call a system topological if (1) boundary resonant modes fully fill all existing spectral gaps whenever the system is halved, and (2) if the boundary spectrum cannot be removed or gapped by any boundary condition. We demonstrate that such topological characteristics can be induced solely through patterning, in a manner entirely independent of the structure of the resonators and the details of the couplings. The existence of such patterns is proven using K theory and exemplified using an experimental platform based on magnetically coupled spinners. Topological metamaterials built on these principles can be easily engineered at any scale, providing a practical platform for applications and devices. 
Pore Configuration Landscape of Granular Crystallization (2017)
Mohammad Saadatfar, Hiroshi Takeuchi, Vanessa Robins, Nicolas Francois, Yisuaki HiraokaAbstract
Emergence and growth of crystalline domains in granular media remains underexplored. Here, the authors analyse tomographic snapshots from partially recrystallized packings of spheres using persistent homology and find agreement with proposed transitions based on continuous deformation of octahedral and tetrahedral voids. 
Hierarchical Structures of Amorphous Solids Characterized by Persistent Homology (2016)
Yasuaki Hiraoka, Takenobu Nakamura, Akihiko Hirata, Emerson G. Escolar, Kaname Matsue, Yasumasa NishiuraAbstract
This article proposes a topological method that extracts hierarchical structures of various amorphous solids. The method is based on the persistence diagram (PD), a mathematical tool for capturing shapes of multiscale data. The input to the PDs is given by an atomic configuration and the output is expressed as 2D histograms. Then, specific distributions such as curves and islands in the PDs identify meaningful shape characteristics of the atomic configuration. Although the method can be applied to a wide variety of disordered systems, it is applied here to silica glass, the LennardJones system, and CuZr metallic glass as standard examples of continuous random network and random packing structures. In silica glass, the method classified the atomic rings as shortrange and mediumrange orders and unveiled hierarchical ring structures among them. These detailed geometric characterizations clarified a real space origin of the first sharp diffraction peak and also indicated that PDs contain information on elastic response. Even in the LennardJones system and CuZr metallic glass, the hierarchical structures in the atomic configurations were derived in a similar way using PDs, although the glass structures and properties substantially differ from silica glass. These results suggest that the PDs provide a unified method that extracts greater depth of geometric information in amorphous solids than conventional methods. 
Morse Theory and Persistent Homology for Topological Analysis of 3D Images of Complex Materials (2014)
O. DelgadoFriedrichs, V. Robins, A. SheppardAbstract
We develop topologically accurate and compatible definitions for the skeleton and watershed segmentation of a 3D digital object that are computed by a single algorithm. These definitions are based on a discrete gradient vector field derived from a signed distance transform. This gradient vector field is amenable to topological analysis and simplification via Forman's discrete Morse theory and provides a filtration that can be used as input to persistent homology algorithms. Efficient implementations allow us to process largescale xray microCT data of rock cores and other materials.