🍩 Database of Original & NonTheoretical Uses of Topology
(found 8 matches in 0.001947s)


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. 
Persistent Homology Advances Interpretable Machine Learning for Nanoporous Materials (2020)
Aditi S. Krishnapriyan, Joseph Montoya, Jens Hummelshøj, Dmitriy MorozovAbstract
Machine learning for nanoporous materials design and discovery has emerged as a promising alternative to more timeconsuming experiments and simulations. The challenge with this approach is the selection of features that enable universal and interpretable materials representations across multiple prediction tasks. We use persistent homology to construct holistic representations of the materials structure. We show that these representations can also be augmented with other generic features such as word embeddings from natural language processing to capture chemical information. We demonstrate our approach on multiple metalorganic framework datasets by predicting a variety of gas adsorption targets. Our results show considerable improvement in both accuracy and transferability across targets compared to models constructed from commonly used manually curated features. Persistent homology features allow us to locate the pores that correlate best to adsorption at different pressures, contributing to understanding atomic level structureproperty relationships for materials design. 
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. 
Quantitative and Interpretable Order Parameters for Phase Transitions From Persistent Homology (2020)
Alex Cole, Gregory J. Loges, Gary ShiuAbstract
We apply modern methods in computational topology to the task of discovering and characterizing phase transitions. As illustrations, we apply our method to four twodimensional lattice spin models: the Ising, square ice, XY, and fullyfrustrated XY models. In particular, we use persistent homology, which computes the births and deaths of individual topological features as a coarsegraining scale or sublevel threshold is increased, to summarize multiscale and highpoint correlations in a spin configuration. We employ vector representations of this information called persistence images to formulate and perform the statistical task of distinguishing phases. For the models we consider, a simple logistic regression on these images is sufficient to identify the phase transition. Interpretable order parameters are then read from the weights of the regression. This method suffices to identify magnetization, frustration, and vortexantivortex structure as relevant features for phase transitions in our models. We also define "persistence" critical exponents and study how they are related to those critical exponents usually considered. 
UltrahighPressure Form of \$\Mathrm\Si\\\mathrm\O\\_\2\\$ Glass With Dense PyriteType Crystalline Homology (2019)
M. Murakami, S. Kohara, N. Kitamura, J. Akola, H. Inoue, A. Hirata, Y. Hiraoka, Y. Onodera, I. Obayashi, J. Kalikka, N. Hirao, T. Musso, A. S. Foster, Y. Idemoto, O. Sakata, Y. OhishiAbstract
Highpressure synthesis of denser glass has been a longstanding interest in condensedmatter physics and materials science because of its potentially broad industrial application. Nevertheless, understanding its nature under extreme pressures has yet to be clarified due to experimental and theoretical challenges. Here we reveal the formation of OSi4 tetraclusters associated with that of SiO7 polyhedra in SiO2 glass under ultrahigh pressures to 200 gigapascal confirmed both experimentally and theoretically. Persistent homology analyses with molecular dynamics simulations found increased packing fraction of atoms whose topological diagram at ultrahigh pressures is similar to a pyritetype crystalline phase, although the formation of tetraclusters is prohibited in the crystalline phase. This critical difference would be caused by the potential structural tolerance in the glass for distortion of oxygen clusters. Furthermore, an expanded electronic band gap demonstrates that chemical bonds survive at ultrahigh pressure. This opens up the synthesis of topologically disordered dense oxide glasses. 
Topological Descriptors Help Predict Guest Adsorption in Nanoporous Materials (2020)
Aditi S. Krishnapriyan, Maciej Haranczyk, Dmitriy MorozovAbstract
Machine learning has emerged as an attractive alternative to experiments and simulations for predicting material properties. Usually, such an approach relies on specific domain knowledge for feature design: each learning target requires careful selection of features that an expert recognizes as important for the specific task. The major drawback of this approach is that computation of only a few structural features has been implemented so far, and it is difficult to tell a priori which features are important for a particular application. The latter problem has been empirically observed for predictors of guest uptake in nanoporous materials: local and global porosity features become dominant descriptors at low and high pressures, respectively. We investigate a feature representation of materials using tools from topological data analysis. Specifically, we use persistent homology to describe the geometry of nanoporous materials at various scales. We combine our topological descriptor with traditional structural features and investigate the relative importance of each to the prediction tasks. We demonstrate an application of this feature representation by predicting methane adsorption in zeolites, for pressures in the range of 1200 bar. Our results not only show a considerable improvement compared to the baseline, but they also highlight that topological features capture information complementary to the structural features: this is especially important for the adsorption at low pressure, a task particularly difficult for the traditional features. Furthermore, by investigation of the importance of individual topological features in the adsorption model, we are able to pinpoint the location of the pores that correlate best to adsorption at different pressure, contributing to our atomlevel understanding of structureproperty relationships.