Quantum Theory of Geometry II: Volume operators

A functional calculus on the space of (generalized) connections was recently introduced without any reference to a background metric. It is used to continue the exploration of the quantum Riemannian geometry. Operators corresponding to volume of three-dimensional regions are regularized rigorously. It is shown that there are two natural regularization schemes, each of which leads to a well-defined operator. Both operators can be completely specified by giving their action on states labelled by graphs. The two final results are closely related but differ from one another in that one of the operators is sensitive to the differential structure of graphs at their vertices while the second is sensitive only to the topological characteristics. (The second operator was first introduced by Rovelli and Smolin and De Pietri and Rovelli using a somewhat different framework.) The difference between the two operators can be attributed directly to the standard quantization ambiguity. Underlying assumptions and subtleties of regularization procedures are discussed in detail in both cases because volume operators play an important role in the current discussions of quantum dynamics.Comment: Latex, 3 figure

MODELING CHAIN PACKING IN COMPLEX PHASES OF SELF-ASSEMBLED BLOCK COPOLYMERS

Block copolymer (BCP) melts undergo microphase seperation and form ordered soft matter crystals with varying domain shapes and symmetries. We study the con- nection between diblock copolymer molecular designs and thermodynamic selection of ordered crystals by modeling features of variable sub-domain geometry filled with individual blocks within non-canonical sphere-like and network phases that together with layered, cylindrical and canonical spherical phases forms “natural forms” of self- assembled amphiphilic soft matter at large. First, we present a model to revise our understanding of optimal Frank-Kasper sphere-like morphologies by advancing the- ory to account for varying domain volumes. We then develop generic approaches to quantify local changes to domain thickness or packing frustration using medial sets and show its application to morphologies with arbitrary domain topologies and sym- metries in both theoretical models and experimental data. We further use medial sets as a proxy for terminal boundaries of blocks within different domains and revise thermodynamic models of BCP assembly in the strong segregation limit. Finally, we use this revised model to study effect of elastic stiffness asymmetry on relaxing packing frustration experienced by BCPs in tubular and matrix domains leading to equilibrium double gyroid network morphology in diblock copolymers

Introduction to Louis Michel’s lattice geometry through group action

Group action analysis developed and applied mainly by Louis Michel to the study of N-dimensional periodic lattices is the central subject of the book. Di¬fferent basic mathematical tools currently used for the description of lattice geometry are introduced and illustrated through applications to crystal structures in two- and three-dimensional space, to abstract multi-dimensional lattices and to lattices associated with integrable dynamical systems. Starting from general Delone sets the authors turn to di¬fferent symmetry and topological classifications including explicit construction of orbifolds for two- and three-dimensional point and space groups

Graph Theory Data for Topological Quantum Chemistry

Topological phases of noninteracting particles are distinguished by global properties of their band structure and eigenfunctions in momentum space. On the other hand, group theory as conventionally applied to solid-state physics focuses only on properties which are local (at high symmetry points, lines, and planes) in the Brillouin zone. To bridge this gap, we have previously [B. Bradlyn et al., Nature 547, 298--305 (2017)] mapped the problem of constructing global band structures out of local data to a graph construction problem. In this paper, we provide the explicit data and formulate the necessary algorithms to produce all topologically distinct graphs. Furthermore, we show how to apply these algorithms to certain "elementary" band structures highlighted in the aforementioned reference, and so identified and tabulated all orbital types and lattices that can give rise to topologically disconnected band structures. Finally, we show how to use the newly developed BANDREP program on the Bilbao Crystallographic Server to access the results of our computation.Comment: v1: 29 Pages, 13 Figures. Explains how to access the data presented in arXiv:1703.02050 v2: Accepted version. References updated, figures improve

Robust recognition and exploratory analysis of crystal structures using machine learning

In den Materialwissenschaften läuten Künstliche-Intelligenz Methoden einen Paradigmenwechsel in Richtung Big-data zentrierter Forschung ein. Datenbanken mit Millionen von Einträgen, sowie hochauflösende Experimente, z.B. Elektronenmikroskopie, enthalten eine Fülle wachsender Information. Um diese ungenützten, wertvollen Daten für die Entdeckung verborgener Muster und Physik zu nutzen, müssen automatische analytische Methoden entwickelt werden. Die Kristallstruktur-Klassifizierung ist essentiell für die Charakterisierung eines Materials. Vorhandene Daten bieten vielfältige atomare Strukturen, enthalten jedoch oft Defekte und sind unvollständig. Eine geeignete Methode sollte diesbezüglich robust sein und gleichzeitig viele Systeme klassifizieren können, was für verfügbare Methoden nicht zutrifft. In dieser Arbeit entwickeln wir ARISE, eine Methode, die auf Bayesian deep learning basiert und mehr als 100 Strukturklassen robust und ohne festzulegende Schwellwerte klassifiziert. Die einfach erweiterbare Strukturauswahl ist breit gefächert und umfasst nicht nur Bulk-, sondern auch zwei- und ein-dimensionale Systeme. Für die lokale Untersuchung von großen, polykristallinen Systemen, führen wir die strided pattern matching Methode ein. Obwohl nur auf perfekte Strukturen trainiert, kann ARISE stark gestörte mono- und polykristalline Systeme synthetischen als auch experimentellen Ursprungs charakterisieren. Das Model basiert auf Bayesian deep learning und ist somit probabilistisch, was die systematische Berechnung von Unsicherheiten erlaubt, welche mit der Kristallordnung von metallischen Nanopartikeln in Elektronentomographie-Experimenten korrelieren. Die Anwendung von unüberwachtem Lernen auf interne Darstellungen des neuronalen Netzes enthüllt Korngrenzen und nicht ersichtliche Regionen, die über interpretierbare geometrische Eigenschaften verknüpft sind. Diese Arbeit ermöglicht die Analyse atomarer Strukturen mit starken Rauschquellen auf bisher nicht mögliche Weise.In materials science, artificial-intelligence tools are driving a paradigm shift towards big data-centric research. Large computational databases with millions of entries and high-resolution experiments such as electron microscopy contain large and growing amount of information. To leverage this under-utilized - yet very valuable - data, automatic analytical methods need to be developed. The classification of the crystal structure of a material is essential for its characterization. The available data is structurally diverse but often defective and incomplete. A suitable method should therefore be robust with respect to sources of inaccuracy, while being able to treat multiple systems. Available methods do not fulfill both criteria at the same time. In this work, we introduce ARISE, a Bayesian-deep-learning based framework that can treat more than 100 structural classes in robust fashion, without any predefined threshold. The selection of structural classes, which can be easily extended on demand, encompasses a wide range of materials, in particular, not only bulk but also two- and one-dimensional systems. For the local study of large, polycrystalline samples, we extend ARISE by introducing so-called strided pattern matching. While being trained on ideal structures only, ARISE correctly characterizes strongly perturbed single- and polycrystalline systems, from both synthetic and experimental resources. The probabilistic nature of the Bayesian-deep-learning model allows to obtain principled uncertainty estimates which are found to be correlated with crystalline order of metallic nanoparticles in electron-tomography experiments. Applying unsupervised learning to the internal neural-network representations reveals grain boundaries and (unapparent) structural regions sharing easily interpretable geometrical properties. This work enables the hitherto hindered analysis of noisy atomic structural data

Detecting hidden spatial and spatio-temporal structures in glasses and complex physical systems by multiresolution network clustering

We elaborate on a general method that we recently introduced for characterizing the "natural" structures in complex physical systems via a multiscale network based approach for the data mining of such structures. The approach is based on "community detection" wherein interacting particles are partitioned into "an ideal gas" of optimally decoupled groups of particles. Specifically, we construct a set of network representations ("replicas") of the physical system based on interatomic potentials and apply a multiscale clustering ("multiresolution community detection") analysis using information-based correlations among the replicas. Replicas may be (i) different representations of an identical static system or (ii) embody dynamics by when considering replicas to be time separated snapshots of the system (with a tunable time separation) or (iii) encode general correlations when different replicas correspond to different representations of the entire history of the system as it evolves in space-time. We apply our method to computer simulations of a binary Kob-Andersen Lennard-Jones system, a ternary model system, and to atomic coordinates in a ZrPt system as gleaned by reverse Monte Carlo analysis of experimentally determined structure factors. We identify the dominant structures (disjoint or overlapping) and general length scales by analyzing extrema of the information theory measures. We speculate on possible links between (i) physical transitions or crossovers and (ii) changes in structures found by this method as well as phase transitions associated with the computational complexity of the community detection problem. We briefly also consider continuum approaches and discuss the shear penetration depth in elastic media; this length scale increases as the system becomes increasingly rigid.Comment: (29 pages, 44 figures

New group theoretical methods for applications in virology and quasicrystals

Non-crystallographic symmetries are ubiquitous in physics, chemistry and biology. Prominent examples are quasicrystals, alloys with long-range order but no translational periodicity in their atomic organisation, and viruses, micro-organisms consisting of a protein shell, the capsid, that in most cases displays icosahedral symmetry. Group theory plays a crucial role in understanding their structures and their physical and geometrical properties. In this thesis new group theoretical methods are introduced, to characterise virus organisation and model structural transitions of icosahedral quasicrystals. It is shown that these problems can be described via the embedding of non-crystallographic groups into the point group of higher dimensional lattices. Indeed, the analysis of orbits of such embeddings, akin to the construction of quasicrystals via the cut-and-project method, provides a rigorous mathematical construction of finite nested point sets with non-crystallographic symmetry at each distinct radial level. In the case of icosahedral symmetry, it is shown that the point arrays thus obtained can be used to provide constraints on the geometry of viral capsids, encoding information on the organisation of the capsid proteins and the genomic material collectively. Moreover, structural transitions of icosahedral quasicrystals can be analysed in a group theoretical framework through continuous rotations in the higher dimensional space connecting distinct copies of the embedded icosahedral group, sharing a common maximal subgroup. These rotations induce in projection continuous transformations between aperiodic point sets that preserve the symmetry described by the common subgroup. Theoretical methods as well as applications are discussed, with emphasis on the computational and geometric aspect of group theory

IST Austria Thesis

We describe arrangements of three-dimensional spheres from a geometrical and topological point of view. Real data (fitting this setup) often consist of soft spheres which show certain degree of deformation while strongly packing against each other. In this context, we answer the following questions: If we model a soft packing of spheres by hard spheres that are allowed to overlap, can we measure the volume in the overlapped areas? Can we be more specific about the overlap volume, i.e. quantify how much volume is there covered exactly twice, three times, or k times? What would be a good optimization criteria that rule the arrangement of soft spheres while making a good use of the available space? Fixing a particular criterion, what would be the optimal sphere configuration? The first result of this thesis are short formulas for the computation of volumes covered by at least k of the balls. The formulas exploit information contained in the order-k Voronoi diagrams and its closely related Level-k complex. The used complexes lead to a natural generalization into poset diagrams, a theoretical formalism that contains the order-k and degree-k diagrams as special cases. In parallel, we define different criteria to determine what could be considered an optimal arrangement from a geometrical point of view. Fixing a criterion, we find optimal soft packing configurations in 2D and 3D where the ball centers lie on a lattice. As a last step, we use tools from computational topology on real physical data, to show the potentials of higher-order diagrams in the description of melting crystals. The results of the experiments leaves us with an open window to apply the theories developed in this thesis in real applications