844 research outputs found
Symmetries of Riemann surfaces and magnetic monopoles
This thesis studies, broadly, the role of symmetry in elucidating structure. In particular, I investigate the role that automorphisms of algebraic curves play in three specific contexts; determining the orbits of theta characteristics, influencing the geometry of the highly-symmetric Bring’s curve, and in constructing magnetic monopole solutions. On theta characteristics, I show how to turn questions on the existence of invariant characteristics into questions of group cohomology, compute comprehensive tables of orbit decompositions for curves of genus 9 or less, and prove results on the existence of infinite families of curves with invariant characteristics. On Bring’s curve, I identify key points with geometric significance on the curve, completely determine the structure of the quotients by subgroups of automorphisms, finding new elliptic curves in the process, and identify the unique invariant theta characteristic on the curve. With respect to monopoles, I elucidate the role that the Hitchin conditions play in determining monopole spectral curves, the relation between these conditions and the automorphism group of the curve, and I develop the theory of computing Nahm data of symmetric monopoles. As such I classify all 3-monopoles whose Nahm data may be solved for in terms of elliptic functions
Digital-analog quantum computing and algorithms
117 p.Esta tesis profundiza en el desarrollo e implementación de algoritmos cuánticos utilizando el paradigma digital-analógico cuántico computacional (DAQC). Proporciona un análisis comparativo del rendimiento de DAQC frente a los enfoques digitales tradicionales, particularmente en presencia de fuentes de ruido de los dispositivos cuánticos de escala intermedia ruidosos (NISQ) actuales. El paradigma DAQC combina las fortalezas de la computación cuántica digital y analógica, y ofrece mayor eficiencia y precisión para implementar algoritmos cuánticos en hardware real. La tesis se centra en la comparación de cuatro algoritmos cuánticos utilizando enfoques digitales y analógico-digitales, y los resultados muestran ventajas significativas a favor de estos últimos. Además, la tesis investiga el efecto de resonancia cruzada para lograr simulaciones hamiltonianas eficientes y de alta precisión. Los hallazgos indican que el paradigma digital-analógico es prometedor para aplicaciones prácticas de computación cuántica. Su capacidad para ofrecer mayor eficiencia y precisión en la implementación de algoritmos cuánticos en hardware real es una ventaja significativa sobre los enfoques digitales tradicionales
Searching for quasicrystals in block copolymer phase separation
Quasicrystals are materials that display long-range order despite lacking translational periodicity. Despite it has been 41 years since its discovery, the stability of quasicrystals remains a perplexing enigma for scientists. Initially discovered in metals, these structures also appear in soft matter systems like block copolymers.
Polymer chains that contain two or more different types of monomer blocks joined together are block copolymers. They can microphase separate to form different patterns and structures in their morphology, including quasicrystals. Different morphological structures are formed depending on the block lengths and the interaction strengths. This soft matter system, akin to a designer material, can be adjusted to autonomously self-assemble into various intriguing morphologies, including quasicrystals.
This thesis proposes two methods for designing block copolymers with the potential to self-assemble into quasicrystals. The stability of morphologies in block copolymers can be determined using well-established phase separation theories: weak segregation theory (WST), self-consistent field theory (SCFT), and strong segregation theory (SST). In this study, we present design criteria for two categories of block copolymers: two-component alternating linear chains and ABC star terpolymers within the context of weak segregation limit. These criteria guide the self-assembly of structures with length-scale ratios conducive to quasicrystals. The second half of the thesis presents a novel framework in strong segregation limit where morphologies in star terpolymers are compared with tiling patterns to study their stability. This framework can incorporate periodic tilings and periodic approximants of aperiodic tilings and develop the phase space for ABC star terpolymers.
The overarching aim is to make experimentally valid predictions on polymer architectures that could lead to stable 2- and 3-dimensional quasicrystals or other structures. Using the two methodologies, we find experimentally feasible composition ranges in the block copolymers we are considering in this thesis that can potentially form quasicrystal or other interesting, complex morphologies
Boundary statistics for the six-vertex model with DWBC
We study the behavior of configurations in the symmetric six-vertex model
with weights in the square with Domain Wall Boundary
Conditions as . We prove that when
, configurations near the boundary have
fluctuations of order and are asymptotically described by the
GUE-corners process of the random matrix theory. On the other hand, when
, the fluctuations are of finite order and configurations are
asymptotically described by the stochastic six-vertex model in a quadrant. In
the special case (which implies ), the limit is expressed as
the -exchangeable random permutation of infinitely many letters, distributed
according to the infinite Mallows measure.Comment: 94 pages, 13 figure
Symmetry and topology of hyperbolic Haldane models
Particles hopping on a two-dimensional hyperbolic lattice feature
unconventional energy spectra and wave functions that provide a largely
uncharted platform for topological phases of matter beyond the Euclidean
paradigm. Using real-space topological markers as well as Chern numbers defined
in the higher-dimensional momentum space of hyperbolic band theory, we
construct and investigate hyperbolic Haldane models, which are generalizations
of Haldane's honeycomb-lattice model to various hyperbolic lattices. We present
a general framework to characterize point-group symmetries in hyperbolic
tight-binding models, and use this framework to constrain the multiple first
and second Chern numbers in momentum space. We observe several topological gaps
characterized by first Chern numbers of value and . The momentum-space
Chern numbers respect the predicted symmetry constraints and agree with
real-space topological markers, indicating a direct connection to observables
such as the number of chiral edge modes. With our large repertoire of models,
we further demonstrate that the topology of hyperbolic Haldane models is
trivialized for lattices with strong negative curvature.Comment: main text (14 pages with 7 figures and 2 tables) + appendices (28
pages with 10 figures and 2 tables) + bibliography (2 pages
The linear system for Sudoku and a fractional completion threshold
We study a system of linear equations associated with Sudoku latin squares.
The coefficient matrix of the normal system has various symmetries arising
from Sudoku. From this, we find the eigenvalues and eigenvectors of , and
compute a generalized inverse. Then, using linear perturbation methods, we
obtain a fractional completion guarantee for sufficiently large and sparse
rectangular-box Sudoku puzzles
Quantum toroidal algebras and solvable structures in gauge/string theory
This is a review article on the quantum toroidal algebras, focusing on their
roles in various solvable structures of 2d conformal field theory,
supersymmetric gauge theory, and string theory. Using -algebras as
our starting point, we elucidate the interconnection of affine Yangians,
quantum toroidal algebras, and double affine Hecke algebras.
Our exploration delves into the representation theory of the quantum toroidal
algebra of in full detail, highlighting its connections to
partitions, -algebras, Macdonald functions, and the notion of
intertwiners. Further, we also discuss integrable models constructed on Fock
spaces and associated -matrices, both for the affine Yangian and
the quantum toroidal algebra of .
The article then demonstrates how quantum toroidal algebras serve as a
unifying algebraic framework that bridges different areas in physics. Notably,
we cover topological string theory and supersymmetric gauge theories with eight
supercharges, incorporating the AGT duality. Drawing upon the representation
theory of the quantum toroidal algebra of , we provide a
rather detailed review of its role in the algebraic formulations of topological
vertex and -characters. Additionally, we briefly touch upon the corner
vertex operator algebras and quiver quantum toroidal algebras.Comment: 151+42 pages, Comments are welcom
Hypertiling -- a high performance Python library for the generation and visualization of hyperbolic lattices
Hypertiling is a high-performance Python library for the generation and
visualization of regular hyperbolic lattices embedded in the Poincar\'e disk
model. Using highly optimized, efficient algorithms, hyperbolic tilings with
millions of vertices can be created in a matter of minutes on a single
workstation computer. Facilities including computation of adjacent vertices,
dynamic lattice manipulation, refinements, as well as powerful plotting and
animation capabilities are provided to support advanced uses of hyperbolic
graphs. In this manuscript, we present a comprehensive exploration of the
package, encompassing its mathematical foundations, usage examples,
applications, and a detailed description of its implementation.Comment: 52 pages, 20 figure
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