scholarly journals Defects in the 3-dimensional toric code model form a braided fusion 2-category

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Liang Kong ◽  
Yin Tian ◽  
Zhi-Hao Zhang
Keyword(s):  
Topological Defects ◽  
Spatial Dimension ◽  
Topological Order ◽  
Mathematical Result ◽  
3 Dimensional ◽  
Code Model ◽  
Model Form ◽  
Degeneracy Condition ◽  
Toric Code

Abstract It was well known that there are e-particles and m-strings in the 3-dimensional (spatial dimension) toric code model, which realizes the 3-dimensional ℤ2 topological order. Recent mathematical result, however, shows that there are additional string-like topological defects in the 3-dimensional ℤ2 topological order. In this work, we construct all topological defects of codimension 2 and higher, and show that they form a braided fusion 2-category satisfying a braiding non-degeneracy condition.

scholarly journals Rényi entropy perspective on topological order in classical toric code models

2015 ◽  
Vol 92 (12) ◽  
Author(s):  
Johannes Helmes ◽  
Jean-Marie Stéphan ◽  
Simon Trebst
Keyword(s):  
Renyi Entropy ◽  
Rényi Entropy ◽  
Topological Order ◽  
Toric Code

scholarly journals LOCALIZED ENDOMORPHISMS IN KITAEV'S TORIC CODE ON THE PLANE

2011 ◽  
Vol 23 (04) ◽  
pp. 347-373 ◽  
Author(s):  
PIETER NAAIJKENS
Keyword(s):  
Representation Theory ◽  
Group Algebra ◽  
Ground State ◽  
Algebraic Approach ◽  
Quantum Spin ◽  
Spin Systems ◽  
Quantum Spin Systems ◽  
Quantum Double ◽  
Code Model ◽  
Toric Code

We consider various aspects of Kitaev's toric code model on a plane in the C*-algebraic approach to quantum spin systems on a lattice. In particular, we show that elementary excitations of the ground state can be described by localized endomorphisms of the observable algebra. The structure of these endomorphisms is analyzed in the spirit of the Doplicher–Haag–Roberts program (specifically, through its generalization to infinite regions as considered by Buchholz and Fredenhagen). Most notably, the statistics of excitations can be calculated in this way. The excitations can equivalently be described by the representation theory of [Formula: see text], i.e. Drinfel'd's quantum double of the group algebra of ℤ2.

Research on Parametric Digital Model Form Based on 3-Dimensional Body Measurement

2013 ◽  
Vol 706-708 ◽  
pp. 972-975
Author(s):  
Li Na Yang ◽  
Tong Zhang ◽  
An Fen Zhang
Keyword(s):  
Modeling Method ◽  
Digital Model ◽  
Design System ◽  
Body Measurement ◽  
3 Dimensional ◽  
Model Form ◽  
Transformation Relation ◽  
Garment Design

The modeling method of 3-dimensional garment model form is the key for the 3-dimensional garment design system. The modeling method of model form will directly have effects on the complexity and presentation effects of the latter design of garment styles. It is also closely related to the garment 3-dimensional al imitation and simulation as well as the transformation relation between 3-dimension and 2-dimension. Therefore, the modeling method of 3-dimensional garment model form is much worthy to study and has a good application prospect.

scholarly journals Spatio-temporal topological relationships between land parcels in cadastral database

2014 ◽  
Vol XL-6 ◽  
pp. 89-92
Author(s):  
W. Song ◽  
F. Zhang
Keyword(s):  
Spatial Dimension ◽  
Temporal Data ◽  
Time Dimension ◽  
3 Dimensional ◽  
Topological Relationships ◽  
Temporal Space ◽  
Temporal Relationships ◽  
Temporal Data Models ◽  
Spatio Temporal ◽  
Spatiotemporal Relationships

There are complex spatio-temporal relationships among cadastral entities. Cadastral spatio-temporal data model should not only describe the data structure of cadastral objects, but also express cadastral spatio-temporal relationships between cadastral objects. In the past, many experts and scholars have proposed a variety of cadastral spatio-temporal data models, but few of them concentrated on the representation of spatiotemporal relationships and few of them make systematic studies on spatiotemporal relationships between cadastral objects. The studies on spatio-temporal topological relationships are not abundant. In the paper, we initially review current approaches to the studies of spatio-temporal topological relationships, and argue that spatio-temporal topological relation is the combination of temporal topology on the time dimension and spatial topology on the spatial dimension. Subsequently, we discuss and develop an integrated representation of spatio-temporal topological relationships within a 3-dimensional temporal space. In the end, based on the semantics of spatiotemporal changes between land parcels, we conclude the possible spatio-temporal topological relations between land parcels, which provide the theoretical basis for creating, updating and maintaining of land parcels in the cadastral database.

scholarly journals Phase transition in a noisy Kitaev toric code model

2019 ◽  
Vol 99 (5) ◽  
Author(s):  
Mohammad Hossein Zarei ◽  
Afshin Montakhab
Keyword(s):  
Phase Transition ◽  
Code Model ◽  
Toric Code

scholarly journals Homological codes and abelian anyons

2019 ◽  
Vol 31 (10) ◽  
pp. 1950038
Author(s):  
Péter Vrana ◽  
Máté Farkas
Keyword(s):  
Homotopy Type ◽  
Ground States ◽  
Chain Complex ◽  
Cochain Complex ◽  
Locally Finite ◽  
Code Model ◽  
Cw Complex ◽  
Localized Excitations ◽  
Proper Homotopy ◽  
Toric Code

We study a generalization of Kitaev’s abelian toric code model defined on CW complexes. In this model, qudits are attached to [Formula: see text]-dimensional cells and the interaction is given by generalized star and plaquette operators. These are defined in terms of coboundary and boundary maps in the locally finite cellular cochain complex and the cellular chain complex. We find that the set of energy-minimizing ground states and the types of charges carried by certain localized excitations depends only on the proper homotopy type of the CW complex. As an application, we show that the homological product of a CSS code with the infinite toric code has excitations with abelian anyonic statistics.

scholarly journals Dispersive toric code model with fusion and defusion

2020 ◽  
Vol 101 (11) ◽  
Author(s):  
Bruno Nachtergaele ◽  
Nicholas E. Sherman
Keyword(s):  
Code Model ◽  
Toric Code
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