The Resource A Computational Noncommutative Geometry Program for Disordered Topological Insulators, by Emil Prodan, (electronic resource)
A Computational Noncommutative Geometry Program for Disordered Topological Insulators, by Emil Prodan, (electronic resource)
 Summary
 This work presents a computational program based on the principles of noncommutative geometry and showcases several applications to topological insulators. Noncommutative geometry has been originally proposed by Jean Bellissard as a theoretical framework for the investigation of homogeneous condensed matter systems. Recently, this approach has been successfully applied to topological insulators, where it facilitated many rigorous results concerning the stability of the topological invariants against disorder. In the first part of the book the notion of a homogeneous material is introduced and the class of disordered crystals defined together with the classification table, which conjectures all topological phases from this class. The manuscript continues with a discussion of electrons{u2019} dynamics in disordered crystals and the theory of topological invariants in the presence of strong disorder is briefly reviewed. It is shown how all this can be captured in the language of noncommutative geometry using the concept of noncommutative Brillouin torus, and a list of known formulas for various physical response functions is presented. In the second part, auxiliary algebras are introduced and a canonical finitevolume approximation of the noncommutative Brillouin torus is developed. Explicit numerical algorithms for computing generic correlation functions are discussed. In the third part upper bounds on the numerical errors are derived and it is proved that the canonicalfinite volume approximation converges extremely fast to the thermodynamic limit. Convergence tests and various applications concludes the presentation. The book is intended for graduate students and researchers in numerical and mathematical physics
 Language
 eng
 Extent
 X, 118 p. 19 illus. in color.
 Contents

 Disordered Topological Insulators: A Brief Introduction
 Homogeneous Materials
 Homogeneous Disordered Crystals
 Classification of Homogenous Disordered Crystals
 Electron Dynamics: Concrete Physical Models
 Notations and Conventions
 Physical Models
 Disorder Regimes
 Topological Invariants
 The NonCommutative Brillouin Torus
 Disorder Configurations and Associated Dynamical Systems
 The Algebra of Covariant Physical Observables
 Fourier Calculus
 Differential Calculus
 Smooth SubAlgebra
 Sobolev Spaces
 Magnetic Derivations
 Physics Formulas
 The Auxiliary C*Algebras
 Periodic Disorder Configurations
 The Periodic Approximating Algebra
 FiniteVolume Disorder Configurations
 The FiniteVolume Approximating Algebra
 Approximate Differential Calculus
 Bloch Algebras
 Canonical FiniteVolume Algorithm
 General Picture
 Explicit Computer Implementation
 Error Bounds for Smooth Correlations
 Assumptions
 First Round of Approximations
 Second Round of Approximations
 Overall Error Bounds
 Applications: Transport Coefficients at Finite Temperature
 The NonCommutative Kubo Formula
 The Integer Quantum Hall Effect
 Chern Insulators
 Error Bounds for NonSmooth Correlations
 The AizenmanMolchanov Bound
 Assumptions
 Derivation of Error Bounds
 Applications II: Topological Invariants
 Class AIII in d = 1
 Class A in d = 2
 Class AIII in d = 3
 References
 Isbn
 9783319550237
 Label
 A Computational Noncommutative Geometry Program for Disordered Topological Insulators
 Title
 A Computational Noncommutative Geometry Program for Disordered Topological Insulators
 Statement of responsibility
 by Emil Prodan
 Language
 eng
 Summary
 This work presents a computational program based on the principles of noncommutative geometry and showcases several applications to topological insulators. Noncommutative geometry has been originally proposed by Jean Bellissard as a theoretical framework for the investigation of homogeneous condensed matter systems. Recently, this approach has been successfully applied to topological insulators, where it facilitated many rigorous results concerning the stability of the topological invariants against disorder. In the first part of the book the notion of a homogeneous material is introduced and the class of disordered crystals defined together with the classification table, which conjectures all topological phases from this class. The manuscript continues with a discussion of electrons{u2019} dynamics in disordered crystals and the theory of topological invariants in the presence of strong disorder is briefly reviewed. It is shown how all this can be captured in the language of noncommutative geometry using the concept of noncommutative Brillouin torus, and a list of known formulas for various physical response functions is presented. In the second part, auxiliary algebras are introduced and a canonical finitevolume approximation of the noncommutative Brillouin torus is developed. Explicit numerical algorithms for computing generic correlation functions are discussed. In the third part upper bounds on the numerical errors are derived and it is proved that the canonicalfinite volume approximation converges extremely fast to the thermodynamic limit. Convergence tests and various applications concludes the presentation. The book is intended for graduate students and researchers in numerical and mathematical physics
 Image bit depth
 0
 Literary form
 non fiction
 Series statement
 SpringerBriefs in Mathematical Physics,
 Series volume
 23
 Label
 A Computational Noncommutative Geometry Program for Disordered Topological Insulators, by Emil Prodan, (electronic resource)
 Antecedent source
 mixed
 Carrier category
 online resource
 Carrier category code
 cr
 Carrier MARC source
 rdacarrier
 Color
 not applicable
 Content category
 text
 Content type code
 txt
 Content type MARC source
 rdacontent
 Contents
 Disordered Topological Insulators: A Brief Introduction  Homogeneous Materials  Homogeneous Disordered Crystals  Classification of Homogenous Disordered Crystals  Electron Dynamics: Concrete Physical Models  Notations and Conventions  Physical Models  Disorder Regimes  Topological Invariants  The NonCommutative Brillouin Torus  Disorder Configurations and Associated Dynamical Systems  The Algebra of Covariant Physical Observables  Fourier Calculus  Differential Calculus  Smooth SubAlgebra  Sobolev Spaces  Magnetic Derivations  Physics Formulas  The Auxiliary C*Algebras  Periodic Disorder Configurations  The Periodic Approximating Algebra  FiniteVolume Disorder Configurations  The FiniteVolume Approximating Algebra  Approximate Differential Calculus  Bloch Algebras  Canonical FiniteVolume Algorithm  General Picture  Explicit Computer Implementation  Error Bounds for Smooth Correlations  Assumptions  First Round of Approximations  Second Round of Approximations  Overall Error Bounds  Applications: Transport Coefficients at Finite Temperature  The NonCommutative Kubo Formula  The Integer Quantum Hall Effect  Chern Insulators  Error Bounds for NonSmooth Correlations  The AizenmanMolchanov Bound  Assumptions  Derivation of Error Bounds  Applications II: Topological Invariants  Class AIII in d = 1  Class A in d = 2  Class AIII in d = 3  References
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 http://library.link/vocab/discovery_link
 {'f': 'http://opac.lib.rpi.edu/record=b4257879'}
 Extent
 X, 118 p. 19 illus. in color.
 File format
 multiple file formats
 Form of item
 electronic
 Isbn
 9783319550237
 Level of compression
 uncompressed
 Media category
 computer
 Media MARC source
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<div class="citation" vocab="http://schema.org/"><i class="fa faexternallinksquare fafw"></i> Data from <span resource="http://link.lib.rpi.edu/portal/AComputationalNoncommutativeGeometryProgram/qgLWdcrHdm4/" typeof="WorkExample http://bibfra.me/vocab/lite/Item"><span property="name http://bibfra.me/vocab/lite/label"><a href="http://link.lib.rpi.edu/portal/AComputationalNoncommutativeGeometryProgram/qgLWdcrHdm4/">A Computational Noncommutative Geometry Program for Disordered Topological Insulators, by Emil Prodan, (electronic resource)</a></span>  <span property="offers" typeOf="Offer"><span property="offeredBy" typeof="Library ll:Library" resource="http://link.lib.rpi.edu/"><span property="name http://bibfra.me/vocab/lite/label"><a property="url" href="http://link.lib.rpi.edu/">Rensselaer Libraries</a></span></span></span></span></div>