A Computational Non-commutative Geometry Program for Disordered Topological Insulators - Emil Prodan

Blick ins Buch

A Computational Non-commutative Geometry Program for Disordered Topological Insulators (eBook)

Emil Prodan (Autor)

eBook Download: PDF

2017 | 1st ed. 2017
X, 118 Seiten
Springer International Publishing (Verlag)
978-3-319-55023-7 (ISBN)

Lese- und Medienproben

Ebook-Leseprobe (PDF)

This work presents a computational program based on the principles of non-commutative 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' 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 non-commutative 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 finite-volume approximation of the non-commutative 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 canonical-finite 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.

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 Non-Commutative Brillouin Torus.- Disorder Configurations and Associated Dynamical Systems.- The Algebra of Covariant Physical Observables.- Fourier Calculus.- Differential Calculus.- Smooth Sub-Algebra.- Sobolev Spaces.- Magnetic Derivations.- Physics Formulas.- The Auxiliary C*-Algebras.- Periodic Disorder Configurations.- The Periodic Approximating Algebra.- Finite-Volume Disorder Configurations.- The Finite-Volume Approximating Algebra.- Approximate Differential Calculus.- Bloch Algebras.- Canonical Finite-Volume 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 Non-Commutative Kubo Formula.- The Integer Quantum Hall Effect.- Chern Insulators.- Error Bounds for Non-Smooth Correlations.- The Aizenman-Molchanov 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.

Erscheint lt. Verlag	17.3.2017
Reihe/Serie	SpringerBriefs in Mathematical Physics
Reihe/Serie	SpringerBriefs in Physics
Zusatzinfo	X, 118 p. 19 illus. in color.
Verlagsort	Cham
Sprache	englisch
Themenwelt	Mathematik / Informatik ► Mathematik
	Naturwissenschaften ► Physik / Astronomie ► Allgemeines / Lexika
	Technik
Schlagworte	Aizenman-Molchanov bound • Bloch algebras • canonical finite-volume algorithm • Chern insulators • disordered topological insulators • homogenous disordered crystals • integer quantum Hall effect • non-commutative Brillouin torus • non-commutative Kubo formula • Sobolev spaces • topological invariants
ISBN-10	3-319-55023-3 / 3319550233
ISBN-13	978-3-319-55023-7 / 9783319550237

Haben Sie eine Frage zum Produkt?

PDF (Wasserzeichen)
Größe: 3,4 MB

DRM: Digitales Wasserzeichen
Dieses eBook enthält ein digitales Wasserzeichen und ist damit für Sie personalisiert. Bei einer missbräuchlichen Weitergabe des eBooks an Dritte ist eine Rückverfolgung an die Quelle möglich.

Dateiformat: PDF (Portable Document Format)
Mit einem festen Seitenlayout eignet sich die PDF besonders für Fachbücher mit Spalten, Tabellen und Abbildungen. Eine PDF kann auf fast allen Geräten angezeigt werden, ist aber für kleine Displays (Smartphone, eReader) nur eingeschränkt geeignet.

Systemvoraussetzungen:
PC/Mac: Mit einem PC oder Mac können Sie dieses eBook lesen. Sie benötigen dafür einen PDF-Viewer - z.B. den Adobe Reader oder Adobe Digital Editions.
eReader: Dieses eBook kann mit (fast) allen eBook-Readern gelesen werden. Mit dem amazon-Kindle ist es aber nicht kompatibel.
Smartphone/Tablet: Egal ob Apple oder Android, dieses eBook können Sie lesen. Sie benötigen dafür einen PDF-Viewer - z.B. die kostenlose Adobe Digital Editions-App.

Buying eBooks from abroad
For tax law reasons we can sell eBooks just within Germany and Switzerland. Regrettably we cannot fulfill eBook-orders from other countries.

Print-Ausgabe

Buch | Softcover

58,84 €