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A Computational Non-commutative Geometry Program for Disordered Topological Insulators

  • Emil Prodan

Part of the SpringerBriefs in Mathematical Physics book series (BRIEFSMAPHY, volume 23)

Table of contents

  1. Front Matter
    Pages i-x
  2. Emil Prodan
    Pages 25-48
  3. Emil Prodan
    Pages 49-61
  4. Emil Prodan
    Pages 63-69
  5. Emil Prodan
    Pages 71-77
  6. Emil Prodan
    Pages 99-107
  7. Emil Prodan
    Pages 109-118

About this book

Introduction

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.

Keywords

Bloch algebras disordered topological insulators homogenous disordered crystals non-commutative Brillouin torus Sobolev spaces canonical finite-volume algorithm non-commutative Kubo formula integer quantum Hall effect Chern insulators Aizenman-Molchanov bound Topological invariants

Authors and affiliations

  • Emil Prodan
    • 1
  1. 1.Department of Physics & Department of Mathematical SciencesYeshiva UniversityNew YorkUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-55023-7
  • Copyright Information The Author(s) 2017
  • Publisher Name Springer, Cham
  • eBook Packages Physics and Astronomy
  • Print ISBN 978-3-319-55022-0
  • Online ISBN 978-3-319-55023-7
  • Series Print ISSN 2197-1757
  • Series Online ISSN 2197-1765
  • Buy this book on publisher's site