# A Computational Non-commutative Geometry Program for Disordered Topological InsulatorsNon-commutative Brillouin Torus

A Computational Non-commutative Geometry Program for Disordered Topological Insulators:... [In this Chapter, we review the fundamental theoretical tools, starting with the space of disordered configurations and its associated dynamical systems, the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^*$$\end{document}-algebra \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal A}_d$$\end{document} of the physical observables, together with its Fourier and differential calculus. The latter is provided by a set of commuting derivations \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial$$\end{document} and a trace \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal T}$$\end{document}. The triple \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\mathcal A}_d,\partial ,{\mathcal T})$$\end{document} defines a non-commutative manifold known as the non-commutative Brillouin torus. We reformulate the topological invariants and other response functions in this new framework. We also introduce the magnetic derivations and investigate the behavior of the correlation functions w.r.t. the magnetic fields. This Chapter also fixes the notation and defines the precise settings for the rest of our calculations.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# A Computational Non-commutative Geometry Program for Disordered Topological InsulatorsNon-commutative Brillouin Torus

Part of the SpringerBriefs in Mathematical Physics Book Series (volume 23)
23 pages

/lp/springer-journals/a-computational-non-commutative-geometry-program-for-disordered-n0bqjir3ig

# References (39)

Publisher
Springer International Publishing
ISBN
978-3-319-55022-0
Pages
25 –48
DOI
10.1007/978-3-319-55023-7_3
Publisher site
See Chapter on Publisher Site

### Abstract

[In this Chapter, we review the fundamental theoretical tools, starting with the space of disordered configurations and its associated dynamical systems, the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^*$$\end{document}-algebra \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal A}_d$$\end{document} of the physical observables, together with its Fourier and differential calculus. The latter is provided by a set of commuting derivations \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial$$\end{document} and a trace \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal T}$$\end{document}. The triple \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\mathcal A}_d,\partial ,{\mathcal T})$$\end{document} defines a non-commutative manifold known as the non-commutative Brillouin torus. We reformulate the topological invariants and other response functions in this new framework. We also introduce the magnetic derivations and investigate the behavior of the correlation functions w.r.t. the magnetic fields. This Chapter also fixes the notation and defines the precise settings for the rest of our calculations.]

Published: Mar 18, 2017

Keywords: Fourier Coefficient; Gibbs Measure; Functional Calculus; Differential Calculus; Bernoulli Shift