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A Computational Non-commutative Geometry Program for Disordered Topological Insulators

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 condens...

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Main Author: Prodan, Emil. (Author, http://id.loc.gov/vocabulary/relators/aut)
Corporate Author: SpringerLink (Online service)
Language:English
Published: Cham : Springer International Publishing : Imprint: Springer, 2017.
Edition:1st ed. 2017.
Series:SpringerBriefs in Mathematical Physics, 23
Subjects:
Physics.
Mathematical physics.
Condensed matter.
K-theory.
Functional analysis.
Mathematical Methods in Physics.
Mathematical Physics.
Condensed Matter Physics.
K-Theory.
Functional Analysis.
Online Access:https://doi.org/10.1007/978-3-319-55023-7
Table of 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 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.

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