A Computational Non commutative Geometry Program for Disordered Topological Insulators 1st Edition by Emil Prodan ISBN 3319550225 978-3319550220

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Product details:

ISBN 10: 3319550225
ISBN 13: 978-3319550220
Author: Emil Prodan

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.

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
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
Applications II: Topological Invariants
Class AIII in d = 1
Class A in d = 2
Class AIII in d = 3
References

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Emil Prodan,Computational,Non commutative,Geometry,Topological,Insulators