A Computational Non-commutative Geometry Program for Disordered Topological Insulators

Nonfiction, Science & Nature, Science, Physics, Solid State Physics, Mathematical Physics
Cover of the book A Computational Non-commutative Geometry Program for Disordered Topological Insulators by Emil Prodan, Springer International Publishing
View on Amazon View on AbeBooks View on Kobo View on B.Depository View on eBay View on Walmart
Author: Emil Prodan ISBN: 9783319550237
Publisher: Springer International Publishing Publication: March 17, 2017
Imprint: Springer Language: English
Author: Emil Prodan
ISBN: 9783319550237
Publisher: Springer International Publishing
Publication: March 17, 2017
Imprint: Springer
Language: English

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.

View on Amazon View on AbeBooks View on Kobo View on B.Depository View on eBay View on Walmart

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.

More books from Springer International Publishing

Cover of the book The Eurasian Wheat Belt and Food Security by Emil Prodan
Cover of the book Total Collapse: The Case Against Responsibility and Morality by Emil Prodan
Cover of the book Advanced Information Technology, Services and Systems by Emil Prodan
Cover of the book Bayesian Statistics from Methods to Models and Applications by Emil Prodan
Cover of the book Modeling Life by Emil Prodan
Cover of the book Pancreatic Islet Isolation by Emil Prodan
Cover of the book Hepatic Critical Care by Emil Prodan
Cover of the book Innovative Mobile and Internet Services in Ubiquitous Computing by Emil Prodan
Cover of the book Communications and Networking by Emil Prodan
Cover of the book Pattern Recognition by Emil Prodan
Cover of the book Common Diagnostic Pitfalls in Thyroid Cytopathology by Emil Prodan
Cover of the book Speech Recognition Using Articulatory and Excitation Source Features by Emil Prodan
Cover of the book Informatics in Control, Automation and Robotics 12th International Conference, ICINCO 2015 Colmar, France, July 21-23, 2015 Revised Selected Papers by Emil Prodan
Cover of the book Cyber Security Intelligence and Analytics by Emil Prodan
Cover of the book Statistical Applications from Clinical Trials and Personalized Medicine to Finance and Business Analytics by Emil Prodan
We use our own "cookies" and third party cookies to improve services and to see statistical information. By using this website, you agree to our Privacy Policy