東華大學圖書館 |

Language: English

Help

回圖書館首頁

手機版館藏查詢

Back

Switch To: Labeled | MARC Mode | ISBD

A computational non-commutative geom...

Prodan, Emil.

Linked to FindBook

Google Book

Amazon

博客來

A computational non-commutative geometry program for disordered topological insulators

Record Type:	Electronic resources : Monograph/item
Title/Author:	A computational non-commutative geometry program for disordered topological insulators/ by Emil Prodan.
Author:	Prodan, Emil.
Published:	Cham :Springer International Publishing : : 2017.,
Description:	x, 118 p. :ill., digital ;24 cm.
[NT 15003449]:	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.
Contained By:	Springer eBooks
Subject:	Noncommutative algebras. -
Online resource:	http://dx.doi.org/10.1007/978-3-319-55023-7
ISBN:	9783319550237

A computational non-commutative geometry program for disordered topological insulators
Prodan, Emil.

A computational non-commutative geometry program for disordered topological insulators[electronic resource] /by Emil Prodan. - Cham :Springer International Publishing :2017. - x, 118 p. :ill., digital ;24 cm. - SpringerBriefs in mathematical physics,v.232197-1757 ;. - SpringerBriefs in mathematical physics ;v.23..

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.

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.

ISBN: 9783319550237

Standard No.: 10.1007/978-3-319-55023-7doiSubjects--Topical Terms:

609929
Noncommutative algebras.

LC Class. No.: QA251.4

Dewey Class. No.: 512.55

A computational non-commutative geometry program for disordered topological insulators
LDR:04234nmm a2200325 a 4500 001 2093107
003 DE-He213
005 20170317163820.0
006 m d
007 cr nn 008maaau
008 171117s2017 gw s 0 eng d
020 $a 9783319550237 $q (electronic bk.)
020 $a 9783319550220 $q (paper)
024 7 $a 10.1007/978-3-319-55023-7 $2 doi
035 $a 978-3-319-55023-7
040 $a GP $c GP
041 0 $a eng
050 4 $a QA251.4
072 7 $a PHU $2 bicssc
072 7 $a SCI040000 $2 bisacsh
082 0 4 $a 512.55 $2 23
090 $a QA251.4 $b .P964 2017
100 1 $a Prodan, Emil. $3 2183213
245 1 2 $a A computational non-commutative geometry program for disordered topological insulators $h [electronic resource] / $c by Emil Prodan.
260 $a Cham : $b Springer International Publishing : $b Imprint: Springer, $c 2017.
300 $a x, 118 p. : $b ill., digital ; $c 24 cm.
490 1 $a SpringerBriefs in mathematical physics, $x 2197-1757 ; $v v.23
505 0 $a 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.
520 $a 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.
650 0 $a Noncommutative algebras. $3 609929
650 0 $a Mathematical physics. $3 516853
650 1 4 $a Physics. $3 516296
650 2 4 $a Mathematical Methods in Physics. $3 890898
650 2 4 $a Mathematical Physics. $3 1542352
650 2 4 $a Condensed Matter Physics. $3 1067080
650 2 4 $a K-Theory. $3 897432
650 2 4 $a Functional Analysis. $3 893943
710 2 $a SpringerLink (Online service) $3 836513
773 0 $t Springer eBooks
830 0 $a SpringerBriefs in mathematical physics ; $v v.23. $3 3227645
856 4 0 $u http://dx.doi.org/10.1007/978-3-319-55023-7
950 $a Physics and Astronomy (Springer-11651)