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Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics 1st ed. 2016 [Hardback]

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  • Formāts: Hardback, 204 pages, height x width: 235x155 mm, weight: 4675 g, 1 Illustrations, black and white; XXII, 204 p. 1 illus., 1 Hardback
  • Sērija : Mathematical Physics Studies
  • Izdošanas datums: 16-Feb-2016
  • Izdevniecība: Springer International Publishing AG
  • ISBN-10: 3319293508
  • ISBN-13: 9783319293509
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Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics 1st ed. 2016Palielināt
  • Formāts: Hardback, 204 pages, height x width: 235x155 mm, weight: 4675 g, 1 Illustrations, black and white; XXII, 204 p. 1 illus., 1 Hardback
  • Sērija : Mathematical Physics Studies
  • Izdošanas datums: 16-Feb-2016
  • Izdevniecība: Springer International Publishing AG
  • ISBN-10: 3319293508
  • ISBN-13: 9783319293509
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9783319293509.html
Keywords:
This monograph offers an overview of rigorous results on fermionic topological insulators from the complex classes, namely, those without symmetries or with just a chiral symmetry. Particular focus is on the stability of the topological invariants in the presence of strong disorder, on the interplay between the bulk and boundary invariants and on their dependence on magnetic fields.The first part presents motivating examples and the conjectures put forward by the physics community, together with a brief review of the experimental achievements. The second part develops an operator algebraic approach for the study of disordered topological insulators. This leads naturally to the use of analytical tools from K-theory and non-commutative geometry, such as cyclic cohomology, quantized calculus with Fredholm modules and index pairings. New results include a generalized Streda formula and a proof of the delocalized nature of surface states in topological insulators with non-trivial inv

ariants. The concluding chapter connects the invariants to measurable quantities and thus presents a refined physical characterization of the complex topological insulators.This book is intended for advanced students in mathematical physics and researchers alike.

Illustration of key concepts in dimension d = 1.- Topological solid state systems: conjectures, experiments and models.- Observables algebras for solid state systems.- K-theory for topological solid state systems.- The topological invariants and their interrelations.- Index theorems for solid state systems.- Invariants as measurable quantities.
1 Illustration of Key Concepts in Dimension d = 1
1(18)
1.1 Periodic Hamiltonian and Its Topological Invariant
1(3)
1.2 Edge States and Bulk-Boundary Correspondence
4(1)
1.3 Why Use K-Theory?
5(5)
1.4 Why Use Non-commutative Geometry?
10(1)
1.5 Disordered Hamiltonian
11(2)
1.6 Why Use Operator Algebras?
13(2)
1.7 Why Use Non-commutative Analysis Tools?
15(2)
1.8 Why Prove an Index Theorem?
17(1)
1.9 Can the Invariants be Measured?
18(1)
2 Topological Solid State Systems: Conjectures, Experiments and Models
19(36)
2.1 The Classification Table
19(3)
2.2 The Unitary Class
22(15)
2.2.1 General Characterization
22(6)
2.2.2 Experimental Achievements
28(1)
2.2.3 Conventions on Clifford Representations
29(2)
2.2.4 Bulk-Boundary Correspondence in a Periodic Unitary Model
31(6)
2.3 The Chiral Unitary Class
37(9)
2.3.1 General Characterization
38(3)
2.3.2 Experimental Achievements
41(2)
2.3.3 Bulk-Boundary Correspondence in a Periodic Chiral Model
43(3)
2.4 Main Hypotheses on the Hamiltonians
46(9)
2.4.1 The Probability Space of Disorder Configurations
46(1)
2.4.2 The Bulk Hamiltonians
47(4)
2.4.3 The Half-space and Boundary Hamiltonians
51(4)
3 Observables Algebras for Solid State Systems
55(30)
3.1 The Algebra of Bulk Observables
55(8)
3.1.1 The Disordered Non-commutative Torus
55(4)
3.1.2 Covariant Representations in the Landau Gauge
59(2)
3.1.3 Covariant Representations in the Symmetric Gauge
61(1)
3.1.4 The Algebra Elements Representing the Hamiltonians
62(1)
3.2 The Algebras of Half-Space and Boundary Observables
63(6)
3.2.1 Definition of the Algebras and Basic Properties
63(2)
3.2.2 The Exact Sequence Connecting Bulk and Boundary
65(1)
3.2.3 The Toeplitz Extension of Pimsner and Voiculescu
65(2)
3.2.4 Half-Space Representations
67(2)
3.2.5 Algebra Elements Representing Half-Space Hamiltonians
69(1)
3.3 The Non-commutative Analysis Tools
69(13)
3.3.1 The Fourier Calculus
69(2)
3.3.2 Non-commutative Derivations and Integrals
71(3)
3.3.3 The Smooth Sub-algebras and the Sobolev Spaces
74(7)
3.3.4 Derivatives with Respect to the Magnetic Field
81(1)
3.4 The Exact Sequence of Periodically Driven Systems
82(3)
4 K-Theory for Topological Solid State Systems
85(28)
4.1 Review of Key Elements of k-Theory
85(12)
4.1.1 Definition and Characterization of K0 Group
86(3)
4.1.2 Definition and Characterization of k1 Group
89(1)
4.1.3 The Six-Term Exact Sequence
90(3)
4.1.4 Suspension and Bott Periodicity
93(2)
4.1.5 The Inverse of the Suspension Map
95(2)
4.2 The k-Groups of the Algebras of Physical Observables
97(8)
4.2.1 The Pimsner-Voiculescu Sequence and Its Implications
97(3)
4.2.2 The Inverse of the Index Map
100(1)
4.2.3 The Generators of the k-Groups
101(4)
4.3 The Connecting Maps for Solid State Systems
105(8)
4.3.1 The Exponential Map for the Bulk-Boundary Correspondence
105(1)
4.3.2 The Index Map for the Bulk-Boundary Correspondence
106(3)
4.3.3 The Bott Map of the Fermi Projection
109(1)
4.3.4 The k-Theory of Periodically Driven Systems
110(3)
5 The Topological Invariants and Their Interrelations
113(32)
5.1 Notions of Cyclic Cohomology
113(3)
5.2 Bulk Topological Invariants Defined
116(2)
5.3 Boundary Topological Invariants Defined
118(4)
5.4 Suspensions and the Volovik-Essin-Gurarie Invariants
122(6)
5.5 Duality of Pairings and Bulk-Boundary Correspondence
128(6)
5.6 Generalized Streda Formulas
134(6)
5.7 The Range of the Pairings and Higher Gap Labelling
140(5)
6 Index Theorems for Solid State Systems
145(28)
6.1 Pairing k-Theory with Fredholm Modules
145(2)
6.2 Fredholm Modules for Solid State Systems
147(8)
6.3 Equality Between Connes-Chern and Chern Cocycles
155(6)
6.4 Key Geometric Identities
161(4)
6.5 Stability of Strong Bulk Invariants Under Strong Disorder
165(4)
6.6 Delocalization of the Boundary States
169(4)
7 Invariants as Measurable Quantities
173(20)
7.1 Transport Coefficients of Homogeneous Solid State Systems
173(3)
7.2 Topological Insulators from Class A in d = 2, 3 and 4
176(3)
7.3 Topological Insulators from Class AIII in d = 1, 2 and 3
179(4)
7.4 Surface IQHE for Exact and Approximately Chiral Systems
183(2)
7.5 Virtual Topological Insulators
185(2)
7.6 Quantized Electric Polarization
187(3)
7.7 Boundary Phenomena for Periodically Driven Systems
190(1)
7.8 The Magneto-Electric Response in d = 3
191(2)
References 193(10)
Index 203
Hermann Schulz-Baldes has been professor at the Department of Mathematics of the University of Erlangen since 2004. Before this he received his PhD from the University of Toulouse under the supervision of Jean Bellissard, and he held several positions at TU Berlin and University of California at Irvine. His research focuses on the quantum Hall effect, quantum transport, and topological insulators. Emil Prodan is full professor of physics at the Yeshiva University. Before this he received his PhD from the Rice University under the supervision of Peter Nordlander, and he has held several positions at University of California Santa Barbara and Princeton University. His research combines rigorous mathematical and computer simulations to study the physics of the condensed matter. He received the NSF CAREER award to support research on topological insulator.