Bulk and Boundary Invariants for Complex Topological Insulators

Description

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 invariants. 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. Front Matter....Pages i-xxii Illustration of Key Concepts in Dimension \(d=1\) ....Pages 1-18 Topological Solid State Systems: Conjectures, Experiments and Models....Pages 19-53 Observables Algebras for Solid State Systems....Pages 55-83 K-Theory for Topological Solid State Systems....Pages 85-111 The Topological Invariants and Their Interrelations....Pages 113-143 Index Theorems for Solid State Systems....Pages 145-172 Invariants as Measurable Quantities....Pages 173-191 Back Matter....Pages 193-204