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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.
This book contains a self-consistent treatment of Besov spaces for
W*-dynamical systems, based on the Arveson spectrum and Fourier
multipliers. Generalizing classical results by Peller, spaces of
Besov operators are then characterized by trace class properties of
the associated Hankel operators lying in the W*-crossed product
algebra. These criteria allow to extend index theorems to such
operator classes. This in turn is of great relevance for
applications in solid-state physics, in particular, Anderson
localized topological insulators as well as topological semimetals.
The book also contains a self-contained chapter on duality theory
for R-actions. It allows to prove a bulk-boundary correspondence
for boundaries with irrational angles which implies the existence
of flat bands of edge states in graphene-like systems. This book is
intended for advanced students in mathematical physics and
researchers alike.
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.
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