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Controlled topological phases and bulk-edge correspondence. (English) Zbl 1357.82013
Summary: In this paper, we introduce a variation of the notion of topological phase reflecting metric structure of the position space. This framework contains not only periodic and non-periodic systems with symmetries in Kitaev’s periodic table but also topological crystalline insulators. We also define the bulk and edge indices as invariants taking values in the twisted equivariant \(K\)-groups of Roe algebras as generalizations of existing invariants such as the Hall conductance or the Kane-Mele \({\mathbb{Z}_2}\)-invariant. As a consequence, we obtain a new mathematical proof of the bulk-edge correspondence by using the coarse Mayer-Vietoris exact sequence. As a new example, we study the index of reflection-invariant systems.

MSC:
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
19L50 Twisted \(K\)-theory; differential \(K\)-theory
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
81V70 Many-body theory; quantum Hall effect
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