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Controlled Topological Phases and Bulk-edge Correspondence

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Abstract

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.

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  1. Graduate School of Mathematical Science, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan

    Yosuke Kubota

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  1. Yosuke Kubota
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Communicated by Y. Kawahigashi

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Kubota, Y. Controlled Topological Phases and Bulk-edge Correspondence. Commun. Math. Phys. 349, 493–525 (2017). https://doi.org/10.1007/s00220-016-2699-3

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