Annales Henri Poincaré

, Volume 17, Issue 4, pp 757–794 | Cite as

On the K-Theoretic Classification of Topological Phases of Matter

  • Guo Chuan Thiang


We present a rigorous and fully consistent K-theoretic framework for studying gapped phases of free fermions. It utilizes and profits from powerful techniques in operator K-theory, which from the point of view of symmetries such as time reversal, charge conjugation, and magnetic translations, is more general and natural than the topological version. In our model-independent approach, the dynamics are only constrained by the physical symmetries, which can be completely encoded using a suitable C *-superalgebra. Contrary to existing literature, we do not use K-theory groups to classify phases in an absolute sense, but to classify topological obstructions between phases. The Periodic Table of Kitaev is exhibited as a special case within our framework, and we prove that the phenomena of periodicity and dimension shifts are robust against disorder and magnetic fields.


Vector Bundle Topological Insulator Covariant Representation Grothendieck Group Grade Operator 
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© Springer Basel 2015

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK
  2. 2.School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia

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