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Communications in Mathematical Physics

, Volume 342, Issue 3, pp 909–963 | Cite as

Bott Periodicity for \({\mathbb{Z}_2}\) Symmetric Ground States of Gapped Free-Fermion Systems

  • R. Kennedy
  • M. R. ZirnbauerEmail author
Open Access
Article

Abstract

Building on the symmetry classification of disordered fermions, we give a proof of the proposal by Kitaev, and others, for a “Bott clock” topological classification of free-fermion ground states of gapped systems with symmetries. Our approach differs from previous ones in that (i) we work in the standard framework of Hermitian quantum mechanics over the complex numbers, (ii) we directly formulate a mathematical model for ground states rather than spectrally flattened Hamiltonians, and (iii) we use homotopy-theoretic tools rather than K-theory. Key to our proof is a natural transformation that squares to the standard Bott map and relates the ground state of a d-dimensional system in symmetry class s to the ground state of a (d + 1)-dimensional system in symmetry class s + 1. This relation gives a new vantage point on topological insulators and superconductors.

Keywords

Vector Bundle Homotopy Class Clifford Algebra Topological Insulator Symmetry Class 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© The Author(s) 2015

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Universität zu KölnCologneGermany

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