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The classification of symmetry protected topological phases of one-dimensional fermion systems

Part of: Selfadjoint operator algebras Groups and algebras in quantum theory

Published online by Cambridge University Press:  16 March 2021

Chris Bourne  and
Yoshiko Ogata
Chris Bourne
Affiliation:
WPI-AIMR, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japan. RIKEN iTHEMS, Wako, Saitama351-0198, Japan; E-mail: chris.bourne@tohoku.ac.jp.
Yoshiko Ogata
Affiliation:
Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Tokyo, 153-8914, Japan; E-mail: yoshiko@ms.u-tokyo.ac.jp.

Abstract

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We introduce an index for symmetry-protected topological (SPT) phases of infinite fermionic chains with an on-site symmetry given by a finite group G. This index takes values in $\mathbb {Z}_2 \times H^1(G,\mathbb {Z}_2) \times H^2(G, U(1)_{\mathfrak {p}})$ with a generalised Wall group law under stacking. We show that this index is an invariant of the classification of SPT phases. When the ground state is translation invariant and has reduced density matrices with uniformly bounded rank on finite intervals, we derive a fermionic matrix product representative of this state with on-site symmetry.

MSC classification

Primary: 81R15: Operator algebra methods
Secondary: 46L30: States
Type
Mathematical Physics
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e25
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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