Geodesic
The classification of symmetry protected topological phases of one-dimensional fermion systems
Forum of Mathematics, Sigma, Tome 9 (2021)

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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. 
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     author = {Chris Bourne and Yoshiko Ogata},
     title = {The classification of symmetry protected topological phases of one-dimensional fermion systems},
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Chris Bourne; Yoshiko Ogata. The classification of symmetry protected topological phases of one-dimensional fermion systems. Forum of Mathematics, Sigma, Tome 9 (2021). doi: 10.1017/fms.2021.19

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