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.

中文翻译：

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一维费米子系统对称保护拓扑相的分类

我们引入了无限费米子链的对称保护拓扑（SPT）相的索引，其现场对称性由有限群给出G. 该索引取值 $\mathbb {Z}_2 \times H^1(G,\mathbb {Z}_2) \times H^2(G, U(1)_{\mathfrak {p}})$ 具有堆叠下的广义沃尔群定律。我们表明，该指数是 SPT 阶段分类的不变量。当基态是平移不变的并且具有在有限间隔上具有均匀有界秩的减小的密度矩阵时，我们推导出具有现场对称性的代表该状态的费米子矩阵乘积。