Abstract
Abstract 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.
Highlights
The notion of symmetry-protected topological (SPT) phases was introduced by Gu and Wen [16]
In [29, 30, 31], an index for SPT phases with on-site finite group symmetry and global reflection symmetry was defined for infinite quantum spin chains in a fully general setting
When the rank of the reduced density matrices of the infinite volume ground state is uniformly bounded, we show that the ground state has a presentation as a fermionic matrix product states (MPS) with on-site symmetry
Summary
The notion of symmetry-protected topological (SPT) phases was introduced by Gu and Wen [16]. In [29, 30, 31], an index for SPT phases with on-site finite group symmetry and global reflection symmetry was defined for infinite quantum spin chains in a fully general setting. In these papers, it was proven that the index is an invariant of the classification of SPT phases. A Z2index to distinguish these phases in infinite systems was introduced in [4] and independently in [24] It was outlined in [4] that this Z2-index is an invariant of the classification of unique parity-invariant gapped ground state phases using techniques from [29] and [28]. The aim of this article is to extend the analysis of fermionic gapped ground states to the case with an on-site symmetry; namely, a classification of one-dimensional fermionic SPT phases
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