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

TL;DR

This paper introduces a new index for classifying symmetry protected topological phases in one-dimensional fermionic systems, capturing their topological invariants and symmetry properties.

Contribution

It defines a novel index taking values in a structured group, providing a complete invariant for classifying SPT phases of infinite fermionic chains with finite group symmetry.

Findings

The index takes values in  imes H^1(G,) imes H^2(G, U(1)_{\u211d})

The index is invariant under stacking of phases

Provides a fermionic matrix product state representation for translation-invariant ground states

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 generalized 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.