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

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.

arXiv commentTo appear in Forum of Mathematics, Sigma

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• A classification of phases of bosonic quantum lattice systems in one dimension
• Non-perturbative constraints from symmetry and chirality on Majorana zero modes and defect quantum numbers in (2+1)D
• An index for two-dimensional SPT states
• The split and approximate split property in 2D systems: stability and absence of superselection sectors
• Elementary derivation of the stacking rules of invertible fermionic topological phases in one dimension
• Generalized Thouless Pumps in 1+1-dimensional Interacting Fermionic Systems
• Rigorous Index Theory for One-Dimensional Interacting Topological Insulators
• An invariant of symmetry protected topological phases with on-site finite group symmetry for two-dimensional Fermion systems
• Duality and Stacking of Bosonic and Fermionic SPT Phases
• Interacting Crystalline Topological Insulators in two-dimensions with Time-Reversal Symmetry
• A ${\mathbb{Z}}_2$-Topological Index for Quasi-Free Fermions
• Non-local order parameters for fermion chains via the partial transpose