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 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.

Lieb-Schultz-Mattis type theorems for Majorana models with discrete symmetries

“…It is known that group cohomology classes corresponding to representations of G f induced by similarity transformations V (g) classify bosonic [22,39,61] and fermionic [55][56][57][58][59]62] SPT phases. Similarly, for a given symmetry group G f and in 1D, fermionic SPT phases are classified by a triplet of indices ([(ν, ρ)],μ).…”

Local Noether theorem for quantum lattice systems and topological invariants of gapped states

“…Such invariants are usually called topological, because they do not vary in suitably defined families. Recently, these methods have been used to define topological invariants of 1d [4,5,6,7] and 2d [8,9,10] gapped lattice systems with symmetries and arbitrarily strong interactions which decay rapidly with distance. In Refs.…”

A classification of invertible phases of bosonic quantum lattice systems in one dimension

“…Recently, Y. Ogata and collaborators [3,4,5] developed an approach to the classification of phases of 1d systems which does not rely on using an injective MPS. Instead they work with arbitrary 1d states satisfying the split property.…”

“…To focus on gapped phases, Refs. [3,4,5] impose the split property. In this paper we focus on states which are "invertible" in the sense of A. Kitaev [12].…”