Complete characterization of spin chains with two Ising symmetries

TL;DR

This paper thoroughly characterizes the 10 distinct topological phases of spin chains with two Ising symmetries, revealing their symmetry protection and dualities, and connecting bosonic and fermionic classifications.

Contribution

It provides a detailed analysis of the physical properties, order parameters, and symmetry protection of all 10 phases, using dualities to relate bosonic and fermionic topological classifications.

Findings

4 phases are topologically protected by Ising symmetries

6 phases break at least one symmetry

Identification of local and non-local order parameters

Abstract

Spin chains with two Ising symmetries are the Jordan-Wigner duals of one-dimensional interacting fermions with particle-hole and time-reversal symmetry. From earlier works on Majorana chains, it is known that this class of models has 10 distinct topological phases. In this paper, we analyze the physical properties of the correspondent 10 phases of the spin model. In particular, thanks to a set of two non-commuting dualities, we determine the local and non-local order parameters of the phases. We find that 4 phases are topologically protected by the Ising symmetries, while the other 6 break at least one symmetry. Our study highlights the non-trivial relation between the topological classifications of interacting bosons and fermions.