From symmetry-protected topological order to Landau order

TL;DR

This paper establishes a mapping between symmetry protected topological phases and symmetry broken phases in one-dimensional spin systems, broadening understanding of topological order and symmetry breaking.

Contribution

It introduces a non-local unitary transformation linking topological and symmetry-broken phases, applicable to various symmetry groups including continuous ones.

Findings

Mapping preserves locality of Hamiltonian.

Applicable to systems with discrete and continuous symmetries.

Identifies subgroups and projective representations for broad classes.

Abstract

Focusing on the particular case of the discrete symmetry group Z_N x Z_N, we establish a mapping between symmetry protected topological phases and symmetry broken phases for one-dimensional spin systems. It is realized in terms of a non-local unitary transformation which preserves the locality of the Hamiltonian. We derive the image of the mapping for various phases involved, including those with a mixture of symmetry breaking and topological protection. Our analysis also applies to topological phases in spin systems with arbitrary continuous symmetries of unitary, orthogonal and symplectic type. This is achieved by identifying suitable subgroups Z_N x Z_N in all these groups, together with a bijection between the individual classes of projective representations.