From symmetry-protected topological order to Landau order

Focusing on the particular case of the discrete symmetry group ${\mathbb{Z}}_N\times {\mathbb{Z}}_N$, 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 nonlocal 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 ${\mathbb{Z}}_N\times {\mathbb{Z}}_N$ in all these groups, together with a bijection between the individual classes of projective representations.