Symmetry-protected topological phases from decorated domain walls

Symmetry-protected topological phases generalize the notion of topological insulators to strongly interacting systems of bosons or fermions. A sophisticated group cohomology approach has been used to classify bosonic symmetry-protected topological phases, which however does not transparently predict their properties. Here we provide a physical picture that leads to an intuitive understanding of a large class of symmetry-protected topological phases in d=1,2,3 dimensions. Such a picture allows us to construct explicit models for the symmetry-protected topological phases, write down ground state wave function and discover topological properties of symmetry defects both in the bulk and on the edge of the system. We consider symmetries that include a Z2 subgroup, which allows us to define domain walls. While the usual disordered phase is obtained by proliferating domain walls, we show that symmetry-protected topological phases are realized when these domain walls are decorated, that is, are themselves symmetry-protected topological phases in one lower dimension. This construction works both for unitary Z2 and anti-unitary time reversal symmetry.