Anyon condensation and a generic tensor-network construction for symmetry-protected topological phases

Bosonic symmetry protected topological (SPT) phases are bosonic analogs of free-fermion topological insulators and superconductors, but require interactions to be realized. Previously, a wide range of bosonic SPT phases protected by on-site symmetries has been systematically investigated, which are found to be related to group cohomology theory. However, a systematic understanding of SPT phases protected by spatial symmetries is still lacking. Here, the authors present systematic constructions of tensor-network wave functions for bosonic SPT phases protected by a general symmetry group SG involving both on-site and spatial symmetries. They find, in spatial dimension d=1, 2, 3, that a wide range of bosonic SPT phases are classified by the group cohomology H^d+1SG, U (1), in which the time-reversal symmetry and mirror symmetries should be treated as anti-unitary operations. They also provide generic tensor-network wave functions for these SPT phases that are useful for numerical simulations. As a by-product, the authors demonstrate a generic connection between rather conventional symmetry-enriched topological phases and SPT phases via an anyon condensation mechanism.