Classifying phases protected by matrix product operator symmetries using matrix product states

We classify the different ways in which matrix product states (MPSs) can stay in-variant under the action of matrix prod-uct operator (MPO) symmetries. This is achieved through a local characterization of how the MPSs, that generate a ground space, remain invariant under a global MPO symmetry. This characterization yields a set of quantities satisfying the cou-pled pentagon equations, associated with a module category over the fusion cat-egory that describes the MPO symme-try. Equivalence classes of these quantities provide complete invariants for an MPO symmetry protected phase: they are ro-bust under continuous deformations of the MPS tensor, and two phases with the same equivalence class can be connected by a symmetric gapped path. Our techniques match and extend the known renormal-ization fixed point classifications and fa-cilitate the numerical study of these sys-tems. For MPO symmetries described by a group, we recover the symmetry protected topological order classification for unique and degenerate ground states. Moreover, we study the interplay between time re-versal symmetry and an MPO symmetry and we also provide examples of our clas-sification, together with explicit construc-tions based on groups. Finally, we elabo-rate on the connection between our setup and gapped boundaries of two-dimensional topological systems, where MPO symme-tries also play a key role.

Automated summary

We classify the different ways in which matrix product states (MPSs) can stay invariant under the action of matrix product operator (MPO) symmetries. This characterization provides a set of quantities satisfying the coupled pentagon equations, which give complete invariants for an MPO symmetry protected phase. Our techniques facilitate the numerical study and match the known renormalization fixed point classifications. Additionally, we recover the symmetry protected topological order classification and explore the interplay between time reversal symmetry and MPO symmetries.