In their work – now published in the open access  journal Quantum – our group members András Molnár and José Garre Rubio, together with Laurens Lootens from the University of Cambridge, have developed a complete classification of quantum phases which remain invariant under non-local symmetries in the framework of one-dimensional (1D) tensor networks. The symmetries studied here, and the results achieved, generalize the established classification of symmetry protected topological (SPT) phases, where symmetries are on-site.

The generalization achieved by the present work is twofold: First, it is not required that the global symmetry is on-site, and second, the symmetry can form a representation of an algebra, and not only of a group. The only restriction imposed on the global symmetry is for it to be realized as a matrix product operator (MPO), that is, a global operator with a 1D tensor network structure. The study of these operators as symmetries is motivated by boundaries of two-dimensional topologically ordered phases, where the boundary system can inherit symmetries from the bulk topological order with strange fusion rules. The authors characterize the distinct phases by a set of quantities that they prove to be the proper invariant of these phases; furthermore, they explicitly construct a path connecting any two systems sharing the same invariants.

For further details, you are very much invited to have a look at the published paper in Quantum 7, 927 (2023) (open access), or at the version on the arXiv.

 This work has received support through the ERC grant SEQUAM.