Matrix product operator algebras II: phases of matter for 1D mixed states

The mathematical classification of topological phases of matter is a crucial step toward comprehending and characterizing the properties of quantum materials. In this study, our focus is on investigating phases of matter in one-dimensional open quantum systems. Our goal is to elucidate the emerging phase diagram of one-dimensional tensor network mixed states that act as renormalization fixed points. These operators hold special significance since, as we prove, they manifest as boundary states of two-dimensional topologically ordered states, encompassing all known two-dimensional topological phases. To achieve their classification we begin by constructing families of such states from C*-weak Hopf algebras, which are algebras with fusion categories as their representations, and we present explicit local fine-graining and coarse-graining quantum channels defining the renormalization procedure. Lastly, we prove that a subset of these states, originating from C*-Hopf algebras, are in the trivial phase.