Exploring two dimensional $${\mathbb{Z}}_{2}$$ invariant phases with time reversal symmetry and their transitions with topological operations

A bstract We use various topological operations to systematically study phase transitions between theories with $${\mathbb{Z}}_{2}$$ and time reversal symmetry in two spacetime dimensions. The phases (and accompanying CFTs) we consider come in two types — bosonic phases that are defined on unorientable manifolds and fermionic phases that are sensitive to a Pin − structure. In both cases, our analysis leads to eight phase diagrams, with the two sets of eight connected by fermionization/bosonization. Starting from a seed CFT, we obtain the CFT that governs each transition. Many of these exhibit symmetry enriched criticality. In addition to showing many symmetry enriched CFTs in their natural habitats, our work discusses the fermionic analogs of the $${\mathbb{Z}}_{2}$$ bosonic operations, which we have not seen discussed in the literature.