On Topology of the Moduli Space of Gapped Hamiltonians for Topological Phases

The moduli space of gapped Hamiltonians that are in the same topological phase is an intrinsic object that is associated to the topological order. The topology of these moduli spaces is used recently in the construction of Floquet codes. We propose a systematical program to study the topology of these moduli spaces. In particular, we use effective field theory to study the cohomology classes of these spaces, which includes and generalizes the Berry phase. We discuss several applications to studying phase transitions. We show that nontrivial family of gapped systems with the same topological order can protect isolated phase transitions in the phase diagram, and we argue that the phase transitions are characterized by screening of topological defects. We argue that family of gapped systems obey a version of bulk-boundary correspondence. We show that family of gapped systems in the bulk with the same topological order can rule out family of gapped systems on the boundary with the same topological boundary condition, constraining phase transitions on the boundary.