Boundary obstructed topological phases

Symmetry protected topological (SPT) phases are gapped phases of matter that cannot be deformed to a trivial phase without breaking the symmetry or closing the bulk gap. Here, we introduce a new notion of a topological obstruction that is not captured by bulk energy gap closings in periodic boundary conditions. More specifically, we say two bulk Hamiltonians belong to distinct boundary obstructed topological ‘phases’ (BOTPs) if they can be deformed to each other on a system with periodic boundaries, but not for symmetric open boundaries without closing the gap at a high symmetry region on the surface. BOTPs are not topological phases of matter in the standard sense since they are adiabatically deformable to each other on a torus but, similar to SPTs, they are associated with surface states or fractional corner charges in the open system. We show that the double-mirror quadrupole model of [Science, 357(6346), 2018] is a prototypical example of such phases, and present a detailed analysis of boundary obstructions in this model. In addition, we introduce several three-dimensional models having boundary obstructions, which are characterized either by surface states or fractional corner charges and discuss their general formulation in terms of Wannier band representations.