Impurity states in crystalline topological materials

Nontrivial topology plays a crucial role in how a system behaves in the presence of impurities and crystalline defects. Topological phases are characterized by a lack of an exponentially localized Wannier representation that respects the local and crystalline symmetries of the system. In this talk, I will relate the lack of localizability to a universal structure of the real part of the local Green's function. I will show that this structure is manifested in defect-bound states with unique qualitative features, independent of the exact nature of the topological phase or which symmetry protects it. Universal defect states can provide critical cues to prove a material is in a topological phase, particularly in phases where crystalline symmetries protect the topology and the boundary states are hard to access.