Ring states in topological materials

Topological obstructions in quantum matter prevent the construction of symmetric and exponentially localized Wannier states from Bloch bands. We show that topological obstructions always lead to bound states at finite ingap energies for local symmetric potentials that exceed the Bloch Hamiltonian's bandwidth, such as lattice vacancies. The resulting ingap states, which we call ring states can serve as a local probe of topology, and are the precursors of topological boundary modes. We demonstrate the origin of ring states as well a boundary states from the pole structure of the local Green's function, present in topological phases protected by any internal or spatial symmetry.