Topological Edge Spectrum Along Curved Interfaces

We prove that if the boundary of a topological insulator divides the plane into two regions, each containing arbitrarily large balls, then it acts as a conductor. Conversely, we construct a counterexample to show that topological insulators that fit within strips do not need to admit conducting boundary modes. This constitutes a new setup where the bulk-edge correspondence is violated. Our proof relies on a seemingly paradoxical and underappreciated property of the bulk indices of topological insulators: they are global quantities that can be locally computed.