Edge states at an intersection of edges of a topological material

We study an exotic state which is localized only at an intersection of edges of a topological material. This edge-of-edge state is shown to exist generically. We construct explicitly generic edge-of-edge states in five-dimensional Weyl semimetals and their dimensional reductions, such as four-dimensional topological insulators of class A and three-dimensional chiral topological insulators of class AIII. The existence of the edge-of-edge state is due to a topological charge of the edge states. The notion of the Berry connection is generalized to include the space of all possible boundary conditions, where Chern-Simons forms are shown to be nontrivial.