# Edge-of-edge States

## TL;DR

This paper introduces and constructs a novel 'edge-of-edge' state localized at the intersection of edges in topological materials, demonstrating its generic existence through explicit models and topological charge analysis.

## Contribution

It explicitly constructs edge-of-edge states in higher-dimensional topological systems and generalizes the Berry connection to include boundary condition spaces, revealing nontrivial Chern-Simons forms.

## Key findings

- Edge-of-edge states exist generically at edge intersections.
- Explicit models in 5D Weyl semimetals and reductions show these states.
- Topological charge underpins the existence of these states.

## Abstract

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 5-dimensional Weyl semimetals and their dimensional reductions, such as 4-dimensional topological insulators of class A and 3-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.