Higher-order topological insulators, topological pumps and the quantum Hall effect in high dimensions

Abstract

Topological insulators are materials with spectral bands associated with an integer-valued index, manifesting through quantized bulk phenomena and robust boundary effects. In this Rapid Communication, we demonstrate that higher-order topological insulators are descendants from a high-dimensional chiral semimetal. Specifically, we apply dimensional reduction to an ancestor four-dimensional Chern insulator, and obtain two-dimensional (2D) second-order topological insulators when the former becomes chiral. Correspondingly, we derive the quantized charge accumulation at the corners of the 2D descendants and relate it to the topological index—the second Chern number—of the ancestor model. Our results provide a clear connection between the boundary states of higher-order topological insulators and topological pumps—the latter being dynamical realizations of the quantum Hall effect in high dimensions.