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

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.