Hopf Insulators and Their Topologically Protected Surface States

TL;DR

This paper introduces a class of 3D topological insulators called Hopf insulators characterized by an integer Hopf index, demonstrating their protected surface states and robustness under perturbations.

Contribution

The authors construct tight-binding models for Hopf insulators with arbitrary Hopf index, showing their topological protection and physical relevance without requiring additional symmetries.

Findings

Hopf insulators possess topologically protected surface states.

These states remain robust under random perturbations.

The models realize all Hopf indices, confirming their topological nature.

Abstract

Three-dimensional (3D) topological insulators in general need to be protected by certain kinds of symmetries other than the presumed $U(1)$ charge conservation. A peculiar exception is the Hopf insulators which are 3D topological insulators characterized by an integer Hopf index. To demonstrate the existence and physical relevance of the Hopf insulators, we construct a class of tight-binding model Hamiltonians which realize all kinds of Hopf insulators with arbitrary integer Hopf index. These Hopf insulator phases have topologically protected surface states and we numerically demonstrate the robustness of these topologically protected states under general random perturbations without any symmetry other than the $U(1)$ charge conservation that is implicit in all kinds of topological insulators.