The $Z_2$ Classification of Dimensional Reduced Hopf Insulators

TL;DR

This paper introduces a new $$ topological classification for dimensional reduced Hopf insulators, demonstrating their unique edge modes and topological properties without symmetry protection.

Contribution

It proposes a novel $$ index for 2D Hamiltonians derived from Hopf insulators and constructs explicit models to illustrate this topological phase.

Findings

Existence of a nontrivial $$ index for 2D Hamiltonians with zero Chern number

Construction of specific model Hamiltonian exhibiting the $$ topological phase

Numerical verification of topologically protected edge modes consistent with the $$ classification

Abstract

The Hopf insulators are characterized by a topological invariant called Hopf index which classifies maps from three-sphere to two-sphere, instead of a Chern number or a Chern parity. In contrast to topological insulator, the Hopf insulator is not protected by any kind of symmetry. By dimensional reduction, we argue that there exists a new type of $\mathbb{Z}_2$ index for 2D Hamiltonian with vanishing Chern number. Specific model Hamiltonian with this nontrivial $\mathbb{Z}_2$ index is constructed. We also numerically calculate the topological protected edge modes of this dimensional reduced Hopf insulator and show that they are consistent with the $\mathbb{Z}_2$ classification.