Symmetry Protected Topological Hopf Insulator and its Generalizations

TL;DR

This paper explores 3D and 4D topological insulators characterized by Hopf maps, introduces a symmetry that classifies them with a $ ext{Z}_2$ invariant, and discusses their relation to Weyl semimetals and potential experiments.

Contribution

It identifies a generalized particle-hole symmetry $ ext{C}^ ext{' }$ that classifies Hopf insulators with a $ ext{Z}_2$ invariant and extends the concept to 4D analogues.

Findings

Hopf insulators have a $ ext{Z}_2$ classification under symmetry $ ext{C}^ ext{' }$.

Minimal models are analogous to Chern insulators combined with $S^1$ or $T^2$.

Relation to Weyl semimetals suggests possible experimental realization.

Abstract

We study a class of $3d$ and $4d$ topological insulators whose topological nature is characterized by the Hopf map and its generalizations. We identify the symmetry $\mathcal{C}^\prime$, a generalized particle-hole symmetry that gives the Hopf insulator a $\mathbb{Z}_2$ classification. The $4d$ analogue of the Hopf insulator with symmetry $\mathcal{C}^\prime$ has the same $\mathbb{Z}_2$ classification. The minimal models for the $3d$ and $4d$ Hopf insulator can be heuristically viewed as "Chern-insulator$ \rtimes S^1$" and "Chern-insulator$\rtimes T^2$" respectively. We also discuss the relation between the Hopf insulator and the Weyl semimetals, which points the direction for its possible experimental realization.