$\mathbb{Z}_{2}$ Spin Hopf Insulator: Helical Hinge States and Returning Thouless Pump

TL;DR

This paper introduces a time-reversal-symmetric spin Hopf insulator with protected helical hinge states and a returning Thouless pump, expanding the understanding of nonmagnetic topological phases with spin-orbit coupling.

Contribution

It presents the first nonmagnetic delicate topological insulator with spin-orbit coupling featuring protected hinge states and a returning Thouless pump.

Findings

Surface $ ext{Z}_2$ invariants are stabilizable by symmetry.

Protected gapless helical hinge states exist.

Returning Thouless pump observed with four-fold rotational symmetry.

Abstract

We introduce a time-reversal-symmetric analog of the Hopf insulator that we call a spin Hopf insulator. The spin Hopf insulator harbors nontrivial Kane-Mele $\Z_2$ invariants on its surfaces, and is the first example of a nonmagnetic delicate topological insulator with spin-orbit coupling. We show that the Kane-Mele $\Z_2$ topology on the surface is generically unstable, but can be stabilized by the addition of a composition of the particle hole and spatial inversion symmetry. Such a symmetry not only protects the surface $\Z_2$ invariant, but also protects gapless helical hinge states on the spin Hopf insulator. Furthermore, we show that in the presence of four-fold rotational symmetry, the spin Hopf insulator exhibits a returning Thouless pump, as well as surface states on sharp boundary terminations.