Z2 spin Hopf insulator: Helical hinge states and returning Thouless pump

Abstract

The authors introduce here a time-reversal-symmetric analog of the Hopf insulator, dubbed as a spin Hopf insulator. The spin Hopf insulator harbors surface Kane-Mele \ensuremath{\mathbb{Z}}${}_{2}$ order and gapless helical hinge modes. Remarkably, the spin Hopf insulator is the first example of a nonmagnetic delicate topological insulator with spin-orbital coupling. In the presence of fourfold rotational symmetry, the spin Hopf insulator also exhibits a returning Thouless pump. These studies of nonmagnetic spin Hopf insulators provide a pathway towards realizing a Hopf material in experiments.