${\mathbb{Z}}_2$ Spin Hopf Insulator: Helical Hinge States and Returning Thouless Pump

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 ${\mathbb{Z}}_2$ 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 ${\mathbb{Z}}_2$ 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 ${\mathbb{Z}}_2$ 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.

9+4 pages, 6+1 figures

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• Representation-protected topology of spin-singlet $s$-wave superconductors
• Quantized crystalline-electromagnetic responses in insulators
• From quantum geometry to non-linear optics and gerbes: Recent advances in topological band theory
• PT-symmetric Non-Hermitian Hopf Metal
• Non-Hermitian Hopf insulators
• Correlated Hopf insulators