<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="double-struck">Z</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math> spin Hopf insulator: Helical hinge states and returning Thouless pump

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