  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.

Mesoscale and Nanoscale Physics;Materials Science