  Band inversion is a known feature in a wide range of topological insulators
characterized by a change of orbital type around a high-symmetry point close to
the Fermi level. In some cases of band inversion in topological insulators, the
existence of quasinodal spheres has been detected, and the change of orbital
type is shown to be concentrated along these spheres in momentum space. To
understand this phenomenon, we develop a local effective fourfold Hamiltonian
that models the band inversion and reproduces the quasinodal sphere. This model
shows that the signal of the spin Hall conductivity, as well as the change of
orbital type, are both localized on the quasinodal sphere, and moreover, that
these two indicators characterize the topological nature of the material. Using
K-theoretical methods, we show that the change of orbital type parametrized by
an odd clutching function is equivalent to the strong Fu-Kane-Mele invariant.
We corroborate these results with ab initio calculations for the materials YH3
and CaTe, where in both cases the signal of the spin Hall conductivity is
localized on the quasinodal spheres in momentum space. We conclude that a
nontrivial spin Hall conductivity localized on the points of change of orbital
type is a good indicator for topological insulation.
