  The existence of bound states induced by local impurities coupled to an
insulating host depends decisively on the global topological properties of the
host's electronic structure. In this context, we consider magnetic impurities
modelled as classical unit-length spins that are exchange-coupled to the
spinful Haldane model on the honeycomb lattice. We investigate the spectral
flow of bound states with the coupling strength $J$ in both the topologically
trivial and Chern-insulating phases. In addition to conventional $k$-space
topology, an additional, spatially local topological feature is available,
based on the space of impurity-spin configurations forming, in case of $R$
impurities, an $R$-fold direct product of two-dimensional spheres. Global
$k$-space and local $S$-space topology are represented by different topological
invariants, the first ($k$-space) Chern number and the $R$-th ($S$-space)
spin-Chern number. We demonstrate that there is a local $S$-space topological
transition as a function of $J$ associated with a change in the spin Chern
number and work out the implications of this for the $J$-dependent local
electronic structure close to the impurities and, in particular, for in-gap
bound states. The critical exchange couplings' dependence on the parameters of
the Haldane model, and thus on the $k$-space topological state, is obtained
numerically to construct local topological phase diagrams for systems with
$R=1$ and $R=2$ impurity spins.
