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The existence of bound states induced by local impurities coupled to aninsulating host depends decisively on the global topological properties of thehost's electronic structure. In this context, we consider magnetic impuritiesmodelled as classical unit-length spins that are exchange-coupled to thespinful Haldane model on the honeycomb lattice. We investigate the spectralflow of bound states with the coupling strength $J$ in both the topologicallytrivial and Chern-insulating phases. In addition to conventional $k$-spacetopology, 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 topologicalinvariants, the first ($k$-space) Chern number and the $R$-th ($S$-space)spin-Chern number. We demonstrate that there is a local $S$-space topologicaltransition as a function of $J$ associated with a change in the spin Chernnumber and work out the implications of this for the $J$-dependent localelectronic structure close to the impurities and, in particular, for in-gapbound states. The critical exchange couplings' dependence on the parameters ofthe Haldane model, and thus on the $k$-space topological state, is obtainednumerically to construct local topological phase diagrams for systems with$R=1$ and $R=2$ impurity spins.

Bound states and local topological phase diagram of classical impurity spins coupled to a Chern insulator