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

Abstract: 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.