Exact solutions of few-magnon problems in the spin-$S$ periodic $XXZ$ chain

We solve the two- and three-magnon problems for a \emph{finite-size} spin-$S$ periodic Heisenberg $XXZ$ chain with single-ion anisotropy through constructing a set of \emph{exact} Bloch states achieving a block diagonalization of the system. The two-magnon (three-magnon) problem with respect to the ferromagnetic reference state is reduced to a single-particle one on a one-dimensional (two-dimensional) effective lattice with size depending linearly (quadratically) on the total number of sites. For parameters lying within certain ranges, various types of multimagnon bound states are manifested and shown to correspond to edge states on the foregoing effective lattices. In the absence of the single-ion anisotropy, we reveal the condition under which exact zero-energy states emerge. As applications of the formalism, we calculate the transverse dynamic structure factor for a higher-spin chain near saturation magnetization and find signatures of the two types of two-magnon bound states revealed in prior studies. We also calculate the real-time three-magnon dynamics from certain localized states, which are relevant to cold-atom quantum simulations, by simulating single-particle quantum walks on the effective lattices. This provides a physically transparent interpretation of the observed dynamics in terms of propagation of bound state excitations. Our method can be directly applied to more general spin or itinerant particle systems possessing translational symmetry.