Interaction between hopping and static spins in a discrete network

We consider a process where a spin hops across a discrete network and at certain sites couples to static spins. While this setting is implementable in various scenarios (e.g. quantum dots or coupled cavities) the physics of such processes is still basically unknown. Here, we take a first step along this line by scrutinizing a two-site and a three-site lattices, each with two static spins. Despite a generally complex dynamics occurs, we show a regime such that the spin dynamics is described by an effective three-spin chain. Tasks such as entanglement generation and quantum state transfer can be achieved accordingly.