We discuss the emergence of an effective low-energy theory for the real-time dynamics of two classical impurity spins within the framework of a prototypical and purely classical model of indirect magnetic exchange: two classical impurity spins are embedded in a host system which consists of a finite number of classical spins localized on the sites of a lattice and interacting via a nearest-neighbor Heisenberg exchange. An effective low-energy theory for the slow impurity-spin dynamics is derived for the regime, where the local exchange coupling between impurity and host spins is weak. To this end, we apply the recently developed adiabatic spin dynamics (ASD) theory. Besides the Hamiltonian-like classical spin torques, the ASD additionally accounts for a novel topological spin torque that originates as a holonomy effect in the close-to-adiabatic-dynamics regime. It is shown that the effective low-energy precession dynamics cannot be derived from an effective Hamilton function and is characterized by a nonvanishing precession frequency even if the initial state deviates only slightly from a ground state. The effective theory is compared to the fully numerical solution of the equations of motion for the whole system of impurity and host spins to identify the parameter regime where the adiabatic effective theory applies. Effective theories beyond the adiabatic approximation must necessarily include dynamic host degrees of freedom and go beyond the idea of a simple indirect magnetic exchange. We discuss an example of a generalized constrained spin dynamics which does improve the description but also fails for certain geometrical setups.

• Received 14 September 2020
• Accepted 12 January 2021

DOI:https://doi.org/10.1103/PhysRevB.103.024449