  We study the onset of collective oscillations at low temperature in a
three-dimensional spin model with non-reciprocal short-range interactions.
Performing numerical simulations of the model, the presence of a continuous
phase transition to global oscillations is confirmed by a finite-size scaling
analysis. By systematically varying the interaction range, we show that
collective oscillations in this spin model actually result from two successive
phase transitions: a mean-field phase transition over finite-size
neighborhoods, which leads to the emergence of local noisy oscillators, and a
synchronization transition of local noisy oscillators, which generates coherent
macroscopic oscillations. Using a Fokker-Planck equation under a local
mean-field approximation, we derive from the spin dynamics coupled Langevin
equations for the complex amplitudes describing noisy oscillations on a
mesoscopic scale. The phase diagram of these coupled equations is qualitatively
obtained from a fully-connected (mean-field) approximation. This analytical
approach allows us to clearly disentangle the onset of local and global
oscillations, and to identify the two main control parameters, expressed as
combinations of the microscopic parameters of the spin dynamics, that control
the phase diagram of the model.
