In active matter systems, external alternating fields, such as electric, magnetic, or optical fields, are widely used to regulate the motion and collective states of self-propelled particles. The presence of inertia introduces a delayed response to such fields, giving rise to complex collective dynamics. Nevertheless, how active particles with rotational inertia behave collectively under an unbiased periodic alternating field remains unclear. In this work, we conduct numerical simulations to study the collective behavior of such particles driven by a time-varying external torque that alternates symmetrically in direction.

Our results show that the frequency of the alternating field plays a decisive role in shaping the collective state of the system. As the frequency increases, the system undergoes a series of different phase transitions. At low frequencies, the particles exhibit synchronized polar order. With frequency rising, inertial delay disrupts this synchronization, driving the system into a disordered state. When the field period matches the intrinsic rotational relaxation time of the particles, stable horizontal or vertical cross-flow bands emerge, within which groups of particles travel in opposite directions. At very high frequencies, the system develops nematic order, characterized by counter-propagating particle streams. The effective diffusion coefficient reaches its peak during the formation of alternating flow bands, indicating enhanced collective transport. These structural transitions are consistently captured by the evolution of global order parameters. In contrast, variations in the particle self-propulsion speed and repulsive interaction strength exert only minor influences on the collective states, highlighting the dominant role of the alternating field frequency. This study elucidates the fundamental mechanism through which periodic alternating fields regulate the collective behavior of inertial active particles via frequency tuning. The results offer new insights into the coupling between external driving fields and particle dynamics in non-equilibrium systems, with potential applications in the design of micromachines and active smart materials.