Finite-time orientational relaxation restructures collective motion in polar active matter

We introduce a Langevin formulation of Vicsek-like active particles, in which orientations evolve through finite-rate relaxation toward the local mean direction, with alignment strength J and rotational diffusivity Dr, thereby combining Vicsek-type local consensus with XY-like orientational dynamics. Using large-scale numerical simulations, we determine the nonequilibrium phase diagram as a function of activity and alignment rate. For example, at intermediate activity and in sufficiently large systems, increasing the alignment rate drives a sequence of transitions from a homogeneous isotropic state to polar bands, a cross-sea phase of intersecting bands, a homogeneous polar state, and ultimately a micro-clustered regime. The isotropic-to-polar transition is strongly first-order, as evidenced by Binder cumulants and bimodal distributions of local polarization and density, indicating coexistence of gas-like and liquid-like regions. Near the onset of collective motion, the band size increases with activity but depends non-monotonically on the alignment rate. Further increasing the alignment rate drives the system through the cross-sea and homogeneous polar phases before enhanced density fluctuations lead to micro-clustering. Our results demonstrate that finite-time orientational relaxation acts as a control parameter that qualitatively restructures collective behavior in polar active matter.