  We show that memory, in the form of underdamped angular dynamics, is a
crucial ingredient for the collective properties of self-propelled particles.
Using Vicsek-style models with an Ornstein-Uhlenbeck process acting on angular
velocity, we uncover a rich variety of collective phases not observed in usual
overdamped systems, including vortex lattices and active foams. In a model with
strictly nematic interactions the smectic arrangement of Vicsek waves giving
rise to global polar order is observed. We also provide a calculation of the
effective interaction between vortices in the case where a telegraphic noise
process is at play, explaining thus the emergence and structure of the vortex
lattices observed here and in motility assay experiments.
