Absence of O(2) symmetry in the Vicsek model.

The phase transition in the Vicsek model is widely believed to be associated with spontaneous symmetry breaking of the two-dimensional rotational symmetry O(2). In this paper, we revisit the original model introduced by Vicsek et al. [Phys. Rev. Lett. 75, 1226 (1995)0031-900710.1103/PhysRevLett.75.1226] and demonstrate that the original angle-based update rule is not invariant under global phase shifts at the level of angular increments when it is implemented using the principal-value arctangent, as in the original definition. As a consequence, we numerically demonstrate that the phase transition reported in the original paper vanishes when the global phase is adaptively chosen.