Absence of O(2) symmetry in the Vicsek model
Phys. Rev. E 114, 014134 – Published 17 July, 2026
DOI: https://doi.org/10.1103/9gcy-ky3z
Abstract
The phase transition in the Vicsek model is widely believed to be associated with spontaneous symmetry breaking of the two-dimensional rotational symmetry . In this paper, we revisit the original model introduced by Vicsek et al. [Phys. Rev. Lett. 75, 1226 (1995)] 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.
