Hohenberg-Mermin-Wagner-Type Theorems for Equilibrium Models of Flocking

We study a class of two-dimensional models of classical hard-core particles with Vicsek type exchange interaction that aligns the directions of motion of nearby particles. By extending the Hohenberg-Mermin-Wagner theorem for the absence of spontaneous magnetization and the McBryan-Spencer bound for correlation functions, we prove that the models do not spontaneously break the rotational symmetry in their equilibrium states at any nonzero temperature. This provides a counterexample to the well-known argument that the mobility of particles is the key origin of the spontaneous symmetry breaking in two-dimensional Vicsek type models. Our result suggests that the origin of the symmetry breaking should be sought in the absence of a detailed balance condition, or, equivalently, in nonequilibrium nature.