Hohenberg-Mermin-Wagner type theorems for equilibrium models of flocking

TL;DR

This paper extends classical theorems to two-dimensional equilibrium models of flocking, showing that spontaneous symmetry breaking does not occur without nonequilibrium conditions, emphasizing the role of detailed balance.

Contribution

It generalizes the Hohenberg-Mermin-Wagner theorem to Vicsek-type models, demonstrating the absence of spontaneous symmetry breaking in equilibrium states at nonzero temperature.

Findings

No spontaneous symmetry breaking in equilibrium models at nonzero temperature

Mobility alone does not cause symmetry breaking in flocking models

Symmetry breaking likely arises from nonequilibrium conditions

Abstract

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. We thus conclude that the mobility of particles alone does not account for the spontaneous symmetry breaking in Vicsek type models. The origin of the symmetry breaking must be sought in the absence of detailed balance condition, or, equivalently, in the nonequilibrium nature.