  We propose a conservative two-dimensional particle model in which particles
carry a continuous and classical spin. The model includes standard
ferromagnetic interactions between spins of two different particles, and a
nonstandard coupling between spin and velocity of the same particle inspired by
the coupling observed in self-propelled hard discs. Because of this coupling
Galilean invariance is broken and the conserved linear momentum associated to
translation invariance is not proportional to the velocity of the center of
mass. Also, the dynamics is not invariant under a global rotation of the spins
alone. This, in principle, leaves room for collective motion and thus raises
the question whether collective motion can arise in Hamiltonian systems. We
study the statistical mechanics of such a system, and show that, in the fully
connected (or mean-field) case, a transition to collective motion does exist in
spite of momentum conservation. Interestingly, the velocity of the center of
mass, which in the absence of Galilean invariance, is a relevant variable, also
feeds back on the magnetization properties, as it acts as an external magnetic
field that smoothens the transition. Molecular dynamics simulations of finite
size systems indeed reveal a rich phase diagram, with a transition from a
disordered to a homogeneous polar phase, but also more complex inhomogeneous
phases with local order interrupted by topological defects.

Statistical Mechanics