Coupling spin to velocity : collective motion of Hamiltonian polar particles

Sigbjørn Løland Bore, Michael Schindler, Khanh-Dang Nguyen Thu Lam, Eric Bertin and Olivier Dauchot

J. Phys. Stat. 3 (2016) 033305

Abstract :

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.