Coupling spin to velocity : collective motion of Hamiltonian polar particles

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.

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