Coupling spin to velocity: collective motion of Hamiltonian polar particles

TL;DR

This paper introduces a Hamiltonian particle model coupling spin and velocity, demonstrating that collective motion can emerge in such conservative systems despite momentum conservation, with complex phase behaviors observed.

Contribution

It presents a novel Hamiltonian model with spin-velocity coupling that exhibits collective motion and complex phase transitions, expanding understanding of non-dissipative active matter systems.

Findings

Existence of a transition to collective motion in mean-field limit

Velocity acts as an external magnetic field influencing phase transition

Rich phase diagram with ordered, disordered, and defect-laden phases

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.