  It was recently shown that wet active matter may form synchronized rotating
vortices in a square lattice, similar to an antiferromagnetic Ising model (by
considering rotation direction as spin projections). In this letter, we
investigate whether such a correlated state occurs for a model of dry active
matter. We achieve that by numerically simulating the dynamics of a system of
active particles in the presence of two identical circular obstacles. Then, we
measure the rotation velocity correlation function of both vortices as a
function of the obstacle diameter, their shortest separation, called gap, and
the particle density. We find that, like the observations of vortex formation
in wet active matter, both vortices can synchronize their rotations in either
opposite or in the same direction; we call such regimes as antiferromagnetic
and ferromagnetic, respectively. We show that, for the antiferromagnetic case,
both vortices keep their motion correlated by exchanging particles through the
region in between them, analogously to synchronized cogs; on the other hand,
for the ferromagnetic regime, both vortices merge in a single rotating cluster,
similar to a belt strapped around the obstacles. Additionallly, we observe the
emergence of uncorrelated states at the transition between correlated states,
in which only a single vortex is present, or in the large gap regime, in which
the vortices are nearly independent on each other.
