  Ensembles of particles rotating in a two-dimensional fluid can exhibit
chaotic dynamics yet develop signatures of hidden order. Such "rotors" are
found in the natural world spanning vastly disparate length scales - from the
rotor proteins in cellular membranes to models of atmospheric dynamics. Here we
show that an initially random distribution of either ideal vortices in an
inviscid fluid, or driven rotors in a viscous membrane, spontaneously self
assembles. Despite arising from drastically different physics, these systems
share a Hamiltonian structure that sets geometrical conservation laws resulting
in distinct structural states. We find that the rotationally invariant
interactions isotropically suppress long wavelength fluctuations - a hallmark
of a disordered hyperuniform material. With increasing area fraction, the
system orders into a hexagonal lattice. In mixtures of two co-rotating
populations, the stronger population will gain order from the other and both
will become phase enriched. Finally, we show that classical 2D point vortex
systems arise as exact limits of the experimentally accessible microscopic
membrane rotors, yielding a new system through which to study topological
defects.
