Abelian Chern-Simons vortices at finite chemical potential

We examine vortices in Abelian Chern-Simons theory coupled to a relativistic scalar field with a chemical potential for particle number or U(1) charge. The Gauss constraint requires chemical potential for the local symmetry to be accompanied by a constant background charge density/magnetic field. Focussing attention on power law scalar potentials $\sim|\Phi|^{2s}$ which do not support vortex configurations in vacuum but do so at finite chemical potential, we numerically study classical vortex solutions for large winding number |n| >> 1. The solutions depending on a single dimensionless parameter $\alpha$, behave as uniform incompressible droplets with radius $\sim \sqrt {\alpha |n|}$ , and energy scaling linearly with |n|, independent of coupling constant. In all cases, the vortices transition from type I to type II at a critical value of the dimensionless parameter, $\alpha_c = s/(s-1)$, which we confirm with analytical arguments and numerical methods.