Abstract
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 ∼ |Φ|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 α, behave as uniform incompressible droplets with radius sim sqrt{alpha left|nright|} , 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=frac{s}{s-1} , which we confirm with analytical arguments and numerical methods.
Highlights
Breaking rotational invariance [9].1 A broader aim for exploring different aspects of finite density physics in Chern-Simons-matter theories is to understand the implications of the associated web [12,13,14] of particle-vortex and Bose-Fermi dualities in 2+1 dimensions [15,16,17,18,19,20,21,22,23]
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
A chemical potential μ for a gauged U(1) symmetry can only be turned on provided a source term representing a classical uniform background charge density is simultaneously introduced
Summary
Our starting point is the Abelian Chern-Simons theory at level k coupled to a relativistic scalar field in 2+1 dimensions. The quartic potential is of general interest because when μ vanishes, the theory flows to the 2+1 dimensional Wilson-Fisher fixed point coupled to a Chern-Simons gauge field. The background charge density J0 is fixed by requiring that the expectation values of A0 and the magnetic field vanish in the ground state: A0 = 0 =⇒ J0 = −2μ |Φ| 2 ,. The ground state conditions are solved by a vanishing source J0 = 0, and an A0 expectation value set by the chemical potential. This latter solution is equivalent to absorbing the chemical potential via a shift in the gauge field leaving the partition function unchanged. We will treat this background value as distinct from the magnetic field carried by the vortex
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