Structure of the Effective Potential in Nonrelativistic Chern-Simons Field Theory

TL;DR

This paper calculates the effective potential for nonrelativistic scalar and fermion fields coupled to an Abelian Chern-Simons gauge field, revealing spontaneous symmetry breaking and potential vortex solutions.

Contribution

It provides the first detailed calculation of the scalar effective potential in nonrelativistic Chern-Simons theories, including gauge independence and radiative effects.

Findings

Conformal symmetry is spontaneously broken except at two coupling values.

The effective potential has a stable minimum near the origin due to radiative corrections.

Results are regularization scheme independent.

Abstract

We present the scalar field effective potential for nonrelativistic self-interacting scalar and fermion fields coupled to an Abelian Chern-Simons gauge field. Fermions are non-minimally coupled to the gauge field via a Pauli interaction. Gauss's law linearly relates the magnetic field to the matter field densities; hence, we also include radiative effects from the background gauge field. However, the scalar field effective potential is transparent to the presence of the background gauge field to leading order in the perturbative expansion. We compute the scalar field effective potential in two gauge families. We perform the calculation in a gauge reminiscent of the $R_\xi$-gauge in the limit $\xi\rightarrow 0$ and in the Coulomb family gauges. The scalar field effective potential is the same in both gauge-fixings and is independent of the gauge-fixing parameter in the Coulomb family gauge. The conformal symmetry is spontaneously broken except for two values of the coupling constant, one of which is the self-dual value. To leading order in the perturbative expansion, the structure of the classical potential is deeply distorted by radiative corrections and shows a stable minimum around the origin, which could be of interest when searching for vortex solutions. We regularize the theory with operator regularization and a cutoff to demonstrate that the results are independent of the regularization scheme.