BPS lumps in the Nonminimal $CP^1$ Maxwell-Chern-Simons Model

TL;DR

This paper studies self-dual solitons in a generalized $CP^1$ model coupled with Maxwell and Chern-Simons fields, revealing how nonminimal interactions influence the structure and properties of BPS lumps.

Contribution

It explicitly constructs the classical mapping from the nonlinear $O(3)$-sigma model to the $CP^1$ formulation, incorporating nonminimal couplings and deriving the conditions for self-duality.

Findings

Magnetic flux remains quantized and fixed by asymptotic behavior.

Finite-energy solutions are lump-like with scalar field vanishing at infinity.

Numerical solutions show regular, localized configurations with confined magnetic flux.

Abstract

We investigate self-dual radially symmetric configurations in the $CP^1$ model coupled to a Maxwell and Chern-Simons (CS) gauge fields through nonminimal interactions. Starting from the nonlinear $O(3)$-sigma model, we explicitly construct its classical mapping to the $CP^1$ formulation, highlighting the emergence of a local $U(1)$ gauge symmetry intrinsically associated with the Fubini-Study geometry of the target space. In the static regime, the combined effects of the Chern-Simons term and the Pauli-like nonminimal coupling modify the effective gauge connection, render the electric sector unavoidable, and give rise to magnetized and electrically polarized BPS lump configurations. By implementing the Bogomolnyi procedure, we determine the self-interaction potential required for self-duality and derive the corresponding BPS equations. We show that the magnetic flux remains quantized and is completely fixed by the asymptotic behavior of the gauge field, even in the presence of the Chern-Simons and nonminimal couplings. A detailed asymptotic analysis further reveals that finite-energy solutions necessarily correspond to lump-like configurations in which the $CP^1$ scalar field vanishes at spatial infinity. Numerical solutions of the BPS equations confirm that the resulting configurations are regular, spatially localized, and free of singularities, exhibiting confined magnetic flux together with a nontrivial localized electric field. These results show that the generalized $CP^1$-Maxwell-CS model supports self-dual solitons whose internal structure is rigidly governed by the target-space geometry rather than by spontaneous symmetry breaking.