Towards spinning $U(1)$ gauged non-topological solitons in the model with Chern-Simons term

TL;DR

The paper constructs localized, finite-energy spinning non-topological solitons in a $(2+1)$-dimensional theory with Maxwell and Chern–Simons gauge terms coupled to a massive complex scalar. Using an axially symmetric ansatz with $U(1)$ frequency $\omega$ and winding $n$, it derives coupled equations for $f(r)$, $G(r)$, and $a(r)$ and analyzes conserved quantities $Q$, $\Phi$, $E$, and $J$, obtaining the exact relation $J = -n Q + \frac{e^2}{4\pi g} Q^2$. Numerically, localized profiles exist for all $n$, with $E(Q)$ growing and satisfying $E < m Q$, indicating bound states, and a thin-wall regime emerges as $\omega \to \omega_{\min}$; $J(Q)$ exhibits a global extremum at $Q_n = \frac{2\pi g n}{e^2}$ and can vanish at $Q = \frac{4\pi g n}{e^2}$, while $\omega \to m$ leads to an isolated point consistent with non-relativistic conformal behavior. The results underscore the stabilizing role of the CS term in achieving finite-energy, spinning solitons and suggest connections to condensed-mmatter analogs in the appropriate limits.

Abstract

We obtain localized field configurations with finite energy in a ($2+1$)-dimensional model with Maxwell and Chern-Simons gauge terms coupled to a massive complex scalar field. These non-topological solitons are characterized by the $U(1)$ frequency and a winding number. Thus, the solutions possess Noether charge and non-trivial angular momentum, which is not quantized in contrast to the topological case. We study the solitons and their integral characteristics numerically and demonstrate that they are kinematically stable. The obtained solutions allow for the thin-wall approximation in some region of frequencies. For each winding number, the Noether charge has a lower bound that coincides with an isolated point, where the non-relativistic conformal symmetry seems to be restored.

Figures (3)

• Figure 1: Profiles of the soliton with $n=0$. Left panel: Amplitudes of the fields $\varphi$, $A^\mu$. Right panel: Amplitudes of the fields $\vec{\mathcal{E}}$, $\mathcal{H}$. Here $g/m=1$, $\lambda/m=2$, $\sigma=3/2$, $e/\sqrt{m}=0.05$, and $\omega/m=0.999$.
• Figure 2: Profiles of the soliton with $n=1$. Left panel: Amplitudes of the fields $\varphi$, $A^\mu$. Right panel: Amplitudes of the fields $\vec{\mathcal{E}}$, $\mathcal{H}$. Here $g/m=1$, $\lambda/m=2$, $\sigma=3/2$, $e/\sqrt{m}=0.05$, and $\omega/m=0.999$.
• Figure 3: Left panel: Soliton energy $E/m$ as a function of its $U(1)$ charge $Q$ for different winding numbers $n$. Right panel: Noether charge $Q$ as a function of frequency $\omega/m$ for the solitonic solutions with different winding numbers $n$. Here $g/m=1$, $\lambda/m=2$, $\sigma=3/2$, and $e/\sqrt{m}=0.05$.