CFT-approach to Rotating Field Lumps in Attractive Potential

TL;DR

The work analyzes rotating, non-topological solitons (Q-tubes) in a 2+1 dimensional complex scalar theory with quartic self-interaction. It derives Q-tube solutions, explores their large-n limit analytically, and studies stability via linear decay modes, highlighting the crucial role of relativistic corrections. In the non-relativistic limit the theory acquires conformal (Schrödinger) symmetry, yielding scale-free solitons with zero Hamiltonian and a fixed maximal charge Q_max that grows with winding number, while relativistic effects reintroduce small exponential instabilities. The results clarify how conformal symmetry governs energy-charge relations and stability and provide a framework for applying these insights to ultralight dark matter contexts and related solitonic systems.

Abstract

We demonstrate the importance of relativistic corrections for the study of the stability of $(2+1)$-dimensional non-topological solitons with quartic self-interaction in the low-energy limit. This result is explained by the restoration of conformal symmetry in the non-relativistic limit. Particularly, the corresponding cubic Gross-Pitaevskii equation supports scale-free non-topological solitons. An unbroken conformal symmetry provides the additional degeneration that allows for the exact result for the energy and angular momentum of the stationary classical solutions. We study the violation of conformal symmetry by relativistic corrections. The emergence of exponentially growing modes on the classical background is demonstrated using analytical approximations and numerical calculations.

Figures (6)

• Figure 1: Comparison of numerical integration with an approximate analytical solution (\ref{['analytical Q-tube']}).
• Figure 2: Integral characteristics of Q-tubes with different values of the parameter n. Calculations were performed by setting $\frac{\lambda}{m}=1$.
• Figure 3: Decay parameter $\gamma_{dec.}/m$ for $\omega/m \in [0.99, 1]$, $n = 0,1,2$.
• Figure 4: Decay mode profile at $\omega/m=0.995$ and $\gamma_{dec.}/m=3.11\cdot10^{-4}$. Scaled soliton profile and ${(-\gamma_{dec.})\frac{\partial f_{\omega}(r)}{\partial \omega}}$ are added for comparison.
• Figure 5: Conformal Q-tube profiles ${\frac{\bar{h}(\bar{r})}{\sqrt{2m}}}$ with winding number $n=0,1,2$.