Vortices of $SO(2)$ gauged Skyrmions in $2+1$ dimensions

Vortices of the $SO(2)$ gauged planar Skyrme model with (a) only Maxwell, (b) only Chern-Simons, and (c) both Maxwell and Chern-Simons dynamics are studied systematically. In cases (a) and (b), where both models feature a single parameter $\lambda$ (the coupling of the potential term), the dependence of the energy on $\lambda$ is analyzed. It is shown that the plots of the energy vs $\lambda$ feature discontinuities and branches. In case (c), the emphasis is on the evolution of the topological charge, taking noninteger values. Throughout, the properties studied are contrasted with those of the corresponding Abelian Higgs models.

Figure 1

$E$ vs $\lambda$ for (M) vortices with $n=1$, 2, 3, $\eta =1/\sqrt{2}$ for the quadratic potential.

Figure 2

Left panel: $E$ vs ${\lambda }^{\prime }$ for (M) vortices with $n=1$, 2, 3, $\eta =1/\sqrt{2}$ for the ‘pion-mass’ potential. Right panel: $a_{\infty }$ vs ${\lambda }^{\prime }$ for the same vortices.

Figure 3

Left panel: $E$ vs $a_{\infty }$ for (CS) vortices with $n=1$, 2, 3, 4, $\eta =1$, $\kappa =1$ and $\lambda =1$. Right panel: $b_{\infty }$ vs $a_{\infty }$ for (CS) vortices with $n=4$, $\eta =1$, $\kappa =1$ and $\lambda =1$.

Figure 4

$E$ vs $\lambda$ for (CS) vortices with $n=1$, 2, 3, $\eta =1$, $\kappa =1$ and $a_{\infty }=0$.

Figure 5

$b_{\infty }$ vs $a_{\infty }$ for (MCS) vortices with $n=1$, $\lambda =1.6$, $\eta =1$, $\kappa =1$, and $\tau =1$.

Figure 6

Left panel: Energy $E$ vs electric charge $Q_e$ for (MCS) vortices with $n=1$, $\lambda =1.6$, $\eta =1$, $\tau =1$, and $\kappa =1$, 5, 10. Right panel: Energy $E$ vs angular momentum $J$ for the same vortices.

Figure 7

Left panel: $E$ vs $q$ for (MCS) vortices with $n=1$, 2, 3, $\lambda =1.6$, $\eta =1$, $\kappa =1$, and $\tau =1$. Right panel: Same for (MCS) vortices with $n=1$, $\lambda =1.6$, $\eta =1$, $\kappa =1$, and $\tau =0$, 1, 5, 10.

Figure 8

$E$ vs $q$ for (MCS) vortices with $n=1$, 2, 3, $\lambda =1.6$, $\eta =1$, $\kappa =1$, and $\tau =1000$.