Excited solutions in a Skyrme--Chern-Simons model in $2+1$ dimensions

Abstract

We study excited solutions in a Skyrme--Chern-Simons theory in $2+1$ dimensions. In particular, we emphasize the necessity of using a Lagrange multiplier method to obtain excited solutions, due to the appearance of a discontinuity when using a constraint compliant parametrization. These solutions are characterized by an integer number $p$, excited solutions corresponding to $p\neq 0$. The dependence of the global charges on the parameters is analyzed, showing non-standard behaviors. We also find that the presence of the Skyrme--Chern-Simons term does not alter significantly the pattern of energy levels, so $p=0$ solutions (fundamental solutions) have always the minimal energy.

Figures (8)

• Figure 1: The profiles of the functions of a typical solution are shown (for $\mu=0.1, \kappa=0.05, n=3, m=2, p=1, c_\infty=0.1, d_\infty=0.2$).
• Figure 2: Functions $S(r)$ (left) and $f(r)$ (right) for solutions with $\mu=0.1, \kappa=0.05, n=3, m=2, p=0,1,2, c_\infty=0.1, d_\infty=0.2$.
• Figure 3: (left) Function $g(r)$ for solutions with $\mu=0.1, \kappa=0.05, n=3, m=2, p=0,1,2, c_\infty=0.1, d_\infty=0.2$. (right) Function $g(r)$ for solutions with $\mu=0.1, \kappa=0.01, n=3, m=2, p=1, (c_\infty=0.1, d_\infty=0.2), (c_\infty=0.2, d_\infty=0.2)$, and $(c_\infty=0.2, d_\infty=0.1)$.
• Figure 4: (left) Energy $E$ vs $d_\infty$ for solutions with $\mu=0.1, \kappa=0.05, n=3, m=2, p=0,1,2, c_\infty=0.1$. (right) Angular momentum $J$ vs $d_\infty$ for solutions with $\mu=0.1, \kappa=0.05, n=3, m=2, p=0,1,2, c_\infty=0.1$.
• Figure 5: (left) Total electric charge $Q_t$ vs $d_\infty$ for solutions with $\mu=0.1, \kappa=0.05, n=3, m=2, p=0,1,2, c_\infty=0.1$. (right) Function $g(r)$ for solutions with $\mu=0.1, \kappa=0.05, n=3, m=2, p=1, c_\infty=0.1, d_\infty=0.2$, both for fully $O(5)$ Skyrme scalar (corresponding to the lower branch in left figure) and for embedded $O(3)$ Skyrme scalar (corresponding to the upper branch in left figure).