Shape modes of $\mathbb{C}P^1$ vortices

Abstract

In this paper we investigate the existence of internal modes of vortices in the gauged $\mathbb{C}P^1$ sigma model. We develop a clean geometric formalism that highlights the symmetries of the Jacobi operator, obtained from the second variation of the energy functional. The formalism and subsequent results fundamentally rely on the Bogomol'nyi decomposition of the energy functional, and can therefore be extended to other models with such a decomposition. We prove the existence of at least one shape mode for a general $\mathbb{C}P^1$ vortex solution on $\mathbb{R}^2$, and find numerically the shape modes and corresponding frequencies of a radially symmetric vortex. A surprising result is that the shape mode eigenvalues are very close to the scattering threshold, suggesting weakly bound shape modes could be characteristic of the $\mathbb{C}P^1$ model.

Figures (5)

• Figure 1: The eigenvalue $\lambda^2$ of the Jacobi operator vs $\tau$ for a North vortex solution with $N=1$, $N=2$ and $N=3$. In each case, the system was solved for $k=0$. The black curve represents the scattering threshold $1-\tau^2$.
• Figure 2: On the left column we present the wavefunction $\psi(r)$ and the potential of the Schr$\ddot{\text{o}}$dinger equation (\ref{['radial_schrodinger_p']}), along with the eigenvalue $\lambda^2$. On the right column we present the radial profiles of the gauge field $a(r)$ and the gauge invariant quantity $\phi_3=\cos{f(r)}$, along with the shape mode perturbations $\psi_1(r)$ and $\psi_2(r)$ given by $\mathscr{S}_1\mathscr{G} \Psi(r)$, see (\ref{['def_psi1']}) and (\ref{['def_psi2']}) for their explicit definition. All quantities were computed for a North vortex solution with $N=1$, $k=0$, and different values of $\tau$, chosen to be $0$, $0.5$ and $0.8$. The eigenvalues were computed numerically to be $\lambda^2\approx 0.99654$, $\lambda^2\approx 0.67116$, $\lambda^2\approx 0.29596$, respectively.
• Figure 3: On the left column we present the wavefunction $\psi(r)$ and the potential of the Schr$\ddot{\text{o}}$dinger equation (\ref{['radial_schrodinger_p']}), along with the eigenvalue $\lambda^2$. On the right column we present the radial profiles of the gauge field $a(r)$ and the gauge invariant quantity $\phi_3=\cos{f(r)}$, along with the shape mode perturbations $\psi_1(r)$ and $\psi_2(r)$ given by $\mathscr{S}_1\mathscr{G} \Psi(r)$, see (\ref{['def_psi1']}) and (\ref{['def_psi2']}) for their explicit definition. All quantities were computed for a North vortex solution with $N=2$, $k=0$, and different values of $\tau$, chosen to be $0$, $0.5$ and $0.8$. The eigenvalues were computed numerically to be $\lambda^2\approx 0.88054$, $\lambda^2\approx 0.50355$, $\lambda^2\approx 0.21058$, respectively.
• Figure 4: On the left column we present the wavefunction $\psi(r)$ and the potential of the Schr$\ddot{\text{o}}$dinger equation (\ref{['radial_schrodinger_p']}), along with the eigenvalue $\lambda^2$. On the right column we present the radial profiles of the gauge field $a(r)$ and the gauge invariant quantity $\phi_3=\cos{f(r)}$, along with the shape mode perturbations $\psi_1(r)$ and $\psi_2(r)$ given by $\mathscr{S}_1\mathscr{G} \Psi(r)$, see (\ref{['def_psi1']}) and (\ref{['def_psi2']}) for their explicit definition. All quantities were computed for a North vortex solution with $N=3$, $k=0$, and different values of $\tau$, chosen to be $0$, $0.5$ and $0.8$. The eigenvalues were computed numerically to be $\lambda^2\approx 0.72042$, $\lambda^2\approx 0.38743$, $\lambda^2\approx 0.15896$, respectively.
• Figure 5: The asymptotic vortex charge $q$ vs $\tau$ for the case $N=1$.

Theorems & Definitions (6)

• Lemma 3.1
• Theorem 3.2
• Proposition 3.3
• Theorem 3.4
• Theorem 3.5
• Theorem 3.6