Classically isospinning Hopf solitons

We perform full three-dimensional numerical relaxations of isospinning Hopf solitons with Hopf charge up to 8 in the Skyrme-Faddeev model with mass terms included. We explicitly allow the soliton solution to deform and to break the symmetries of the static configuration. It turns out that the model with its rich spectrum of soliton solutions, often of similar energy, allows for transmutations, formation of new solution types, and the rearrangement of the spectrum of minimal-energy solitons in a given topological sector when isospin is added. We observe that the shape of isospinning Hopf solitons can differ qualitatively from that of the static solution. In particular, the solution type of the lowest energy soliton can change. Our numerical results are of relevance for the quantization of the classical soliton solutions.

Figure 1

Position (blue tube) and linking (red tube) curve for $\mu =1$ Hopf solitons with Hopf charge $1\le N\le 8$ and potential $V_I$. We label each configuration by its Hopf charge $N$ and its solution type. The corresponding energy values can be found in Table . (a) ${1\mathcal{A}}_{1,1}$, (b) ${2{\mathcal{A}}_2}_{,1}$, (c) ${\widetilde{3\mathcal{A}}}_{3,1}$, (d) ${3\mathcal{A}}_{3,1}$, (e) ${\widetilde{4\mathcal{A}}}_{4,1}$, (f) ${4\mathcal{A}}_{2,2}$, (g) ${4\mathcal{A}}_{4,1}$, (h) ${5\mathcal{L}}_{1,2}^{1,1}$, (i) $5{\tilde{\mathcal{A}}}_{5,1}$, (j) $5{\mathcal{A}}_{5,1}$, (k) $6{\mathcal{A}}_{3,2}$, (l) $6{\mathcal{L}}_{2,2}^{1,1}$, (m) $6{\mathcal{L}}_{3,1}^{1,1}$, (n) $6{\tilde{\mathcal{A}}}_{6,1}$, (o) $6{\mathcal{A}}_{6,1}$, (p) $7{\mathcal{K}}_{3,2}$, (q) $7{\mathcal{K}}_{2,3}$, (r) $7{\tilde{\mathcal{A}}}_{7,1}$, (s) $8{\tilde{\mathcal{A}}}_{4,2}$, (t) $8{\mathcal{K}}_{3,2}$, (u) $8{\tilde{\mathcal{A}}}_{8,1}$.

Figure 2

The normalized energies $M_N^{\star }=M_N/(M_1{N}^{3/4})$ for different minimal energy, massive Hopf soliton solutions as a function of the Hopf charge $N$. The mass parameter $\mu$ is chosen to be 1. Here, our global minima for $1\le N\le 8$ are represented by filled circles, bent unknots $N{\tilde{\mathcal{A}}}_{N,1}$ by triangles, and rotationally symmetric unknots $N{\mathcal{A}}_{N,1}$ by diamonds, and the remaining local energy minima are displayed as open circles. The dashed line shows our linear fit to the bent unknots $N=1–5$ (${\mathcal{A}}_{1,1}$, ${\mathcal{A}}_{2,1}$ included), whereas the dash-dotted line represents a quadratic fit to the rotationally symmetric unknots $N=1–6$. The expected $N^{3/4}$ power growth is represented by the horizontal line. The corresponding plots for the massless Faddeev-Skyrme model can be found in Refs. 8, 9. (a) Normalized energies for Hopf solitons in the Faddeev-Skyrme model modified by potential $V_I$ vs Hopf charge $N$. (b) Same as (a) but for potential $V_{II}$.

Figure 3

Energy and moment of inertia of isospinning Hopf solitons with $N=1–3$. For $N=1$ the results for both potential choices are shown. (a) $E_{\mathrm{tot}}$ as function of $\omega$. (b) $E_{\mathrm{tot}}$ as function of $K$. (c) $U_{33}$ as function of $\omega$.

Figure 4

Total energy $E_{\mathrm{tot}}$ and moment of inertia $U_{33}$ of isospinning 4-Hopf solitons calculated with potential $V_I$. The isospinning $4{\mathcal{A}}_{2,2}$ soliton (blue curve) deforms into $4{\mathcal{L}}_{1,1}^{1,1}$ (red curve). The transition occurs at $\omega \approx 0.606$, $K\approx 23$. The bent configuration $4{\tilde{\mathcal{A}}}_{4,1}$ (green curve) exists for all $\omega \in [0,1)$, and its size is growing with $\omega$. (a) $E_{\mathrm{tot}}$ as function of $\omega$. (b) $E_{\mathrm{tot}}$ as function of $K$. (c) $U_{33}$ as function of $\omega$.

Figure 5

Deformation of the isospinning $4{\mathcal{A}}_{2,2}$ Hopf configuration into $4{\mathcal{L}}_{1,1}^{1,1}$. Results are plotted for potential $V_I$ (results for potential $V_{II}$ show the same qualitative behavior). We visualize the linking structure by plotting tubelike isosurfaces ${\varphi }_1=-0.9$ (red tube) and ${\varphi }_3=-0.9$ (blue tube). Recall that the angular momentum $K$ is given in units of $4\pi$. (a) $K=0(\omega =0)$, (b) $K=10(\omega =0.292)$, (c) $K=20(\omega =0.554)$, (d) $K=21(\omega =0.572)$, (e) $K=22(\omega =0.586)$, (f) $K=25(\omega =0.609)$, (g) $K=30(\omega =0.654)$, (h) $K=40(\omega =0.755)$.

Figure 6

Total energy $E_{\mathrm{tot}}$ of isospinning charge 5-Hopf solitons as a function of $\omega$ and $K$ ($V=V_I$). The energy curve $E_{\mathrm{tot}}(\omega )$ of the link $5{\mathcal{L}}_{1,2}^{1,1}$ (green curve) crosses the one of $5{\tilde{\mathcal{A}}}_{5,1}$ (purple curve at $\omega \approx 0.33$. The lowest-energy, isospinning soliton is of type $5{\mathcal{L}}_{1,2}^{1,1}$ for $\omega \in [0,0.33)$ and $5{\tilde{\mathcal{A}}}_{5,1}$ for $\omega \in [0.33,1)$. For comparison, we also show the axial, unstable $5{\mathcal{A}}_{5,1}$ solution (blue curve). (a) $E_{\mathrm{tot}}$ as function of $\omega$. (b) $E_{\mathrm{tot}}$ as function of $K$.

Figure 7

Total energy $E_{\mathrm{tot}}$ for different, isospinning 6-Hopf soliton configurations ($V=V_I$). The isospinning solution of type $6{\mathcal{A}}_{3,2}$ (blue curve) deforms into $6{\mathcal{L}}_{2,2}^{1,1}$ (green curve) at $\omega \approx 0.56$ ($K\approx 35$). The ${\tilde{\mathcal{A}}}_{6,1}$ configuration (purple curve) exists for all $\omega \in [0,1)$, but is of higher energy. (a) $E_{\mathrm{tot}}$ as a function of $\omega$, (b) $E_{\mathrm{tot}}$ as a function of $K$.

Figure 8

Deformation of the isospinning $6{\mathcal{A}}_{3,2}$ Hopf soliton solution into $6{\mathcal{L}}_{2,2}^{1,1}$. First row: We display isosurfaces ${\varphi }_1=-0.98$ (red tube) and ${\varphi }_3=-0.65$ (blue tube) to illustrate the change of the solution types. Second row: We show the linking curves for ${\varphi }_1=-0.98$, separately. (a) $K=0(\omega =0)$, (b) $K=20(\omega =0.367)$, (c) $K=30(\omega =0.502)$, (d) $K=45(\omega =0.677)$.

Figure 9

Linking curves of the isospinning $6{\mathcal{L}}_{3,1}^{1,1}$ Hopf soliton solution. The linking structure is visualized by the isosurfaces ${\varphi }_1=-0.95$ (red tube) and ${\varphi }_3=-0.65$ (blue tube). (a) $K=0(\omega =0)$, (b) $K=40(\omega =0.619)$, (c) $K=45(\omega =0.677)$, (d) $K=59(\omega =0.829)$.

Figure 10

The total energy $E_{\mathrm{tot}}$ for isospinning 8-Hopf solitons as a function of $\omega$ and $K$ ($V=V_I$). The isospinning $8{\tilde{\mathcal{A}}}_{4,2}$ configurations deform into the $8{\mathcal{K}}_{3,2}$ knot at $\omega \approx 0.68$ ($K\approx 49$). (a) $E_{\mathrm{tot}}$ as a function of $\omega$, (b) $E_{\mathrm{tot}}$ as a function of $K$.

Figure 11

Deformation of the isospinning $8{\tilde{\mathcal{A}}}_{4,2}$ Hopf configuration into $8{\mathcal{K}}_{3,2}$. The linking structure is visualized by the isosurfaces ${\varphi }_1=-0.97$ (red tube) and ${\varphi }_3=-0.8$ (blue tube). (a) $K=0(\omega =0)$, (b) $K=30(\omega =0.504)$, (c) $K=40(\omega =0.615)$, (d) $K=60(\omega =0.780)$.