Global monopoles of charge 2

Global two-monopoles are unstable in their simplest formulation. We construct a model with a metriclike prefactor which we show can stabilize the global two-monopoles. The stabilizing construction is realized with a Skyrme sector where the metriclike prefactor is a function of the Skyrmion profile.

Figure 1

(a) profile function $h$, (b) energy density and (c) energy density on loglog scale for the hedgehog ($Q=1$) monopole.

Figure 2

Interaction energy of (a) the type A monopole-monopole pair and (b) the type A monopole-antimonopole pair as a function of separation distance $2d$ for a cutoff value $L=20$. The blue dashed line shows linear expansions in $d$.

Figure 3

Interaction energy of the type B monopole-monopole pair (which is the same as the type B monopole-antimonopole) as a function of separation distance $2d$ for a cutoff value $L=20$. The blue dashed line shows the linear expansion in $d$.

Figure 4

Interaction energy of the type C monopole-antimonopole as a function of separation distance $2d$ for a cutoff value $L=20$. The blue dashed line shows the linear expansion in $d$.

Figure 5

A global monopole inside a Skyrmion. The figure shows isosurfaces of the monopole charge and Skyrmion charge densities, respectively, at half their respective maximum values. The gray cloud is the Skyrmion charge isosurface and the colored isosurface is the monopole charge. The coloring is made using a hue-saturation-lightness (HSL) map from the monopole field such that $\arg ({\mathrm{\Phi }}_1/{\mathrm{\Phi }}_2)$ is mapped to the hue and $|{\mathrm{\Phi }}_3|$ determines the lightness. The parameters are chosen as: $c_2=1$, $c_4=4$, $m=1$, $v=1/4$, $\lambda =128$, $b=1/2$ and $\alpha =4$, and the numerically integrated charges are $B=0.99993$ and $Q=0.96707$. The calculation is made on a ${201}^3$ cubic lattice.

Figure 6

Topological charges for the global monopole inside a Skyrmion; (a) is the monopole charge and (b) is the Skyrmion charge. The figures show slices in the $xy$ plane for $z=0$. The numerically integrated charges are $B=0.99993$ and $Q=0.96707$.

Figure 7

Energies for the global monopole inside a Skyrmion; (a) is the kinetic term of the monopole (with the prefactor $\mathcal{G}$), (b) is the Skyrmion kinetic energy, (c) is the potential for the monopole (again with the prefactor $\mathcal{G}$), (d) is the pion-mass term of the Skyrmion sector, (e) is the total energy of the configuration and (f) is the logarithm of the total energy. The figures show slices in the $xy$ plane for $z=0$.

Figure 8

The global two-monopole of type A inside a single Skyrmion. The figure shows isosurfaces of the monopole charge and Skyrmion charge densities, respectively, at half their respective maximum values. The gray cloud is the Skyrmion charge isosurface and the colored isosurface of the shape of a torus is the monopole charge. The coloring is made using an HSL (hue-saturation-lightness) map from the monopole field such that $\arg ({\mathrm{\Phi }}_1/{\mathrm{\Phi }}_2)$ is mapped to the hue and $|{\mathrm{\Phi }}_3|$ determines the lightness. The parameters are chosen as: $c_2=1$, $c_4=4$, $m=1$, $v=1/4$, $\lambda =128$, $b=1/2$ and $\alpha =4$, and the numerically integrated charges are $B=0.99993$ and $Q=1.9863$. The calculation is made on a ${201}^3$ cubic lattice.

Figure 9

Topological charges for the global two-monopole inside a single Skyrmion; (a) is the monopole charge and (b) is the Skyrmion charge. The figures show slices in the $xy$ plane for $z=0$. The numerically integrated charges are $B=0.99993$ and $Q=1.9863$.

Figure 10

Energies for the global two-monopole inside a single Skyrmion; (a) is the kinetic term of the monopole (with the prefactor $\mathcal{G}$), (b) is the Skyrmion kinetic energy, (c) is the potential for the monopole (again with the prefactor $\mathcal{G}$), (d) is the pion-mass term of the Skyrmion sector, (e) is the total energy of the configuration and (f) is the logarithm of the total energy. The figures show slices in the $xy$ plane for $z=0$.

Figure 11

Asymmetry in the monopole charge, $Q_x/Q_y$ as function of relaxation time $\tau$, for various values of the deformation parameter $a$.

Figure 12

The global two-monopole with a deformed initial condition ($a=5$) inside a single Skyrmion. The figures show isosurfaces of the monopole charge and Skyrmion charge densities, respectively, at half their respective maximum values. The blue surface is the Skyrmion charge and the magenta surface is the monopole charge. As the relaxation time progresses, the solution converges towards that of Fig. 8. The calculation is made on a ${121}^3$ cubic lattice.

Figure 13

The global two-monopole with a deformed initial condition ($a=5$) inside a single Skyrmion. The figures show contours of the norm of $\mathrm{\Phi }$, i.e. $\sqrt{{\mathrm{\Phi }}_1^2+{\mathrm{\Phi }}_2^2+{\mathrm{\Phi }}_3^2}$ on an $xy$ slice at $z=0$. The blue circles in the middle of (a)–(f) show two separated zeros in the monopole field. As the relaxation time progresses, the two distinct zeros clearly merge to a single zero and the solution converges towards that of Fig. 8.

Figure 14

The global two-monopole of type B inside a single Skyrmion. The figures show isosurfaces of the monopole charge and Skyrmion charge densities, respectively, at half their respective maximum values. The blue surface is the Skyrmion charge and the magenta surface is the monopole charge. As the relaxation time progresses, the solution diverges and the monopoles eventually escape off to infinity. The calculation is made on a ${121}^3$ cubic lattice.

Figure 15

The global two-monopole of type A with deformation parameter $d=h_x$. The figures show isosurfaces of the monopole charge at half its maximum value. As the relaxation time progresses, the solution diverges and the monopoles eventually escape off to infinity. The parameters are chosen as $v=1/4$ and $\lambda =128$. The calculation is made on a ${121}^3$ cubic lattice.