Spontaneous symmetry breaking in dual-core baby-Skyrmion systems

We introduce a system composed of two ($2+1$)-dimensional baby-Skyrmion models (BSMs) set on parallel planes and linearly coupled by tunneling of fields. This system can be realized in a dual-layer ferromagnetic medium. Unlike dual-core models previously studied in nonlinear optics and Bose-Einstein condensates, here the symmetry-breaking bifurcation (SBB) in solitons (baby Skyrmions) occurs with the increase of the intercore coupling ($\kappa$), rather than with its decrease, due to the fact that, even in the uncoupled system, neither core may be empty. Prior to the onset of the symmetry breaking between the two components of the solitons, they gradually separate in the opposite directions, due to the increase of $\kappa$, which is explained in an analytical form by means of an effective interaction potential. Such evolution scenarios are produced for originally symmetric states with topological charges in the two cores, $B^{(1)}=B^{(2)}=1$, 2, 3, 4. The evolution of mixed states, of the $(B^{(1)},B^{(2)})=(1,2)$ and (2,4) types, with the variation of $\kappa$ is studied too.

Figure 1

Field components ${\varphi }_1^{(2)}$ and ${\varphi }_3^{(2)}$ of the (1,1) configuration along the positive $y$ axis compactified onto the unit interval, $Y=y/(1+y)\in$ $[0,1]$ [cf., Eq. (8)] in the model with potential (7) at $\kappa =0$; 0.2; 0.4; 1.0; 1.5; 1.9; 2.0.

Figure 2

The upper and lower rows display contour plots of the energy density in the two sectors of the (1,1) configuration in the model with potential (7) at $\kappa =0$, 0.1, 1.0, 1.5, 2.0 (from left to right).

Figure 3

Contour plots of coupled components ${\varphi }_3^{(1)}$ and ${\varphi }_1^{(2)}$ (the upper and lower rows, respectively) of the (1,1) configuration in the model with potential (7) at $\kappa =0$, 0.1, 1.0, 1.5, 2.0 (from left to right).

Figure 4

Contour plots of the energy density of the sectors in the (2,2) configuration in the model with potential (7) at $\kappa =0$, 0.2, 0.7, 1.0, 2.0 (from left to right).

Figure 5

Contour plots of coupled components ${\varphi }_3^{(1)}$ and ${\varphi }_1^{(2)}$ (the upper and lower rows, respectively) of the (2,2) configuration in the model with potential (7) at $\kappa =0$, 0.2, 0.7, 1.0, 2.0 (from left to right).

Figure 6

Contour plots of the energy density of the sectors in the (3,3) configuration in the model with potential (7) at $\kappa =0$, 0.4, 0.8, 1.5, 2.0 (from left to right).

Figure 7

Contour plots of coupled components ${\varphi }_3^{(1)}$ and ${\varphi }_1^{(2)}$ (the upper and lower rows, respectively) of the (3,3) configuration in the model with potential (7) at $\kappa =0$, 0.4, 0.8, 1.5, 2.0 (from left to right).

Figure 8

Contour plots of the energy density of the components of the (4,4) configuration in the model with potential (7) at $\kappa =0$, 0.4, 0.8, 1.5, 2.0 (from left to right).

Figure 9

Contour plots of coupled components ${\varphi }_3^{(1)}$ and ${\varphi }_1^{(2)}$ (the upper and lower rows, respectively) of the (4,4) configuration in the model with potential (7) at $\kappa =0$, 0.4, 0.8, 1.5, 2.0 (from left to right).

Figure 10

(a) The symmetry measure, defined as per Eq. (14), for the (1,1), (2,2), (3,3) and (4,4) configurations, versus the coupling constant, $\kappa$. (b) The effective potential of interaction between the components of the (1,1) configuration, defined as per Eq. (13), versus the separation between them, $d$.

Figure 11

Contour plots of the energy density of the sectors in the (1,2) configuration in the model with potential (7) at $\kappa =0$, 0.4, 1.0, 1.5, 2.0 (from left to right).

Figure 12

Contour plots of coupled components ${\varphi }_3^{(1)}$ and ${\varphi }_1^{(2)}$ (the upper and lower rows, respectively) of the (1,2) configuration in the model with potential (7) at $\kappa =0$, 0.4, 1.0, 1.5, 2.0 (from left to right).

Figure 13

Contour plots of the energy density of the sectors of the (2,4) configuration in the model with potential (7) at $\kappa =0$, 0.4, 0.8, 1.5, 2.0 (from left to right).

Figure 14

Contour plots of coupled components ${\varphi }_3^{(1)}$ and ${\varphi }_1^{(2)}$ (the upper and lower rows, respectively) of the (2,4) configuration in the model with potential (7), at $\kappa =0$, 0.4, 0.8, 1.5, 2.0 (from left to right).

Figure 15

Energy density plots of the two components in the (1,2) and (3,3) configurations (the top and bottom panels, respectively) for $\kappa =2$, in the model with potential (7) and ${\mu }_1={\mu }_2=1$ (the first and third rows), or ${\mu }_1={\mu }_2=0$ (the second and forth rows).

Figure 16

Contour plots of the energy density of the components of the (1,1) configuration in the model with potential (15), at $\kappa =0$, 0.2, 1.0, 2.0 (from left to right).

Figure 17

The energy density of components of the (2,2) configuration in the model with potential (15), at $\kappa =0$, 0.2, 1.0, 2.0 (from left to right).

Figure 18

The energy density of components of the (3,3) configuration in the model with potential (15), at $\kappa =0$, 0.3, 1.0, 2.0 (from left to right).