Baryonic handles: Skyrmions as open vortex strings on a domain wall

We consider the BEC Skyrme model, which is based on a Skyrme-type model with a potential motivated by Bose-Einstein condensates (BECs), and, in particular, we study the Skyrmions in proximity of a domain wall that inhabits the theory. The theory turns out to have a rich flora of Skyrmion solutions that manifest themselves as twisted vortex rings or vortons in the bulk and vortex handles attached to the domain wall. The latter are linked open vortex strings. We further study the interaction between the domain wall and the Skyrmion and between the vortex handles themselves as well as between the vortex handle and the vortex ring in the bulk. We find that the domain wall provides a large binding energy for the solitons and it is energetically preferred to stay as close to the domain wall as possible; other configurations sticking into the bulk are metastable. We find that the most stable 2-Skyrmion is a torus-shaped braided string junction ending on the domain wall, which is produced by a collision of two vortex handles on the wall, but there is also a metastable configuration that is a doubly twisted vortex handle produced by a collision of a vortex handle on the wall and a vortex ring from the bulk.

Figure 1

The vortex string ending on the domain wall from the $\otimes$ phase in the $2+4$ model: (a) energy isosurface(s), the maximum of the energy is between the two surfaces; (b) isosurfaces of $0.1|{\varphi }_1|$ (magenta) and $0.1|{\varphi }_2|$ (blue), showing the surfaces close to the vacua. The typical logarithmic bending of the domain wall characteristic of a boojum is clearly visible. The parameters $c_4=1$ and $M=3$ were used for this calculation.

Figure 2

Skyrmion as a vortex handle with a single twist on the domain wall in the $2+4$ model. (a) The energy density is shown with blue transparent isosurfaces illustrating the domain wall, and the baryon charge density is shown with an isosurface with the color scheme described in the text. (b) The blue isosurface (bottom) represents the zeros of ${\varphi }_2$, and the magenta isosurface (top) is the zeros of ${\varphi }_1$. The baryon charge is added transparently. (c) The vortex (red) and antivortex (blue) pair of ${\varphi }_2$. (d) The vortex (red) and antivortex (blue) pair of ${\varphi }_1$. In this figure, we have taken $M=3$.

Figure 3

The Skyrmion as a vortex handle with a single twist on the domain wall in the $2+6$ model. (a) The energy density is shown with blue transparent isosurfaces illustrating the domain wall, and the baryon charge density is shown with an isosurface with the color scheme described in the text. (b) The blue isosurface (bottom) represents the zeros of ${\varphi }_2$, and the magenta isosurface (top) is the zeros of ${\varphi }_1$. The baryon charge is added transparently. (c) The vortex (red) and antivortex (blue) pair of ${\varphi }_2$. (d) The vortex (red) and antivortex (blue) pair of ${\varphi }_1$. In this figure, we have taken $M=3$.

Figure 4

Skyrmion as a bulk vortex ring in the $\otimes$ phase interacting with the domain wall in the $2+4$ model. The four columns display (1) the energy isosurfaces and baryon charge isosurface, (2) the zeros of ${\varphi }_2$ (blue) and ${\varphi }_1$ (magenta), (3) the vortex (red) and antivortex (blue) pair of ${\varphi }_2$, and (4) the vortex (red) and antivortex (blue) pair of ${\varphi }_1$. The rows correspond to the (imaginary) relaxation time of the simulation. The rows are not equidistant in relaxation time. In this figure, we have taken $M=3$.

Figure 5

2-Skyrmion as a vortex handle with double twist on the domain wall in the $2+4$ model. (a) The energy density is shown with blue transparent isosurfaces illustrating the domain wall, and the baryon charge density is shown with an isosurface with the color scheme described in the text. (b) The blue isosurface (bottom) represents the zeros of ${\varphi }_2$, and the magenta isosurface (top) is the zeros of ${\varphi }_1$. The baryon charge is added transparently. (c) The vortex (red) and antivortex (blue) pair of ${\varphi }_2$. (d) The vortex (red) and antivortex (blue) pairs of ${\varphi }_1$. In this figure, we have taken $M=3$.

Figure 6

2-Skyrmion as a vortex handle with double twist on the domain wall in the $2+6$ model. (a) The energy density is shown with blue transparent isosurfaces illustrating the domain wall, and the baryon charge density is shown with an isosurface with the color scheme described in the text. (b) The blue isosurface (bottom) represents the zeros of ${\varphi }_2$, and the magenta isosurface (top) is the zeros of ${\varphi }_1$. The baryon charge is added transparently. (c) The vortex (red) and antivortex (blue) pair of ${\varphi }_2$. (d) The vortex (red) and antivortex (blue) pairs of ${\varphi }_1$. In this figure, we have taken $M=7$.

Figure 7

Skyrmion-Skyrmion interaction in the domain wall: the vortex handle interacts with another vortex handle in the attractive channel in the $2+4$ model. The four columns display (1) the energy isosurfaces and baryon charge isosurface, (2) the zeros of ${\varphi }_2$ (blue) and ${\varphi }_1$ (magenta), (3) the vortex (red) and antivortex (blue) pairs of ${\varphi }_2$, and (4) the vortex (red) and antivortex (blue) pairs of ${\varphi }_1$. The rows correspond to the (imaginary) relaxation time of the simulation. The rows are not equidistant in relaxation time. In this figure, we have taken $M=7$.

Figure 8

Same as the second column of Fig. 7, but with a titled view point. The time evolution of the simulation has the following order: upper left, upper right, lower left, and lower right.

Figure 9

Skyrmion-Skyrmion interaction in the domain wall: the vortex handle interacts with another vortex handle in the repulsive channel in the $2+4$ model. The four columns display (1) the energy isosurfaces and baryon charge isosurface, (2) the zeros of ${\varphi }_2$ (blue) and ${\varphi }_1$ (magenta), (3) the vortex (red) and antivortex (blue) pairs of ${\varphi }_2$, and (4) the vortex (red) and antivortex (blue) pairs of ${\varphi }_1$. In this figure, we have taken $M=7$.

Figure 10

Bulk Skyrmion-domain wall Skymion interaction: a Skyrmion vortex ring in the $\otimes$ bulk interacts with a vortex handle in the $2+4$ model. The four columns display (1) the energy isosurfaces and baryon charge isosurface, (2) the zeros of ${\varphi }_2$ (blue) and ${\varphi }_1$ (magenta), (3) the vortex (red) and antivortex (blue) pair of ${\varphi }_2$, and (4) the vortex (red) and antivortex (blue) pair of ${\varphi }_1$. The rows correspond to the (imaginary) relaxation time of the simulation. The rows are not equidistant in relaxation time. In this figure, we have taken $M=7$.

Figure 11

3-Skyrmion as a vortex handle with three twists on the domain wall in the $2+4$ model. (a) The energy density is shown with blue transparent isosurfaces illustrating the domain wall, and the baryon charge density is shown with an isosurface with the color scheme described in the text. (b) The blue isosurface (bottom) represents the zeros of ${\varphi }_2$, and the magenta isosurface (top) is the zeros of ${\varphi }_1$. The baryon charge is added transparently. (c) The vortex (red) and antivortex (blue) pair of ${\varphi }_2$. (d) The vortex (red) and antivortex (blue) pair of ${\varphi }_1$. In this figure, we have taken $M=3$.

Figure 12

4-Skyrmion as a vortex handle with four twists on the domain wall in the $2+4$ model. (a) The energy density is shown with blue transparent isosurfaces illustrating the domain wall, and the baryon charge density is shown with an isosurface with the color scheme described in the text. (b) The blue isosurface (bottom) represents the zeros of ${\varphi }_2$, and the magenta isosurface (top) is the zeros of ${\varphi }_1$. The baryon charge is added transparently. (c) The vortex (red) and antivortex (blue) pair of ${\varphi }_2$. (d) The vortex (red) and antivortex (blue) pair of ${\varphi }_1$. In this figure, we have taken $M=3$.

Figure 13

5-Skyrmion as a double vortex junction with twists on the domain wall next to a 1-Skyrmion as a handle, in the $2+4$ model. (a) The energy density is shown with blue transparent isosurfaces illustrating the domain wall, and the baryon charge density is shown with an isosurface with the color scheme described in the text. (b) The blue isosurface (bottom) represents the zeros of ${\varphi }_2$, and the magenta isosurface (top) is the zeros of ${\varphi }_1$. The baryon charge is added transparently. (c) The vortex (red) and antivortex (blue) pairs of ${\varphi }_2$. (d) The vortex (red) and antivortex (blue) pairs of ${\varphi }_1$. In this figure, we have taken $M=3$.

Figure 14

7-Skyrmion as a double vortex junction with twists on the domain wall in the $2+4$ model. (a) The energy density is shown with blue transparent isosurfaces illustrating the domain wall, and the baryon charge density is shown with an isosurface with the color scheme described in the text. (b) The blue isosurface (bottom) represents the zeros of ${\varphi }_2$, and the magenta isosurface (top) is the zeros of ${\varphi }_1$. The baryon charge is added transparently. (c) The vortex (red) and antivortex (blue) pairs of ${\varphi }_2$. (d) The vortex (red) and antivortex (blue) pairs of ${\varphi }_1$. In this figure, we have taken $M=3$.