Domain wall Skyrmions

Skyrmions of different dimensions are related by domain walls. We obtain explicit full numerical solutions of various Skyrmion configurations trapped inside a domain wall. We find for the quadratic mass term that multi-Skyrmions are ring shaped and conjecture for the linear mass term, that the lowest-energy state of multi-Skyrmions will consist of charge-2 rings accommodated in a lattice.

Figure 1

(a) Three-dimensional view of isosurfaces for the domain wall with a domain line on which a Skyrmion resides; the magenta surfaces represent the energy isosurfaces at a third of the maximum of the energy and the blue surface at the center shows the baryon charge isosurface, at half its maximum value. (b) and (c) show respectively the energy density and baryon charge density at a $yz$-slice in the middle of the domain wall (at $x=0$). The calculation is done on a ${129}^3$ cubic lattice, $B^{\text{numerical}}=0.992$ and the potential used is (13) with $m_4=4$, $m_3=2$, $m_2=1$. (a) isosurfaces; (b) energy density; (c) baryon charge density.

Figure 2

The domain wall with a domain line on which half a Skyrmion resides. For details see Fig. 1. The calculation is done on a ${129}^3$ cubic lattice, $B^{\text{numerical}}=0.492$ and the potential used is (16) with $m_4=4$, $m_3=2$, $m_2=1$. (a) isosurfaces; (b) energy density; (c) baryon charge density.

Figure 3

The domain wall with a trapped Skyrmion. For details see Fig. 1. The calculation is done on a ${129}^3$ cubic lattice, $B^{\text{numerical}}=0.998$ and the potential used is (19) with $m_4=4$, $m_3=2$. (a) isosurfaces; (b) energy density; (c) baryon charge density.

Figure 4

The domain wall with trapped $B=2$, 3, 4 Skyrmions. For details see Fig. 1. The calculations are done on a ${129}^3$ cubic lattice and the potential used is (19) with $m_4=4$, $m_3=2$. (a) isosurfaces; (b) energy density; (c) baryon charge density.

Figure 5

The domain wall with a minimal lattice structure of two trapped $B=2$ Skyrmions giving the total of a $B=4$ configuration. For details see Fig. 1. The calculation is done on an ${81}^3$ cubic lattice, $B^{\text{numerical}}=3.990$, $E=4\times 73.07$ and the potential used is (19) with $m_4=4$, $m_3=2$. (a) isosurfaces; (b) energy density; (c) baryon charge density.

Figure 6

The domain wall with a bound state of a 2-ring and a deformed 3-ring forming a $B=5$ configuration. For details see Fig. 1. The calculation is done on an ${81}^3$ cubic lattice, $B^{\text{numerical}}=4.988$, $E=5\times 73.61$ and the potential used is (19) with $m_4=4$, $m_3=2$. (a) isosurfaces; (b) energy density; (c) baryon charge density.

Figure 7

The domain wall with a symmetric initial condition for the 5-ring which collapses to the shown bound state of single Skyrmions forming a metastable $B=5$ configuration. For details see Fig. 1. The calculation is done on an ${81}^3$ cubic lattice, $B^{\text{numerical}}=4.98$, $E=5\times 75.58$ and the potential used is (19) with $m_4=4$, $m_3=2$. (a) isosurfaces; (b) energy density; (c) baryon charge density.

Figure 8

The domain wall with a trapped Skyrmion in a quadratic potential. Although having a flat top, the energy density is not a ringlike structure. For details see Fig. 1. The calculation is done on a ${129}^3$ cubic lattice, $B^{\text{numerical}}=0.998$ and the potential used is (23) with $m_4=4$, $m_3=2$. (a) isosurfaces; (b) energy density; (c) baryon charge density.

Figure 9

The domain wall with 2, 3, 4 sine-Gordon kinks living on trapped $B=2$, 3, 4 Skyrmions. For details see Fig. 1. The calculations are done on ${129}^3$, ${129}^3$, ${81}^2$ cubic lattices and the potential used is (25) with $m_4=4$, $m_3=2$, $m_2=1$. (a) isosurfaces; (b) energy density; (c) baryon charge density.

Figure 10

The domain wall with five sine-Gordon kinks living on a trapped $B=5$ Skyrmion. For details see Fig. 1. The calculation is done on an ${81}^3$ cubic lattice and the potential used is (25) with $m_4=4$, $m_3=2$, $m_2=1$. (a) isosurfaces; (b) energy density; (c) baryon charge density.

Figure 11

The domain wall with a trapped Skyrmion in a quadratic potential (with a kink mass). The energy density and baryon charge density on the middle of the wall ($x=0$) and the Skyrmion in the potential (25) are shown with those of Fig. 8 as a reference in a transparent shade. The calculation is done on an ${81}^3$ cubic lattice and $m_4=4$, $m_3=2$, $m_2=1$. (a) isosurfaces; (b) energy density; (c) baryon charge density.

Figure 12

The domain wall with a domain line on which a Skyrmion resides in the theory without the Skyrme term. The calculation is done on a ${129}^3$ cubic lattice, $B^{\text{numerical}}=0.9995$ and the potential used is (13) with $m_4=4$, $m_3=2$, $m_2=1$. (a) isosurfaces; (b) energy density; (c) baryon charge density.

Figure 13

The domain wall with a domain line on which half a Skyrmion resides. The calculation is done on a ${129}^3$ cubic lattice, $B^{\text{numerical}}=0.49993$ and the potential used is (16) with $m_4=4$, $m_3=2$, $m_2=1$. (a) isosurfaces; (b) energy density; (c) baryon charge density.

Figure 14

Contours of constant energy in the $yz$ plane of the double-winding baby Skyrmion on a wall (at $x=0$) in a linear potential as function of relaxation time $\tau$. It is observed that the two separated baby Skyrmions attract each other and find a minimum of the energy in a ringlike solution. This calculation was done on an ${81}^3$ cubic lattice and the potential used is (19) with $m_4=4$, $m_3=2$.