Figure 1

A single domain wall in two-component BECs. (a) The amplitude of ${\Psi }_1$ for a domain wall. (b) The pseudospin texture of the single domain wall perpendicular to the $x^1$ axis in real space. The arrows denote points in the target space $S^2$. The gradient and interaction energies are localized around the wall, which is shaded schematically. The arrows on the wall imply the phase ${\phi }_0$ which the wall possesses.Reuse & Permissions

Figure 2

The $S^2$ target space where the north and south poles are denoted by $\odot$ and $\otimes$, respectively. (a) The path connecting the north and south poles represents the map from the path in the domain wall in Fig. 1b along the $x^1$ axis in real space from $x^1\to -\infty$ to $x^1\to +\infty$. The path in the $S^2$ target space passes through one point on the equator, which is represented by “$\leftarrow$” in Fig. 1b in this example. In general, the $\text{U}\left(1\right)$ zero mode is localized on the wall. (b) The path in the target space $S^2$ for a domain wall and an antidomain wall. The path represents the map from the path along the $x^1$ axis from $x^1\to -\infty$ to $x^1\to +\infty$ in real space in Fig. 3b.Reuse & Permissions

Figure 3

A pair of domain wall and antidomain wall in two-component BECs. (a) The amplitude of ${\Psi }_1$ for the wall and antiwall configuration. (b) The pseudospin texture of the wall and antiwall configuration in real space. The arrows denote the pseudospin. ${\Psi }_1$ (${\Psi }_2$) is filled outside (between) the walls, where the other component is zero. In the upper (lower) region outside the walls, the phase of ${\Psi }_1$ is fixed to be zero ($\pi$).Reuse & Permissions

Figure 4

Decaying processes of the wall-antiwall pair. (a) and (b) The loop in the pseudospin space is unwound in two ways. (c) and (d) A bridge is created between the wall and antiwall. In this process there are two possibilities of spin structure along the bridge. (e) and (f) The upper and lower regions are connected with a passage through the bridge being formed. (g) and (h) The ${\Psi }_1$ component is filled inside the passage, and the phase of ${\Psi }_1$ component is connected anticlockwise or clockwise.Reuse & Permissions

Figure 5

Stable and unstable wall rings. (a) and (b) Stable wall rings. The domain wall phase winds once (the winding number is $\pm 1$) along the rings. Total configurations are 2D skyrmions with a nontrivial element $\pm 1$ of the second homotopy group ${\pi }_2$. The phase ${\Psi }_1$ winds $\pm 1$ so that it is a vortex, while ${\Psi }_2$ is filled inside the core. (c) and (d) Unstable wall rings. The domain wall phase does not wind (the winding number is 0) along the rings. The phase ${\Psi }_1$ does not wind so that it cannot be a vortex, while ${\Psi }_2$ is filled inside the ring. They decay into ground state (up pseudospin).Reuse & Permissions

Figure 6

Decay of a domain wall pair in three dimensions. Domain walls are two-dimensional objects. (a) Two kinds of holes labeled by $\uparrow$ and $\downarrow$ are created after a domain-wall pair decay. (b) Along the boundary of these two kinds of holes, there appear vortex lines (denoted by dotted lines), which in general making vortex loops.Reuse & Permissions

Figure 7

The typical D-brane soliton in two-component BECs. (a) Schematic illustration of the wall-vortex soliton configuration viewed from the length scale larger than the domain-wall width and the vortex core size. The two-component BECs ${\Psi }_1$ ($x^1<0$) and ${\Psi }_2$ ($x^1>0$) are separated by the domain wall. A single vortex located at $x^1<0$ (${\Psi }_1$ component) is connected to the domain wall. (b) The isosurface of $s_3=0$ for the solution Eq. (27) of the NL$\sigma$M, where $M=1$, $x_0^1=0$, ${\phi }_0=0$, $N_{v1}=1$, $N_{v2}=0$, and $z_1^{\left(1\right)}=0$. The corresponding spin textures $\mathbf{s}$ in the $z=0$ plane and $y=0$ plane are shown in (c) and (d), respectively. The magnitude of $s_3$ is denoted by color. We have $\mathbf{s}=(0,0,-1)$ along the vortex core.Reuse & Permissions

Figure 8

The D-brane soliton to which two vortices attach. (a) Schematic illustration of the configuration in which each component has a single vortex connected to the wall. (b) The isosurface of $s_3=0$ for the solution Eq. (27) of the NL$\sigma$M, where $M=1$, $x_0^1=0$, ${\phi }_0=0$, $N_{v1}=1$, $N_{v2}=1$, $z_1^{\left(1\right)}=-2$, and $z_1^{\left(2\right)}=2$. The corresponding spin textures $\mathbf{s}$ in the $z=0$ plane and $y=0$ plane are shown in (c) and (d), respectively. The magnitude of $s_3$ is denoted by color. The wall becomes asymptotically flat due to the balance between the tensions of the attached vortices.Reuse & Permissions

Figure 9

(a) A pair of a D-brane (domain wall) and an anti-D-brane (antidomain wall) stretched by a string (vortex) in two-component BECs. The branes are perpendicular to the $x^1$ axis and the string is placed along the $x^1$ axis. The arrows denote pseudospins. The ${\Psi }_1$ (${\Psi }_2$) component is filled outside (between) the branes, where the other component is zero. In the upper (lower) region outside the branes, the phase of ${\Psi }_1$ is fixed to be zero ($\pi$). In the middle region, the phase of ${\Psi }_2$ has winding around the vortex placed at the $x^1$ axis. Accordingly, the pseudospin rotates once (anti)clockwise at the endpoint of string on the (anti)brane. The profile of the ${\Psi }_1$ component along a line parallel to the $x^1$ axis at $(x^2,x^3)\ne 0$ represents two kinks in the left panel, while the profile of the ${\Psi }_1$ component along the $x^1$ axis shows the coincident two kinks, that is, a dark soliton. The dot in the center denotes the point $({\Psi }_1,{\Psi }_2)=0$, which corresponds to a singularity in the NL$\sigma$M approximation ($\rho =\text{const}$) in (b). (b) The isosurface of $s_3=0$ of an approximate solution in Eq. (30) with a domain wall and an antidomain wall stretched by a vortex in the $\text{O}\left(3\right)$ NL$\sigma$M, where $M=1$, $x_1^1=-3$, $x_2^1=3$, ${\phi }_1=0$, ${\phi }_2=\pi$. (c) The pseudospin texture of an approximate solution in Eq. (30).Reuse & Permissions

Figure 10

Loops in the wall-vortex systems. While the loop A yields an untwisted vortex ring in Fig. 11a, the loop B (C) yields a vorton, that is, a vortex ring twisted once (twice). A vorton with twisted once is shown in Fig. 11b.Reuse & Permissions

Figure 11

(a) The pseudospin texture of an untwisted vortex ring, and (b) and (c) the pseudospin and phase textures of a twisted vortex ring, that is, a vorton, after the brane-antibrane annihilation. (b) The pseudospin structure of the vorton. The torus divides the regions of ${\Psi }_1$ and ${\Psi }_2$ which repel each other: ${\Psi }_1$ (${\Psi }_2$) are filled outside (inside) the torus. The vertical section of the torus by the $x^1$-$x^2$ plane is a pair of 2D skyrmions and antiskyrmions. While they rotate along the $x^1$ axis their pseudospins are twisted. This spin texture is equivalent to that of a knot [22, 28, 29]. (c) The phase structure of the vorton. The arrows denote the phase of ${\Psi }_1$ and ${\Psi }_2$. The circle denotes the core of vorton where ${\Psi }_2$ is filled and ${\Psi }_1$ is zero. The phase of ${\Psi }_2$ winds once along that circle. The square of the dotted line denotes a loop where ${\Psi }_2$ is zero. The phase of ${\Psi }_1$ winds once along that loop. Note that the loops of ${\Psi }_1$ and ${\Psi }_2$ are zero, respectively, make a link. Along the zeros of ${\Psi }_1$ (${\Psi }_2$), the phase of ${\Psi }_2$ (${\Psi }_1$) winds once.Reuse & Permissions

Figure 12

A numerical simulation of a vorton creation. Surfaces are defined by $n_1-n_2=0.18({\mu }_1/g)$ (${\mu }_1=g=1,\hslash =m=1$), while color represents the phase of the ${\Psi }_2$ component. (a) First, we prepare a pair of a domain wall and an antidomain wall at coincident limit with a vortex winding in the ${\Psi }_2$ component. (b) Holes are created in the wall annihilation. (c) A vorton is created in the center.Reuse & Permissions

Figure 13

The equivalence between the vorton and the 3D skyrmion. The arrows denote the phase of ${\Psi }_1$ along a large loop (of the square of the dotted line) going to the boundary where ${\Psi }_2$ is zero, making a link with the vorton core. The left panel represents the configuration of the vorton from the brane-antibrane annihilation [see Fig. 11a], while the right panel represents the configuration of a 3D skyrmion. They are topologically isomorphic to each other.Reuse & Permissions