Three-dimensional dimeron as a stable topological object

Searching for novel topological objects is always an intriguing task for scientists in various fields. We study a three-dimensional (3D) topological structure called a 3D dimeron in trapped two-component Bose-Einstein condensates. The 3D dimeron differs from the conventional 3D skyrmion for the condensates hosting two interlocked vortex rings. We demonstrate that the vortex rings are connected by a singular string and the complexity constitutes a vortex molecule. The stability of the 3D dimeron is examined in two different models using the imaginary time evolution method. We find that the stable 3D dimeron can be naturally generated from a vortex-free Gaussian wave packet incorporating a synthetic non-Abelian gauge potential into the condensates.

Figure 1

(a) Isosurface of density of ${\psi }_1$ (red [gray]) and ${\psi }_2$ (blue [dark gray]) from the ansatz wave function (1) with a Thomas-Fermi density profile. Each component hosts a ring-shaped interlocked vortex. The 2D density profiles of ${\psi }_1$ are (b) the $x\text{−}y$ plane and (c) the $x\text{−}z$ plane.

Figure 2

Spin texture of the 3D dimeron at different radii: (a) $r=3$ and (b) $r=10$. The color of the arrows indicate variation of the global $U\left(1\right)$ phase. At large enough distance from the center, all the arrows uniformly point in the $x$ direction.

Figure 3

Texture of $\mathbf{S}\left(\mathbf{r}\right)$ in (a) the $z=0$ plane and (b) the $y=0$ plane. The color of the arrows specifies the magnitude of $S_z$.

Figure 4

Isosurface of the relative phase between the two components: (a) $\vartheta =0$, (b) $\vartheta =\pi /2$, (c) $\vartheta =4\pi /3$, and (d) $\vartheta =-3\pi /2$ and $\vartheta =3\pi /2$ (green [gray]), respectively. The red and blue loops (gray and dark gray) are the cores of the vortex ring in each component. The purple (dark gray) line is a common segment of all isosurfaces, which implies it is not well defined and constitutes a singular string.

Figure 5

The knotted topology of the 3D dimeron: The torus is the preimage isosurface of $S_x=0$. The variations of color on the torus specify the angle between $S_z$ and $S_y$ which reflect chirality of the knot. The yellow (light gray), red (gray), and green (dark gray) loops are the preimages of $\mathbf{S}=(0,1,0)$ and $\mathbf{S}=(0,0,1)$, and $\mathbf{S}=(-1,0,0)$, respectively. They are pairwise interlocked.

Figure 6

The spatial profile of a stable 3D dimeron generated by the imaginary-time evolution of the GPEs with the non-Abeian gauge potential (6). (a) Isosurface of the density for ${\psi }_1$ (red [gray]) and ${\psi }_2$ (blue [dark gray]), where two interlocked vortex rings are evident. The 2D density profiles of ${\psi }_1$ in (b) the $x\text{−}y$ plane and (c) the $x\text{−}z$ plane are similar to those in Fig. 1.