Skyrmion-like states in two- and three-dimensional dynamical lattices

We construct, in discrete two-component systems with cubic nonlinearity, stable states emulating Skyrmions of the classical field theory. In the two-dimensional case, an analog of the baby Skyrmion is built on the square lattice as a discrete vortex soliton of a complex field [whose vorticity plays the role of the Skyrmion’s winding number (WN)], coupled to a radial “bubble” in a real lattice field. The most compact quasi-Skyrmion on the cubic lattice is composed of a nearly planar complex-field discrete vortex and a three-dimensional real-field bubble; unlike its continuum counterpart which must have $\mathrm{WN}=2$, this stable discrete state exists with $\mathrm{WN}=1$. Analogs of Skyrmions in the one-dimensional lattice are also constructed. Stability regions for all these states are found in an analytical approximation and verified numerically. The dynamics of unstable discrete Skyrmions (which leads to the onset of lattice turbulence) and their partial stabilization by external potentials are explored too.

Figure 1

(Color online) The one-dimensional discrete analog of the baby Skyrmion. Left and right panels in the top row show norms $N_1={\sum }_{\mathbf{n}}\mid u_{\mathbf{n}}{\mid }^2$ (solid line) and $N_2={\sum }_{\mathbf{n}}(\Lambda -\mid v_{\mathbf{n}}{\mid }^2)$ (dashed line), as well as the two most unstable eigenvalues vs coupling $C$ (the vertical dashed line is the analytically predicted instability onset). Left and right panels in the middle and bottom rows display, respectively, the solutions $u_n$ (solid line) and $v_n$ (dashed line) for $C=0.05$ and $0.15$, and the corresponding spectral planes $({\lambda }_r,{\lambda }_i)$ of the eigenvalues, $\lambda ={\lambda }_r+i{\lambda }_i$. In this figure and below, results are presented for $\beta =1∕4$ and $\Lambda =2$. This case is generic, as revealed by systematic simulations.

Figure 2

(Color online) Same as in Fig. 1 for the lattice analog of the two-dimensional (baby) Skyrmion. Left and right paired panels pertain, respectively, to $C=0.025$ and $0.075$, displaying, respectively, stable and unstable solutions. The top, middle, and bottom rows of the solution profiles display contours of real and imaginary parts of the complex field, and the real field, respectively.

Figure 3

(Color online) A three-dimensional lattice structure emulating the toroidal Skyrmion, for $C=0.05$. The left and right panels show, respectively, contours of the complex field, at $\mathrm{Re}\left(u\right)=\pm 1$ (blue/red in the color version and dark gray/gray in black and white) and $\mathrm{Im}\left(u\right)=\pm 1$ (green/yellow in the color version and light gray/very light gray in black and white), and of the real field, at $v=(1,0,-1)$ (blue, green, and red, i.e., contours from the inside out, respectively).

Figure 4

(Color online) Development of the instability of one-dimensional lattice quasi-Skyrmions. The top two panels show space-time contours of the absolute value of the complex field and the real one. The two bottom panels show the spatial distribution of the respective fields at $t=1100$.

Figure 5

(Color online) The most unstable eigenvalues for the one-dimensional lattice quasi-Skyrmion (top), and an example of a restabilized field configuration, $u_n$ (solid line) and $v_n$ (dashed line), and the respective eigenvalues for $C=0.115$ (middle and bottom) in the presence of the trapping potential with $\Omega =0.1$.