Bound States of Skyrmions and Merons near the Lifshitz Point

We study topological defects in anisotropic ferromagnets with competing interactions near the Lifshitz point. We show that Skyrmions and bimerons are stable in a large part of the phase diagram. We calculate Skyrmion-Skyrmion and meron-meron interactions and show that Skyrmions attract each other and form ring-shaped bound states in a zero magnetic field. At the Lifshitz point merons carrying a fractional topological charge become deconfined. These results imply that unusual topological excitations may exist in weakly frustrated magnets with conventional crystal lattices.

Figure 1

(a) $J_2-J_3$ phase diagram of the square lattice frustrated magnet with a weak magnetic anisotropy. Stripes show the stability region of the $Q=1$ Skyrmion in the FM phase, for $K={10}^{-3}$. (b) $J_2-K$ stability diagram of Skyrmions and bimerons with $Q=1$, for $J_3=0$.

Figure 2

(a)–(c) Elementary Skyrmion at the points 1,2,3 on the phase diagram Fig. 1. Arrows show in-plane spin components, color indicates $S_z$. (d)–(f) The corresponding contour plots of the topological density, ${\rho }_Q(x,y)$.

Figure 3

(a) Potential energy $U_{12}(r)$ of interaction between two Skyrmions as a function of distance between the centers of the Skyrmions for equal helicities ${\chi }_1={\chi }_2$ (blue line) and for opposite helicities ${\chi }_1-{\chi }_2=\pi$ (red line). (b) Energy per Skyrmion $E_Q/Q$ in the Skyrmion ring. The calculations were preformed for $J_2=0.2$, $J_3=0.149$, and $K=0.01$.

Figure 4

(a) The spin configuration of the Skyrmion ring with $Q=-6$ in the frustrated magnet with an easy-axis anisotropy $K=0.01$ and (c) the corresponding topological charge density distribution ${\rho }_Q(x,y)$. (b) A meron cluster with a square lattice of vortices and antivortices minimizing energy for topological charge $Q=-8$ and the easy-plane anisotropy $K=-0.01$; (d) the corresponsing ${\rho }_Q(x,y)$. Other parameters of these simulations are $J_2=0.2$ and $\rho =3\times {10}^{-3}$.

Figure 5

(a)–(c) Deconfinement of meron pairs in a frustrated magnet with an easy-plane anisotropy $K<0$ for $J_2$ and $J_3$ at the points (1,2,3) on the phase diagram in Fig. 1. (d)–(f) Topological density ${\rho }_Q$ for the spin configurations in (a)–(c). In panels (a),(d) and (c),(f) the system is close to the FM-spiral phase boundary, which results in the topological “fractionalization,” i.e., a spatial separation of merons.

Figure 6

(a) Optimal distance between the centers of merons $R_{12}$ vs spin stiffness. $R_{12}$ increases near the LP $\rho =0$ resulting in a fractionalization of topological charge. (b) The meron-meron interaction potential $U_{12}$ vs the distance between merons. (c) The “non-Coulomb” part of the interaction potential $U_{12}-U_C$ vs meron-meron distance. The model parameters are $K=-5\times {10}^{-3}$ and $J_3=0.1$. The minimum of the potential energy curves is shifted to zero.