Stable topological textures in a classical two-dimensional Heisenberg model

Abstract

We show that stable localized topological soliton textures (skyrmions) with ${\ensuremath{\pi}}_{2}$ topological charge $\ensuremath{\nu}\ensuremath{\ge}1$ exist in a classical two-dimensional Heisenberg model of a ferromagnet with uniaxial anisotropy. For this model the soliton exists only if the number of bound magnons exceeds some threshold value ${N}_{\text{cr}}$ depending on $\ensuremath{\nu}$ and the effective anisotropy constant ${K}_{\text{eff}}$. We define soliton phase diagram as the dependence of threshold energies and bound magnons number on anisotropy constant. The phase boundary lines are monotonous for both $\ensuremath{\nu}=1$ and $\ensuremath{\nu}>2$ while the solitons with $\ensuremath{\nu}=2$ reveal peculiar nonmonotonous behavior, determining the transition regime from low to high topological charges. In particular, the soliton energy per topological charge (topological energy density) achieves a minimum neither for $\ensuremath{\nu}=1$ nor high charges but rather for intermediate values $\ensuremath{\nu}=2$ or $\ensuremath{\nu}=3$. We show that this peculiarity is related to the character of convergence of integrals defining soliton energy and number of bound magnons at different $\ensuremath{\nu}$.