Isolated Skyrmions in the $CP^2$ nonlinear $\sigma$-model with a Dzyaloshinskii-Moriya type interaction

TL;DR

This paper investigates two-dimensional soliton solutions in a $CP^2$ nonlinear sigma model with Dzyaloshinskii-Moriya interaction, deriving exact solutions and exploring their properties and numerical counterparts.

Contribution

It introduces a $CP^2$ nonlinear sigma model with Dzyaloshinskii-Moriya interaction, deriving exact soliton solutions and analyzing their asymptotic behavior.

Findings

Exact soliton solutions derived for specific parameters

Numerical solutions match the asymptotic decay of analytical solutions

Vacuum state interpreted as a spin nematic state

Abstract

We study two dimensional soliton solutions in the $CP^2$ nonlinear $\sigma$-model with a Dzyaloshinskii-Moriya type interaction. First, we derive such a model as a continuous limit of the $SU(3)$ tilted ferromagnetic Heisenberg model on a square lattice. Then, introducing an additional potential term to the derived Hamiltonian, we obtain exact soliton solutions for particular sets of parameters of the model. The vacuum of the exact solution can be interpreted as a spin nematic state. For a wider range of coupling constants, we construct numerical solutions, which possess the same type of asymptotic decay as the exact analytical solution, both decaying into a spin nematic state.