# Magnetic Skyrmion Lattice by Fourier Transform Method

## TL;DR

This paper introduces a rapid Fourier-based numerical method for studying skyrmion lattices in magnetic materials, accurately predicting phase transitions and discovering new metastable states.

## Contribution

A novel Fourier transform method for efficiently analyzing skyrmion and spiral phases in magnetic systems with Dzyaloshinsky-Moriya interaction, including metastable state prediction.

## Key findings

- Reproduced known critical fields for phase transitions with high accuracy.
- Predicted new metastable skyrmion lattice states stabilized by coupling with superconducting vortices.
- Method adaptable to other micromagnetic systems.

## Abstract

We demonstrate a fast numerical method of theoretical studies of skyrmion lattice or spiral order in magnetic materials with Dzyaloshinsky-Moriya interaction. The method is based on the Fourier expansion of the magnetization combined with a minimization of the free energy functional of the magnetic material in Fourier space, yielding the optimal configuration of the system for any given set of parameters. We employ a Lagrange multiplier technique in order to satisfy micromagnetic constraints. We apply this method to a system that exhibits, depending on the parameter choice, ferromagnetic, skyrmion lattice, or spiral (helical) order. Known critical fields corresponding to the helical-skyrmion as well as the skyrmion-ferromagnet phase transitions are reproduced with high precision. Using this numerical method we predict new types of excited (metastable) states of the skyrmion lattice, which may be stabilized by coupling the skyrmion lattice with a superconducting vortex lattice. The method can be readily adapted to other micromagnetic systems.