# Fast and Robust Algorithm for the Energy Minimization of Spin Systems Applied in an Analysis of High Temperature Spin Configurations in Terms of Skyrmion Density

## TL;DR

This paper introduces a geometric energy minimization algorithm for magnetic systems, effectively analyzing thermal configurations and skyrmion densities, with improved convergence speed demonstrated in ferromagnetic and itinerant magnet models.

## Contribution

A novel geometric algorithm for energy minimization that preserves magnetic moment length and accelerates convergence using LBFGS, applied to skyrmion analysis in magnetic systems.

## Key findings

- Skyrmions appear within a narrow temperature range.
- The algorithm accurately identifies local energy minima.
- LBFGS significantly speeds up convergence.

## Abstract

An algorithm for the minimization of the energy of magnetic systems is presented and applied to the analysis of thermal configurations of a ferromagnet to identify inherent structures, i.e. the nearest local energy minima, as a function of temperature. Over a rather narrow temperature interval, skyrmions appear and reach a high temperature limit for the skyrmion density. In addition, the performance of the algorithm is further demonstrated in a self-consistent field calculation of a skyrmion in an itinerant magnet. The algorithm is based on a geometric approach in which the curvature of the spherical domain is taken into account and as a result the length of the magnetic moments is preserved in every iteration. In the limit of infinitesimal rotations, the minimization path coincides with that obtained using damped spin dynamics while the use of limited-memory quasi-newton minimization algorithms, such as the limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) algorithm, significantly accelerates the convergence.