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Energy Minimisation

Three interactive simulations, from a valley on a map to a skyrmion in a magnet

POLYTOPO · Interactive Simulations

Energy Minimisation

Stable magnetic textures, such as domain walls and skyrmions, are states in which the energy of the magnet has a minimum. To find them on a computer, one starts from a guess and walks downhill on the energy landscape, step by step, until the slope vanishes. How the next step is chosen can make the difference between a few steps and many thousands. These three interactive simulations follow the walk from the simplest case to a model magnet: two variables on a map, eleven in a chain, and finally 2,304 spins, where the walk ends in a skyrmion.

Series · 3 simulations
Part 1 · Two variables
The Energy Landscape

Drag the starting point over a curved valley and watch seven methods walk down to the minimum, from plain gradient descent to Newton’s method. Every step is drawn on the map, and a leaderboard shows how many steps each method needs.

Open Part 1 → Part 2 · Eleven variables
The Minimiser Race

No map can show eleven dimensions, so the seven methods race instead. Lengthen the chain of variables and see how fast the work grows, and why a very small energy does not always mean that a method has arrived.

Open Part 2 → Part 3 · 2,304 spins
Relaxing a Skyrmion

The energy of a model magnet. Start from a rough guess, let three minimisers walk downhill, and a magnetic skyrmion appears. Raise the field until it collapses, or start from random spins and count how many skyrmions are left.

Open Part 3 →

New to the subject? Start with Part 1 and follow the parts in order.

Based on

The series grew out of the C++ program written by Philipp Rybakov (University of Uppsala, Sweden) for his practical session “Numerical energy minimization in magnetics” at the POLYTOPO Training School on Physics and Mathematics of Topological Textures, Budapest, 12–13 May 2025. Parts 1 and 2 adapt it for the web: the test energy and the implementation of all seven minimisation methods are taken from it. Part 3 applies the same ideas to a spin model that is not part of the code. The original source code is available for download and is used here with the author’s permission. We thank him for sharing it.

Created by

Yorgis Androulakis, IACM / FORTH: web adaptation of Parts 1 and 2; concept, design & development of Part 3.

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Energy Minimisation