These are the reports for a research project in Computational Physics and Magnetism about

Skyrmion - Monte Carlo: Simulated annealing with the Metropolis algorithm for a 3D spin lattice

The julia notebook in the source directory of this repository does not contain the final version of the code, but only served as a playground for the initial development.

The actual code can be found here.

We give a general introduction to Monte Carlo methods with a special focus on simulated annealing and the Metropolis algorithm. This report is conceived as an ab-initio guide to implement and properly analyse simulated annealing by the example of a three-dimensional lattice spin model. This system can be used to simulate different phases of chiral magnets such as MnSi at finite temperature in an external magnetic field. A thorough theoretical understanding of the requirements, applicability and errors is developed and illustrated by reference to the specific example. All code is publicly available for reproducability. Common pitfalls are identified and we provide clear and practical guidelines on how to avoid them.

We use the three-dimensional spin lattice Monte Carlo code Sky-MoCa to investigate the skyrmion phase in chiral magnets non-perturbatively. We derive the appropriate lattice Hamiltonian and discuss the relevant interactions as well as their consequences. The thermodynamic phases and transitions between them can be analyzed by means of Bragg intensity patterns and thermodynamic properties such as the specific heat and the susceptibility. We specifically demonstrate that the model does indeed exhibit the helical, conical and skyrmion phase for appropriate choices of the temperature and the external field. Subsequently we study the temperature driven transition from the unordered random phase to the helical phase. Moreover, we cool the system into the skyrmion phase and simulate both transitions driven by a change of the magnetic field. First, we decrease the field until the skyrmion tubes start to touch and gradually merge eventually resulting in the helical phase. Secondly, we increase the magnetic field until the skyrmion tubes simply vanish and the system transitions into the fully aligned phase. All observed phenomena qualitatively match experimental data.

To compile the tex files, navigate either to the directory report_magnetism or report_computational and simply type make.