Computer Simulation of Skyrmions on a Square Lattice
Aleksander Perzhua,b, Egor Vasilieva,b , Dmitrii Kapitana,b , Alexey Rybina,b , Alena Korola,b,
Konstantin Nefedev a,b and Vitalii Kapitana,b
a
  Department of Computer Systems, School of Natural Sciences, Far Eastern Federal University, Vladivostok,
  690950, 8 Sukhanova St., Russian Federation
b
  Institute of Applied Mathematics, Far Eastern Branch, Russian Academy of Science, Vladivostok, 690041, 7
  Radio St., Russian Federation


                 Abstract
                 We studied a square lattice of spins in frame of Heisenberg model with direct exchange and
                 Dzyaloshinskii-Moriya interaction. For the analysis of data obtained during the Monte Carlo
                 simulation, a convolutional neural network was used for the recognition of different phases of
                 the spin system, which was dependent on simulation parameters such as DMI and external
                 magnetic field (Hz). Based on these data, the phase diagram (Hz,D) was plotted. The various
                 states of the systems under observation were visualized, and the boundaries between the
                 different phases were defined as spirals, skyrmions and others. We proposed the controlling
                 method for movement of skyrmions using by Monte Carlo simulation.

                 Keywords 1
                 Monte Carlo method, convolutional neural network, magnetic skyrmion, Heisenberg model,
                 Dzyaloshinskii-Moriya interaction

1. Introduction
    Spintronics or magnetic electronics is continually evolving, and new promising materials for new
storage and processing data devices are emerging. Skyrmions are attractive candidates for information
carriers in a new type of non-mechanical magnetic medium - a racetrack memory - because they are
only a few nanometers in size, very stable, and can be driven by pulses of spin-polarized currents. At
a fundamental level, skyrmions are model systems for topologically protected spin structures and can
be regarded as an analogue of topologically protected states, emphasizing the role of topology in the
formation of complex states of condensed matter [1]. The creation, detection and control of individual
skyrmions have become especially relevant topics in connection with the possible implementation of
physical devices based on skyrmions in spintronics. For the development of control methods for
magnetic skyrmions in a magnetic nanostrip, it is necessary to conduct a detailed analysis of the
simulation parameters and the correlations between them in order to select the optimal parameters for
further studies of magnetic skyrmions.
    In our paper, the conditions for the nucleation and stable existence of magnetic skyrmions in two-
dimensional magnetic films were considered in the frame of the classical Heisenberg model. For
computer simulation, we used the Metropolis algorithm. For the analysis of the data obtained during
the Monte Carlo simulation, a convolutional neural network (CNN) was used for the recognition of
different phases of the spin system, depending on simulation parameters such as Dzyaloshinskii-
Moriya interaction and the external magnetic field (Hz). Based on these data, the phase diagram
(Hz,D) was plotted. Also, we proposed the controlling method for the movement of skyrmions.



VI International Conference Information Technologies and High-Performance Computing (ITHPC-2021),
September 14–16, 2021, Khabarovsk, Russia
EMAIL: perzhu_av@dvfu.ru (A. 1); kapitan.vyu@dvfu.ru (A. 7);
ORCID: 0000-0002-5068-8910 (A. 7)
            ©️ 2020 Copyright for this paper by its authors.
            Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
            CEUR Workshop Proceedings (CEUR-WS.org)


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2. Model and algorithms

2.1.    Mathematical model
     In 1960 Dzyaloshinskii presented a model to describe weak ferromagnetism [2]. Based on
symmetries he introduced an asymmetrical term which later on was clarified by Moriya [3]. The
Dzyaloshinskii-Moriya (DMI) interaction is a microscopic characteristic of interacting spins that
occurs in a system that lacks inversion symmetry and has a strong spin-orbit coupling. The
Heisenberg model is one of the models used in statistical physics to model ferromagnetism. It is used
in the study of critical points and phase transitions of different magnetic systems. We used the lattice
Hamiltonian, consisting of Heisenberg exchange (HJ) and DMI interaction (HD) terms for the
microscopic description of a chiral helimagnet [4-6], see formulas (1-3).

                                  𝐻 = (𝐻𝐽 + 𝐻𝑧 + 𝐻𝐴 ) + 𝐻𝐷 ,                                         (1)

                          𝐻𝐽 = −𝐽 ∑𝑟 𝑆⃗𝑟 ⋅ (𝑆⃗𝑟+𝑥̂ + 𝑆⃗𝑟+𝑦̂ ) − 𝐻𝑧 ∑𝑟 𝑆⃗𝑟 − 𝐻𝐴 ∑𝑟 ⃗⃗⃗⃗
                                                                                  |𝑆𝑟 |2 ,           (2)

                                  𝐻𝐷 = −𝐷 ∑𝑟 𝑆⃗𝑟 × 𝑆⃗𝑟+𝑥̂ ⋅ 𝑥̂ + 𝑆⃗𝑟 × 𝑆⃗𝑟+𝑦̂ ⋅ 𝑦̂,                  (3)
where 𝑆⃗𝑟 is the atomic spin, 𝐽 is the value of ferromagnetic short-range exchange interaction, 𝐷 is the
value of DMI, Hz – an external magnetic field and a magnetic anisotropy coefficient is HA.

2.2.    Metropolis algorithm
   The Metropolis algorithm is used to determine the global minimum. The main idea is to uniformly
sample the state space with a given distribution probability. At each iteration of the sample, the
configuration of the system changes due to a change in the orientation of a randomly selected spin.
The configuration is accepted and becomes the initial one for the next step if the new energy value is
greater than the previous one (𝐸1 > 𝐸2 ); otherwise, it is accepted with the probability:
                                                            𝑃(𝐸 )
                                     𝑃(𝐸𝑖 → 𝐸𝑗 ) = 𝑚𝑖𝑛 (𝑃(𝐸 𝑖 ) , 1)                                 (4)
                                                               𝑗
   Due to this, the algorithm avoids getting stuck in local minima. Convergence is achieved after
passing a given number of Monte Carlo steps until the moment when the standard deviation reaches a
specified minimum, depending on the problem being solved [7-10]. C++ and Rust programming
languages were used for software development, providing possibilities for the independent calculation
of the properties of the spin systems. We used dimensionless quantities in J units for the simulation.
The software has been verified for the Heisenberg model [11,12].

2.3.    Convolutional neural network for states classification
    We used configurations of spin systems obtained at different simulation parameters for the training
and subsequent classification of them in a neural network. To date, the most accurate analysis results
are demonstrated by neural networks based on convolutional architecture. We used the TensorFlow
library to create a convolutional neural network [13] and to classify our spin systems to different
phases [14].
    In our research, we have reduced the problem of determining the phases of spin systems to the
problem of image classification - in fact, to the main problem area in which neural networks are used.
For recognising images, CNN accepts them in the RGB format as a three-dimensional matrix. In our
case, the convolutional neural network received as input a three-dimensional array representing the
components of a three-dimensional spin in the frame of the Heisenberg model.




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    Following this, the convolutional neural network learned, using the training dataset, to highlight
the features inherent in one or another spin configuration. Our CNN consists of next layers (main
ones), see Figure 1:
    1. Input layer
    Input data (configurations of spins), each of the neurons (spins) of which is assigned an initial
random weight. The components of a three-dimensional vector were fed to the network input (i.e., the
components of Heisenberg spin). The dataset was prepared using Monte Carlo simulation data for
training the neural network in state recognition.




Figure 1: The architecture of the convolutional neural network

   2. Convolutional layer with 3×3 filter
   When neurons are connected to only a few neurons in the next layer, the layer is said to be
convolutional. The convolutional layer acts as a filter that discards the least informative parts of the
input data. Each layer has filters (i.e., matrices with weight values). When the filter moves along the
matrix of the previous layer, each filter element is multiplied by the value of the neuron, and the
values are summed up and written to the feature map.
   3. Pooling layer for reducing the dimensions of the data
   4. Fully connected layer
   Fully connected layers are used for classification. All layers before the fully connected layer are
used to highlight various features that are fed to the input of the classifier. This layer can also be used
as the final (output) CNN layer, the result of which is the probability of the input configuration of
spins belonging to a certain class.

3. Results and discussions
    We studied different phases that appeared depending on the magnitude of the Dzyaloshinskii-
Moriya interaction D and the external magnetic field Hz at fixed temperature T, see Figure 2. The
convolutional neural network was used to analyze the data obtained from the Monte Carlo simulations
for the recognition of the different phases of the spin system, dependent on the simulation parameters.
    In a magnetic film, with an increase of the magnetic field strength and DMI, various phases were
observed for the flat Heisenberg spin systems: Spiral (Sp), Spiral-skyrmion (SpSk) Skyrmion (Sk),
Skyrmion-ferromagnetic (SkF) and Ferromagnetic (FM) phases, see Figure 3. In Skyrmion phase, due
to the alignment of the stripes against the magnetic field, stable skyrmions are formed in the system.



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In these skyrmions, the spins of the nucleus are directed against the magnetic field. In this study,
skyrmions of the Bloch type were formed.




Figure 2: (𝐻𝑧 , 𝐷) phase diagram for the square lattice.




Figure 3: a) Stripe configuration, b) Mixed state, c) Skyrmion’s lattice.




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   We could also “push” the skyrmion from one side of the sample to another using the increasing of
a magnetic anisotropy. This is a rather precise method, see Figure 4. In frame of this numerical
experiment, we have an anisotropy gradient from 0.9 on the left side of the sample to 0.1 on the other
side. We have an incline plane of anisotropy gradient. And we increase the value of anisotropy step
by step for pushing the skyrmion. In the physical experiment, it is possible to control the anisotropy
applying the voltage to the sample.




Figure 4: Movement of the skyrmion.

4. Conclusion
    In the frame of the classical two-dimensional Heisenberg model, a spin system with direct short-
range exchange was modelled, and a study of its competition with the Dzyaloshinskii-Moriya
interaction was carried out. Due to the direct exchange interaction, the neighboring spins of the
system are collinearly aligned and, in turn, the Dzyaloshinskii-Moriya interaction contributes to the
deviation of the spins from parallel orientation. As a result, competition results between collinear and
noncollinear alignments of spins, which leads to the transition of the system of spins from a
ferromagnetic to a spiral ground state. In the presence of an external magnetic field, stable topological
structures - magnetic skyrmions - are generated in such systems.
    In this paper, we proposed a method for manipulation of the position of a skyrmion using by a
control of anisotropy gradient.
  We performed MC simulation and the convolutional neural network was used for the recognition of
the different phases of the spin systems, depending on the simulation parameters. For the visualisation
and analysis of research data, the phase diagram (Hz,D) was plotted.
  The data obtained in the numerical experiments will be used in our further studies to determine the
model parameters of the system for the formation of a stable skyrmion state, both in the form of
individual skyrmions and skyrmion lattices and for the development of methods for controlling
skyrmions in magnetic stripes.

5. Acknowledgements
   This work was supported in frame of joint program "Mikhail Lomonosov" of the Ministry of
Science and Higher Education of Russia and German Academic Exchange Service (DAAD) (#2298-
21). The studies were carried out using the resources of the Center for Shared Use of Scientific

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Equipment "Center for Processing and Storage of Scientific Data of the Far Eastern Branch of the
Russian Academy of Sciences" [15], funded by the Russian Federation represented by the Ministry of
Science and Higher Education of the Russian Federation under project No. 075-15-2021-663.
Additional computing resources were provided by the FEFU supercomputer cluster (cluster.dvfu.ru).

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