Supervised learning approach for recognizing magnetic skyrmion phases

We propose and apply simple machine learning approaches for recognition and classification of complex noncollinear magnetic structures in two-dimensional materials. The first approach is based on the implementation of the single-hidden-layer neural network that only relies on the $z$ projections of the spins. In this setup, one needs a limited set of magnetic configurations to distinguish ferromagnetic, skyrmion, and spin spiral phases, as well as their different combinations in transitional areas of the phase diagram. The network trained on the configurations for the square-lattice Heisenberg model with Dzyaloshinskii-Moriya interaction can classify the magnetic structures obtained from Monte Carlo calculations for a triangular lattice and vice versa. The second approach we apply, a minimum distance method, performs a fast and cheap classification in cases when a particular configuration is to be assigned to only one magnetic phase. The methods we propose are also easy to use for analysis of the numerous experimental data collected with spin-polarized scanning tunneling microscopy and Lorentz transmission electron microscopy experiments.

Figure 1

Schematic representation of the machine learning process. (a) The skyrmion magnetic structure as obtained from the classical Monte Carlo simulations for a two-dimensional ferromagnet with Dzyaloshinskii-Moriya interaction at finite temperature and magnetic field. Black arrows indicate the in-plane $xy$ spin components. (b) The matrix containing the $z$ projection of the spin structure to be classified. (c) Neural network having a single hidden layer of sigmoid neurons. The values of the input neurons are equal to $z$ components of the spins of the magnetic configuration.

Figure 2

Phase diagram in terms of Dzyaloshinskii-Moriya interaction and magnetic field. The abbreviations Sk, FM, and Sp denote skyrmion lattice state, ferromagnetic, and spin spiral state, respectively. The phase diagram was obtained at $T=0.02$. All the parameters are given in units of $J$. Black ovals denote the phase areas used for supervised learning.

Figure 3

Phase triptych obtained by using the neural network with 64 hidden neurons. Color intensities indicate the values of the output neurons for different phases; dark and light colors correspond to 1 and 0, respectively. The Dzyaloshinskii-Moriya interaction was chosen to be $D=0.72$. All the parameters are given in units of $J$. White circles denote the phase boundaries defined with the spin structure factors.

Figure 4

Hidden-layer arguments as a function of the $z$-oriented magnetization of the simulated spin configurations for (top) 8- and (bottom) 64-hidden-neuron network.

Figure 5

Hidden-layer arguments as a function of the skyrmion number of the simulated spin configurations for (top) 8- and (bottom) 64-hidden-neuron network. The results have been obtained for the pure DMI model having $J=0$.

Figure 6

(Left) $z$-projection of the skyrmion magnetic configuration obtained with the parameters $J=1$, $D=0.2$, $T=0.02$, and $B=0.02$. (Right) Visualization of the arguments of the specific hidden layer neurons.

Figure 7

(a) Example of a spiral state ($D=1.4$, $B=0.02$, $T=0.05$, and $J=1)$ used for training the network. (b) Example of a complex spiral configuration from a test set obtained with $D=0.72$, $B=0.03$, $T=0.22$, and $J=1$. (c) The output neurons values in the case of configuration (b) depending on the number of hidden neurons. Numbers in blue, orange, and green circles correspond to values of skyrmion, spiral, and FM outputs, respectively.

Figure 8

Examples of (a) skyrmion, (b) spin spiral, and (c) ferromagnetic configurations stabilized on the triangular lattice and recognized with the neural network trained on square-lattice data. Numbers in blue, orange, and green circles correspond to values of skyrmion, spiral, and ferromagnetic outputs, respectively.

Figure 9

Comparison of centroids calculated for different phases. Left panels are the examples of skyrmion, spin-spiral, paramagnetic, and ferromagnetic configurations. The arrows denote in-plane orientations of the magnetic moments. Middle panels are the corresponding $z$ projections of the example magnetic configurations. Right panels represent two-dimensional visualizations of the centroids calculated with Eq. (2) for training datasets.

Figure 10

Comparison of centroids calculated with (b) test set configurations on the triangular lattice, and (a,c) training set on the square lattice.

Figure 11

One-dimensional visualization of a training set comprising 4000 magnetic configurations. Yellow circles, red squares, blue diamonds, and green triangles denote magnetic configurations belonging to paramagnetic, spiral, skyrmion, and ferromagnetic phases, respectively. They are distributed with respect to the distance from the origin in 2304-dimensional space ($48\times 48$ spins in total for each configuration).

Figure 12

Examples of pure skyrmion (a), mixed skyrmion-bimeron (b), and pure spiral (c) magnetic configurations obtained at low temperature ($T=0.02J$) and corresponding spin-structure factors. Numbers in blue, orange, and green circles correspond to values of skyrmion, spiral, and FM outputs, respectively. Skyrmion numbers of these configurations are equal to 32, 28, and 0 from left to right.

Figure 13

Examples of nonperiodic skyrmions (a) obtained at low temperature ($T=0.02J$), pure skyrmions (b), and mixed skyrmion-bimeron (c) magnetic configurations obtained at high temperature ($T=0.4J$) and corresponding spin-structure factors. Numbers in blue, orange, and green circles correspond to values of skyrmion, spiral, and FM outputs, respectively. Skyrmion numbers of the presented configurations are equal to 15, 19, and 15 from left to right.

Figure 14

Schematic representation of a constructed neural network with a single hidden layer. We used sigmoid as an activation function of hidden and output neurons. All the notations are described in the text.