DOI: 10.5281/zenodo.18459494
title: Topological Magnetic Configurations in a 2D Heisenberg Model with Dzyaloshinskii–Moriya Interaction: Monte Carlo Dataset

description: Abstract

This dataset provides Monte Carlo–generated magnetic configurations of a two-dimensional classical Heisenberg model on a 200 × 200 square lattice with periodic boundary conditions, including Dzyaloshinskii–Moriya interaction (DMI) and an external magnetic field applied perpendicular to the plane. The exchange interaction is fixed to unity, while the strengths of the Dzyaloshinskii–Moriya interaction and the magnetic field are varied to explore the phase space of the model.

Spin configurations were obtained using simulated annealing employing Monte Carlo simulations, resulting in low-temperature magnetic configurations. The simulations were performed using the Uppsala Atomistic Spin Dynamics (UppASD) software package. Each configuration consists of three-component spin vectors defined on lattice sites and is stored in a machine-learning–friendly numerical format (npz).

The dataset captures several magnetic phases, including ferromagnetic (FM), spin-spiral (SP), and skyrmion-lattice (SK) states, is intended for applications in machine learning and data-driven condensed-matter physics, such as phase classification or parameter prediction, unsupervised phase discovery, representation learning, and generative modeling of topological spin textures.

The dataset is divided into two complementary subsets. The phase-diagram dataset consists of magnetic configurations sampled on a regular grid in the (D, B) parameter space and does not include phase labels. The training dataset was constructed by randomly sampling configurations in the vicinity of representative FM, SK, and SP states and includes phase labels for supervised learning.

Model Hamiltonian

The magnetic system is described by a classical two-dimensional Heisenberg Hamiltonian defined on a square lattice. The Hamiltonian includes isotropic exchange interaction, Dzyaloshinskii–Moriya interaction (DMI), and an external magnetic field applied perpendicular to the plane. Only nearest-neighbour interactions are considered.

The Hamiltonian expressed in units of the exchange interaction $J$ reads:

$${\cal H} = -J\, \sum_{\langle i, j \rangle} \mathbf{s}_i \cdot \mathbf{s}_j - \sum_{\langle i, j \rangle} \mathbf{D}_{ij} \cdot (\mathbf{s}_i \times \mathbf{s}_j) - B\, \sum_i s_{z,i}\,,$$

where $\mathbf{s}_i$ is a unit vector representing the direction of the magnetic moment at lattice site $i$, while the parameters $J$ and $B$ denote the strength of the exchange interaction and the external magnetic field, respectively. $\mathbf{D}_{ij}$ is the Dzyaloshinskii–Moriya vector describing DMI between spin at lattice site $i$ and $j$. $\langle i, j \rangle$ means the summation over the  nearest neighbors.

DMI is assumed to obey $C_{4v}$ symmetry appropriate for a square lattice with broken inversion symmetry. The DMI vector for a pair of nearest-neighbour sites $i$ and $j$ is defined as

$$\mathbf{D}_{ij} = D\, (\mathbf{e}_z \times \mathbf{u}_{ij})$$

where $\mathbf{u}_{ij}$ is the unit vector pointing from site $i$ to site $j$, and $\mathbf{e}_z = (0, 0, 1)$ is the unit vector perpendicular to the lattice plane. This symmetry reduces the DMI to a single scalar parameter $D$, favoring chiral spin textures.

In our simulations we set $J = 1$. Thus parameters $D$ and $B$ are expressed in the units of the exchange interaction $J$.

Numerical simulations

Spin configurations were generated using simulated annealing employing Monte Carlo simulations as implemented in the Uppsala Atomistic Spin Dynamics (UppASD) package. The annealing procedure yields low-temperature magnetic configurations representative of near–ground-state spin textures for given values of $D$ and $B$.

Dataset content

The dataset consists of two main compressed NumPy archive files (.npz):



phase diagram grid dataset: heisenberg2D_DMI_B_grid.npz

training dataset: heisenberg2D_DMI_B_train.npz


Both files contain optimized magnetic configurations and their respective parameters $D$ and $B$.

All magnetic configurations are stored as NumPy arrays representing three-component spin vectors defined on a two-dimensional square lattice of size 200 × 200.

Each configuration is associated with a pair of model parameters (D, B), stored as NumPy arrays of parameter values. The parameters correspond to the strength of the Dzyaloshinskii–Moriya interaction and the external magnetic field, both expressed in units of the exchange interaction.

For easier analysis of the configurations, we also estimated total magnetization and topological charge (M, Q) for each configuration in the dataset. Magnetization is calculated as a magnitude of total spin per site, while the topological charge is calculated following the method presented in Yin et al.

In the training dataset, phase labels are additionally provided as string identifiers, which correspond to one of the main phase



"FM" - ferromagnetic configuration (200 samples)

"SK" - skyrmion lattice (300 samples)

"SP" - spin spiral configuration (200 samples)


These labels can be used for training machine-learning models to recognize the three fundamental magnetic phases present in the phase diagram. 

Structure of the .npz files

The structure of the NumPy archive files is following:

Phase diagram grid dataset

Phase diagram grid dataset contains magnetic configurations sampled on a regular grid in the (D, B) parameter space. Parameter $D$ is sampled in the range from 0.0 to 2.0 with discretization step 0.05. Magnetic field is sampled between 0.0 and 1.0011 with discretization step 0.0213. For each pair of parameters we provide five independent configurations. All together the test dataset contains 9840 samples.

This subset is intended for validation of classification or regression models trained using the training dataset. Since it contains multiple configurations for each $(D, B)$ pair, it can be split and used for supervised training for prediction of model parameters or physical properties. In addition, it can be also used for unsupervised training of generative models.

heisenberg2D_DMI_B_grid.npz contains following numpy arrays



configs: array of shape (N, 200, 200, 3)Three-component unit vectors aligned with the direction of magnetic moments on a regular 2D lattice of size $200\times 200$ with periodic boundary conditions.

values: array of shape (N, 2)Parameters (D, B) associated with each configuration. Both parameters are expressed in the units of exchange interaction ($J=1$).

properties: array of shape (N, 2)Total magnetization and topological charge (M, Q) estimated for each configuration. 


N=9840 is the number of samples in the test dataset.

Training dataset

Training dataset contains spin configurations randomly sampled in the vicinity of representative magnetic phases (FM, SP, and SK).

This subset is intended for training models for prediction magnetic phase, model parameters (D, B) or physical quantities (M, Q).

heisenberg2D_DMI_B_train.npz contains following numpy arrays:



configs: array of shape (N, 200, 200, 3)Three-component unit vectors aligned with the direction of magnetic moments on a regular 2D lattice.

values: array of shape (N, 2)Parameters (D, B) associated with each configuration. Both parameters are expressed in the units of exchange interaction ($J=1$).

properties: array of shape (N, 2)Total magnetization and topological charge (M, Q) estimated for each configuration. 

labels: array of shape (N,)Strings expressing the phase associated with each configuration.


N=900 is the total number of samples in the training dataset.

Example usage

To load the training dataset follow 

import numpy as np

data = np.load("heisenberg2D_DMI_B_train.npz")

configs= data["configs"]        # (N, 200, 200, 3)labels = data["labels"]          # (N,)values = data["values"]          # (N, 2) -> (D, B)properties = data["properties"]  # (N, 2) -> (M, Q)

An example Jupyter notebook demonstrating data loading, visualization, and basic analysis is provided with the dataset.

Acknowledgements

Computational resources were provided by the e-INFRA CZ project (ID:90254), supported by the Ministry of Education, Youth and Sports of the Czech Republic.

References



B. Skubic, J. Hellsvik, L. Nordström, and O. Eriksson, A method for atomistic spin dynamics simulations: implementation and examples, J. Phys.: Condens. Matter 20, 315203 (2008). DOI

H. Yin et al., Topological charge analysis of ultrathin magnetic films, Phys. Rev. B 93, 174403 (2016). DOI