Frustrated magnetic systems such as spin ice are key platforms for novel metamaterials. However, identifying their ground states in finite arrays is a formidable challenge, as boundary sensitivity and metastable states trap conventional optimization methods. We introduce a virtuous-cycle AI pipeline where a genetic algorithm explores the latent space of a variational autoencoder (VAE), with the best candidates progressively refining the VAE's representation. Applied to Kagome spin ice, this method reveals how the boundary magnetism is determined: boundaries break the symmetry of the 3 & times;3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{3\,}\times \sqrt{3\,}$$\end{document} magnetic superstructure while the bulk superstructure order in the interior maintains. Furthermore, it demonstrates that high geometric confinement induces a novel quasi-ferromagnetic phase, which breaks the interior superstructure order. Our work provides a predictive framework for designing frustrated materials and demonstrates a powerful AI approach for boundary-sensitive physical systems.

Layered magnetic systems with interlayer antiferromagnetic coupling are emerging as a promising platform for energy-efficient spintronic technologies, particularly in the form of van der Waals (vdW) magnets. The direct observation and manipulation of topological spin textures in perpendicularly magnetized 2D antiferromagnets, however, remain highly challenging due to their compensated magnetic states and rigid interlayer exchange, thereby hindering direct insight into spin ordering and functional exploration. Here, using spin-polarized low-energy electron microscopy, we directly visualize antiferromagnetic domains and emergent topological textures in the vdW antiferromagnet (Fe1-xCox)3GaTe2. We uncover tuneable stripe-like domain configurations that can be reconfigured into antiferromagnetic skyrmions under external stimuli, despite the absence of remanent magnetization. These results provide direct insight into spin ordering in a perpendicularly magnetized layered two-dimensional antiferromagnet and open new avenues for next-generation antiferromagnetic spintronic devices.

This study proposes a novel approach to extract topological properties, specifically the Euler characteristic, from input images using neural networks without relying on large pre-existing datasets but with a single geometric image. Inspired by solid-state physics, where topological properties of magnetic structures are derived from spin field analysis, our model generates a unit vector field from an image, interpreted as a spin configuration. The Euler characteristic is then predicted by computing the skyrmion number of this generated spin configuration. Remarkably, the network learns to construct chiral magnetic textures without access to ground-truth chiral spin configurations, relying instead on only a single, simple geometric image and the straightforward skyrmion number computation. Furthermore, spin configurations generated by independently trained networks can be non-unique due to inherent degrees of freedom. To constrain these degrees of freedom and further refine the spin configuration, we incorporate a magnetic Hamiltonian, comprising exchange interaction, Dzyaloshinskii-Moriya (DM) interaction, and anisotropy, as an additional, physics-informed loss function. We validate the model's efficacy on complex geometrical shapes and demonstrate its applicability to practical tasks.

Magnetic skyrmionic structures, including magnetic skyrmions and antiskyrmions, are characterized by swirling spin textures with non-trivial topologies. They are featured with specific topological charges, Q, which are of crucial importance in determining their topological properties. Owing to the invariance of the chiral nature, it is generally believed that Q is conserved in a given magnetic skyrmionic structure and is hard to alter. Here, we experimentally realize the control of Q of magnetic skyrmionic structures at room temperature in a Dzyaloshinskii-Moriya interaction (DMI) platform with spatially alternating signs. Depending on how many times it crosses the interfaces between DMI regions with opposite signs, the magnetic skyrmionic structures possess different Q. Modifying the DMI energy landscape through chemisorbed oxygen, a magnetic topological transition is realized. This creation and manipulation of magnetic skyrmionic structures with controllable Q, in particular the DMI-stabilized thin-film antiskyrmions and high-Q skyrmionic structures, enables a new degree of freedom to control their dynamics via a novel DMI confinement effect. Our findings open up an unexplored avenue on various topological magnetic skyrmionic structures and their potential applications.

Our study investigates the method to obtain topological properties of input images with neural networks, not requiring training datasets. In the field of solid-state physics, research has been conducted to obtain topological properties of magnetic structures by analyzing the spin fields. Utilizing the approaches, our model generates a unit vector field interpreted as spin fields from various images and predicts the Euler characteristic of input images by computing the skyrmion number of the generated vector field. Even if the networks are trained by a single image of a fixed Euler characteristic, they successfully predict the Euler characteristics of the various images. The resulting spin configurations from independently trained neural networks are not unique due to the remaining degrees of freedom in the spin configuration. To further control the spin configuration by confining these degrees of freedom, we incorporate a magnetic Hamiltonian as an additional loss function, which includes exchange Interaction, Dzyaloshinskii-Moriya (DM) Interaction, and anisotropy. We validate the model on more complex geometrical shapes and apply it to practical tasks.

We introduce a novel computational methodology for indexing the Euler characteristics of [Formula: see text]-dimensional objects by overlaying ([Formula: see text]+1)-dimensional chiral vector fields. Analogous to how the skyrmion number characterizes a two-dimensional magnetic skyrmion through the integration of the solid angle of its spin field, we generalize this principle to arbitrary dimensions. By iteratively applying a simple numerical process, we generate ([Formula: see text]+1)-dimensional chiral vector fields on [Formula: see text]-dimensional objects. The Euler characteristics of these objects are calculated by aggregating the local solid angles subtended by neighboring chiral vectors. In this study, we focus on verifying our method in two and three dimensions. For dimensions higher than three, we conduct preliminary experiments on simple objects to explore potential applicability. Although our method shows promising potential in higher dimensions, further investigation is required to fully understand its applicability beyond three dimensions.

The spin structures, including magnetic skyrmions, have attracted great attention due to their interesting properties caused by their topology. We introduce a magnetic structure tensor as a powerful tool for characterizing topological properties in two-dimensional spin textures. We derived an order parameter from the two orthogonal components of the magnetic structure tensor. The order parameter effectively analyzes the structures of the spin patterns and identifies topological defects such as the end or branching points of linear chiral structures and magnetic skyrmions, as a pole or multipoles composed of topological quantized charges. We present several spin configurations interpreted as topological monopoles, dipoles, and quadrupoles formed by the combination of these topological poles. This approach provides valuable insights into defect distribution and interactions, validated through both simulations and experimental data. Moreover, the approach can be generally applied for effective analysis of self-organized patterns such as fingerprints.

The phase transition phenomenon is an important research topic in various physical studies. However, it is difficult to define the order parameters in many complex systems involving self-organized structures. We propose a method to define order parameters using a variational autoencoder network. To demonstrate these capabilities, we trained a deep learning network with a dataset composed of spin configurations in a chiral magnetic system at various temperatures. It removes thermal fluctuations from the input data and leaves the remaining structural information with a spin magnitude. We define an order parameter with magnitude of output spins and compare the results with those of conventional analysis. The comparison indicates similar results. Using the order parameter, the thermal properties of the chiral magnetic system were investigated by varying the physical parameters and data size.

Topological analysis is widely adopted in various research fields to unveil intricate features and structural relationships implied in geometrical objects. Especially, in the fields of data analysis, exploring the topological properties of various images offers rich insights into the intrinsic geometrical information within them. In this study, a novel approach is proposed to investigate the topological properties of arbitrary grayscale images by employing a straightforward procedure used in 2D magnetism studies to calculate topological numbers. This method utilizes machine learning techniques to transfer chiral magnetic textures onto the images. Then, the topological number is then computed directly from the converted images by integrating the solid angles formed by adjacent spin vectors. The method successfully identifies the topological numbers of various grayscale images, showing stable performances against small noises. Furthermore, two applications of the method: are demonstrated topological analysis of the Modified National Institute of Standards and Technology (MNIST) dataset and the counting of blood cells in microscopic images.

The Dzyaloshinskii-Moriya antisymmetric exchange interaction (DMI) stabilises topological spin textures with promising future spintronics applications. According to crystal symmetry, the DMI can be categorized as four different types that favour different chiral textures. Unlike the other three extensively-investigated types, out-of-plane DMI, as the last type that favours in-plane chirality, remained missing so far. Here we apply point-group-dependent DMI matrix analysis to show that out-of-plane DMI exists under reduced crystal symmetry. Through strain and structure engineering, we show how Cs symmetry is realized in ultrathin magnets and observe the out-of-plane DMI stabilised in-plane chirality using spin-polarized electron microscopy. Our results show that extremely low out-of-plane DMI strengths at µeV/atom are sufficient to stabilise topological spin textures, including merons and bimerons. We also demonstrate field-induced reversible control of the in-plane chirality and merons. Our findings open up untapped paths on topological magnetic textures and their potential applications.

We construct a deep neural network to enhance the resolution of spin structure images formed by spontaneous symmetry breaking in the magnetic systems. Through the deep neural network, an image is expanded to a super-resolution image and reduced to the original image size to be fitted with the input feed image. The network does not require ground truth images in the training process. Therefore, it can be applied when low-resolution images are provided as training datasets, while high-resolution images are not obtainable due to the intrinsic limitation of microscope techniques. To show the usefulness of the network, we train the network with two types of simulated magnetic structure images; one is from self-organized maze patterns made of chiral magnetic structures, and the other is from magnetic domains separated by walls that are topological defects of the system. The network successfully generates high-resolution images highly correlated with the exact solutions in both cases. To investigate the effectiveness and the differences between datasets, we study the network's noise tolerance and compare the networks' reliabilities. The network is applied with experimental data obtained by magneto-optical Kerr effect microscopy and spin-polarized low-energy electron microscopy.

Recently, deep generative models using machine intelligence are widely utilized to investigate scientific systems by generating scientific data. In this study, we experiment with a hybrid model of a variational autoencoder (VAE) and a generative adversarial network (GAN) to generate a variety of plausible two-dimensional magnetic topological structure data. Due to the topological properties in the system, numerous and diverse metastable magnetic structures exist, and energy and topological barriers separate them. Thus, generating a variety of plausible spin structures avoiding those barrier states is a challenging problem. The VAE-GAN hybrid model can present an effective approach to this problem because it brings the advantages of both VAE's diversity and GAN's fidelity. It allows one to perform various applications including searching a desired sample from a variety of valid samples. Additionally, we perform a discriminator-driven latent sampling (DDLS) using our hybrid model to improve the quality of generated samples. We confirm that DDLS generates various plausible data with large coverage, following the topological rules of the target system.

Magnetic skyrmions are topologically nontrivial spin textures with envisioned applications in energy-efficient magnetic information storage. Toggling the presence of magnetic skyrmions via writing/deleting processes is essential for spintronics applications, which usually require the application of a magnetic field, a gate voltage or an electric current. Here we demonstrate the reversible field-free writing/deleting of skyrmions at room temperature, via hydrogen chemisorption/desorption on the surface of Ni and Co films. Supported by Monte-Carlo simulations, the skyrmion creation/annihilation is attributed to the hydrogen-induced magnetic anisotropy change on ferromagnetic surfaces. We also demonstrate the role of hydrogen and oxygen on magnetic anisotropy and skyrmion deletion on other magnetic surfaces. Our results open up new possibilities for designing skyrmionic and magneto-ionic devices.

Chiral magnetic domains are topological spin textures in which the Dzyaloshinskii–Moriya interaction assigns a given chirality to the domain walls. Notably, despite rapid progress in chiral magnetic research, one fundamental issue that remains unclear is how the chirality of chiral magnetic domains change as a magnetic field deforms the spin texture. Using spin‐polarized low energy electron microscopy, the evolution of Fe/Ni chiral magnetic stripe domains are investigated in single‐crystalline Fe/Ni/Cu/Co/Cu(001) multilayers in which the interlayer magnetic coupling between the Co and Fe/Ni films serves as an in‐plane magnetic field. Contrary to theoretical works, it is found that the chirality of the Néel walls results in a parallel alignment of the magnetic stripes with the in‐plane magnetic field direction. The transformation of chiral Néel walls into achiral Bloch walls can be precisely controlled by tuning the Cu spacer layer thickness. In addition, the domain wall exhibits a spontaneous asymmetry within the in‐plane magnetic field, leading to an unbalanced chirality between the left‐handed and right‐handed Bloch walls. These new results foster a better understanding of the chiral domain properties within a magnetic field.

We propose a strategy for optimizing physical quantities based on exploring in the latent space of a variational autoencoder (VAE). We train a VAE model using various spin configurations formed on a two-dimensional chiral magnetic system. Three optimization algorithms are used to explore the latent space of the trained VAE. The first algorithm, the single-code modification algorithm, is designed for improving the local energetic stability of spin configurations to generate physically plausible spin states. The other two algorithms, the genetic algorithm and the stochastic algorithm, aim to optimize the global physical quantities, such as topological index, magnetization, energy, and directional correlation. The advantage of our method is that various optimization algorithms can be applied in the latent space containing the abstracted representation constructed by the trained VAE model. Our method based on latent space exploration is utilized for efficient physical quantity optimization.

Autoencoder, an artificial neural network, was adopted to generate spin structures that interpolate and extrapolate between two distinct magnetic chiral states, the labyrinth structure and the skyrmion structure. We trained the autoencoder using two distinct magnetic chiral structures. Each input data is encoded through a deep learning process into a latent code, a new representation of the information in a reduced dimensional space. We investigated the latent space to acquire information on the structure of the latent code distribution. With the acquired information, we successfully produced various magnetic structures that exhibit plausible properties under various external fields not provided in the training data. The latent codes were modified by two algorithms. The first algorithm utilizes inversion and translation operation in the latent space and the second algorithm uses recursive flow with a modification bias. The first produced structures preserving the chiral structure of original data and the second produced statistically plausible states.

We investigated spin wave (SW) propagation and emission in thin film systems with strong interfacial Dzyaloshinskii-Moriya interaction (DMI) utilizing micromagnetic simulation. The effect of DMI on SW propagation is analogous to the flow of magnetic medium leading to the spin Doppler effect, and a spin-polarized current can enhance or suppress it. We demonstrate that for a Doppler velocity exceeding a critical value, a shock-wave-like emission of SWs with a cone-shape emerges from a magnetically irregular point as the cone apex. The cone angle is quantitatively determined by the DMI-induced Doppler velocity. Combining the interfacial DMI and the spin-polarized current, a constant SW emission by a static source is demonstrated, which provides a promising route to efficiently generate SWs with tunable frequency.

Searching for the ground state of a given system is one of the most fundamental and classical questions in scientific research fields. However, when the system is complex and large, it often becomes an intractable problem; there is essentially no possibility of finding a global energy minimum state with reasonable computational resources. Recently, a novel method based on deep learning techniques was devised as an innovative optimization method to estimate the ground state. We apply this method to one of the most complicated spin-ice systems, aperiodic Penrose P3 patterns. From the results, we discover new configurations of topologically induced emergent frustrated spins, different from those previously known. Additionally, a candidate of the ground state for a still unexplored type of Penrose P3 spin-ice system is first proposed through this study. We anticipate that the capabilities of the deep learning techniques will not only improve our understanding on the physical properties of artificial spin-ice systems, but also bring about significant advances in a wide range of scientific research fields requiring computational approaches for optimization.

Two-dimensional (2D) magnetic van der Waals crystals have received great attentions recently owing to their novel electronic and magnetic properties and rich potentials for low-dimensional and high-density spintronic devices. We investigated the magnetization dynamics of CrI3 and CrBr3 bulk single crystals by utilizing broadband ferromagnetic resonance (FMR) in a frequency range of 1 - 40 GHz and over a wide temperature range of 10 - 300 K. Complex features observed in the FMR spectra are quantitatively described by the multi-domain FMR theory. The minimum resonance frequency f(min) in FMR spectra was recognized and analyzed as characteristics of the transition of the domain structure. Micromagnetic simulation on the domain structure and spin dynamics explains that the multi-domain structure exists under strong in-plane magnetic field for CrI3 and CrBr3 and significantly influences the magnetic dynamics properties. Our results also suggest that the linewidths of FMR spectra in CrI3 and CrBr3 are dominated by the magnetic inhomogeneous broadening.

Numerical generation of physical states is essential to all scientific research fields. The role of a numerical generator is not limited to understanding experimental results; it can also be employed to predict or investigate characteristics of uncharted systems. A variational autoencoder model is devised and applied to a magnetic system to generate energetically stable magnetic states with low local deformation. The spin structure stabilization is made possible by taking the explicit magnetic Hamiltonian into account to minimize energy in the training process. A significant advantage of the model is that the generator can create a long-range ordered ground state of spin configuration by increasing the role of stabilization even if the ground states are not necessarily included in the training process. It is expected that the proposed Hamiltonian-guided generative model can bring about great advances in numerical approaches used in various scientific research fields.