金属磁体的低能模型：机器学习辅助推导

费米子与经典场的相互作用囊括了各种领域的知识，例如量子化学、凝聚态物理以及高能物理。在这些领域中，磁性金属的阻挫问题吸引了不少学者的关注，因为磁性金属中的四体自旋相互作用可以稳定非共面的磁有序，从而诱导非零的贝利曲率，在转变温度下产生明显的拓扑霍尔效应。例如，f-电子磁体中存在磁场诱导的斯格明子晶体（SkXs），可以实现拓扑霍尔效应。

针对SkXs，已有不同方法对产生机制等行为进行了研究，包括基于传统微扰理论的三角晶格RKKY模型，三角近藤晶格模型（KLM）等。随之，越来越多的有效四体相互作用形式被提出，但是这些多数还是基于唯象理论，没有考虑所有对称性允许的相互作用。

来自田纳西大学物理与天文系的Vikram Sharma等，提出了一种获得自旋哈密顿量的方法，可适用于传统方法失效时的场景，比如具有非解析相互作用的三角KLM求解。

Fig. 3 T = 0 phase diagrams of the KLM at J/t = 0.5 and nc = 0.0586.

作者证明，利用机器学习辅助协议生成的低能有效模型可以准确地预测大部分的相边界，并且98%的点没有包含在训练集中，很好地体现出了模型的通用性。

该方法不仅计算资源消耗小，同时具有非常好的准确性，所以在计算原始KLM相图方面具有广泛的应用前景。相关论文近期发布于npj Computational Materials 9: 192 (2023)。

Fig. 5 Evolution of parameters as the strength of L1 regularization is increased.

Editorial Summary

Lattice models of fermions interacting with classical fields encompass different areas of knowledge, including quantum chemistry, condensed matter, and high-energy physics. In these fields, the frustration of magnetic metals attracts the attention of scientists. For instance, four-spin interactions can stabilize non-coplanar orderings that induce nonzero Berry curvature of the reconstructed bands, leading to a large topological Hall effect below the magnetic ordering temperature. An outstanding example is the search for field-induced skyrmion crystals (SkXs) in f-electron magnets. Different methods have been used to study the SkXs, including the triangular lattice RKKY model based on the traditional perturbation theory, and the triangular Kondo lattice model (KLM), etc. Since then, more effective four-spin interactions have been proposed, but most of them are phenomenological, because they do not consider all the symmetry-allowed four-spin interactions.

Vikram Sharma et al. from the Department of Physics and Astronomy, University of Tennessee, proposed a pathway to derive spin Hamiltonians when conventional methods fail, such as a triangular KLM which gives rise to effective four-spin interactions that are non-analytic functions of J/t. The authors demonstrated that the low-energy effective model generated by the machine learn-assisted protocol can accurately predict the main phase boundaries of the phase diagram, and over 98% of the points are not included in the training set, thus demonstrating the generalizability of the model. This method has a much lower numerical cost relative to the original KLM while keeps the great accuracy. Therefore, the protocol presented in this work will have a wide application prospect in the calculation of phase diagram of original KLM. This article was recently published in npj Computational Materials 9: 192 (2023).

原文Abstract及其翻译

Machine learning assisted derivation of minimal low-energy models for metallic magnets (机器学习辅助推导金属磁体的最小低能模型)

Vikram Sharma, Zhentao Wang & Cristian D. Batista 

Abstract We consider the problem of extracting a low-energy spin Hamiltonian from a triangular Kondo Lattice Model (KLM). The non-analytic dependence of the effective spin-spin interactions on the Kondo exchange excludes the use of perturbation theory beyond the second order. We then introduce a Machine Learning (ML) assisted protocol to extract effective two- and four-spin interactions. The resulting spin model reproduces the phase diagram of the original KLM as a function of magnetic field and single-ion anisotropy and reveals the effective four-spin interactions that stabilize the field-induced skyrmion crystal phase. Moreover, this model enables the computation of static and dynamical properties with a much lower numerical cost relative to the original KLM. A comparison of the dynamical spin structure factor in the fully polarized phase computed with both models reveals a good agreement for the magnon dispersion even though this information was not included in the training data set.

摘要我们考虑从一个三角近藤晶格模型（KLM）中提取低能自旋哈密顿量。有效自旋–自旋相互作用对近藤交换的非解析依赖性，使得超越二阶的微扰理论失效。对此，我们引入了一种机器学习（ML）辅助协议，来提取有效的二体和四体自旋相互作用。所得到的自旋模型再现了以磁场和离子各向异性为变量的原始KLM相图，同时揭示了能够稳定场诱导斯格明子的有效四体自旋相互作用。相比于原始KLM，该模型在计算静态和动态特性时可以节省更多资源。利用两种模型计算完全极化相中的动态自旋结构因子可以发现，尽管相关信息未包含在训练数据集中，但在磁振子色散方面依旧可以展现出良好的一致性。