費米子與經典場的相互作用囊括了各種領域的知識，例如量子化學、凝聚態物理以及高能物理。在這些領域中，磁性金屬的阻挫問題吸引了不少學者的關注，因為磁性金屬中的四體自旋相互作用可以穩定非共面的磁有序，從而誘導非零的貝利曲率，在轉變溫度下產生明顯的拓撲霍爾效應。例如，f-電子磁體中存在磁場誘導的斯格明子晶體（SkXs），可以實現拓撲霍爾效應。

針對SkXs，已有不同方法對產生機制等行為進行了研究，包括基于傳統微擾理論的三角晶格RKKY模型，三角近藤晶格模型（KLM）等。隨之，越來越多的有效四體相互作用形式被提出，但是這些多數還是基于唯象理論，沒有考慮所有對稱性允許的相互作用。

來自田納西大學物理與天文系的Vikram Sharma等，提出了一種獲得自旋哈密頓量的方法，可適用于傳統方法失效時的場景，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.

作者證明，利用機器學習輔助協議生成的低能有效模型可以準確地預測大部分的相邊界，并且98%的點沒有包含在訓練集中，很好地體現出了模型的通用性。

該方法不僅計算資源消耗小，同時具有非常好的準確性，所以在計算原始KLM相圖方面具有廣泛的應用前景。相關論文近期發布于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及其翻譯

Machine learning assisted derivation of minimal low-energy models for metallic magnets (機器學習輔助推導金屬磁體的最小低能模型)

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.

摘要我們考慮從一個三角近藤晶格模型（KLM）中提取低能自旋哈密頓量。有效自旋–自旋相互作用對近藤交換的非解析依賴性，使得超越二階的微擾理論失效。對此，我們引入了一種機器學習（ML）輔助協議，來提取有效的二體和四體自旋相互作用。所得到的自旋模型再現了以磁場和離子各向異性為變量的原始KLM相圖，同時揭示了能夠穩定場誘導斯格明子的有效四體自旋相互作用。相比于原始KLM，該模型在計算靜態和動態特性時可以節省更多資源。利用兩種模型計算完全極化相中的動態自旋結構因子可以發現，盡管相關信息未包含在訓練數據集中，但在磁振子色散方面依舊可以展現出良好的一致性。