費(fèi)米子與經(jīng)典場(chǎng)的相互作用囊括了各種領(lǐng)域的知識(shí)，例如量子化學(xué)、凝聚態(tài)物理以及高能物理。在這些領(lǐng)域中，磁性金屬的阻挫問(wèn)題吸引了不少學(xué)者的關(guān)注，因?yàn)榇判越饘僦械乃捏w自旋相互作用可以穩(wěn)定非共面的磁有序，從而誘導(dǎo)非零的貝利曲率，在轉(zhuǎn)變溫度下產(chǎn)生明顯的拓?fù)浠魻栃?yīng)。例如，f-電子磁體中存在磁場(chǎng)誘導(dǎo)的斯格明子晶體（SkXs），可以實(shí)現(xiàn)拓?fù)浠魻栃?yīng)。

針對(duì)SkXs，已有不同方法對(duì)產(chǎn)生機(jī)制等行為進(jìn)行了研究，包括基于傳統(tǒng)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(lái)越多的有效四體相互作用形式被提出，但是這些多數(shù)還是基于唯象理論，沒(méi)有考慮所有對(duì)稱性允許的相互作用。

來(lái)自田納西大學(xué)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(tǒng)方法失效時(shí)的場(chǎng)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(jī)器學(xué)習(xí)輔助協(xié)議生成的低能有效模型可以準(zhǔn)確地預(yù)測(cè)大部分的相邊界，并且98%的點(diǎn)沒(méi)有包含在訓(xùn)練集中，很好地體現(xiàn)出了模型的通用性。

該方法不僅計(jì)算資源消耗小，同時(shí)具有非常好的準(zhǔn)確性，所以在計(jì)算原始KLM相圖方面具有廣泛的應(yīng)用前景。相關(guān)論文近期發(fā)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(jī)器學(xué)習(xí)輔助推導(dǎo)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(gè)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(duì)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(duì)此，我們引入了一種機(jī)器學(xué)習(xí)（ML）輔助協(xié)議，來(lái)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(xiàn)了以磁場(chǎng)和離子各向異性為變量的原始KLM相圖，同時(shí)揭示了能夠穩(wěn)定場(chǎng)誘導(dǎo)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(jì)算靜態(tài)和動(dòng)態(tài)特性時(shí)可以節(jié)省更多資源。利用兩種模型計(jì)算完全極化相中的動(dòng)態(tài)自旋結(jié)構(gòu)因子可以發(fā)現(xiàn)，盡管相關(guān)信息未包含在訓(xùn)練數(shù)據(jù)集中，但在磁振子色散方面依舊可以展現(xiàn)出良好的一致性。