基于符号回归方法探索磁性斯格明子结构近似解析式

史猛; 王伟伟; 杜海峰

doi:10.7498/aps.73.20231473

摘要

磁性斯格明子是一种非平庸的拓扑磁结构, 它能够在具有Dzyaloshinskii-Moriya (DM)相互作用的手性磁体中稳定存在, 其静态以及动态特性与其结构特征息息相关. 然而, 一般情况下斯格明子结构解析式并不存在. 因此, 许多研究者给出了近似解析式. 本文介绍了使用符号回归算法探索磁性斯格明子结构解析式的新方法, 在考虑DM相互作用和外部磁场对磁性斯格明子结构的影响下, 使用符号回归算法得到了两个较为合适的近似解析式, 其适用范围与占主导地位的相互作用有关. 研究结果验证了符号回归算法在探索磁性斯格明子结构解析式的强大能力, 为磁结构的解析式探索提供了新思路.

关键词:

Abstract

Magnetic skyrmion is a kind of nontrivial topological magnetic structure, which can exist stably in chiral magnet with Dzyaloshinskii-Moriya (DM) interaction, and its static and dynamic properties are closely related to its structural characteristics. However, there are no general analytical expressions for skyrmion profiles. Therefore, many researchers have provided approximate solutions. In this paper, a new approach to exploring magnetic skyrmion structures is introduced by using a symbolic regression approach. Considering the influence of DM interaction and external magnetic field on magnetic skyrmion structure, two suitable approximate expressions are obtained through symbolic regression algorithms. The applicability of these expressions depends on the dominant interaction. The research results in this work validate the powerful capability of symbolic regression algorithms in exploring the magnetic skyrmion profiles. So, the present study provides a new method for finding the analytical expressions for magnetic structure.

Keywords:

作者及机构信息

1.
中国科学院合肥物质科学研究院, 强磁场科学中心, 合肥　230031

2.
中国科学技术大学, 合肥　230026

3.
安徽大学物质科学与信息技术研究院, 合肥　230601

通信作者: 杜海峰, duhf@hmfl.ac.cn

基金项目: 国家重点研发计划(批准号: 2022YFA1403603)、中国科学院战略性先导科技专项(批准号: XDB33030100)、国家自然科学基金(批准号: 12241406)、国家杰出青年科学基金(批准号: 52325105)、中国科学院装备发展项目(批准号: YJKYYQ20180012)和中国科学院青年基础研究项目(批准号: YSBR-084)资助的课题.

Authors and contacts

1.
High Magnetic Field Laboratory, Hefei Institutes of Physical Science, Chinese Academy of Sciences, Hefei 230031, China

2.
University of Science and Technology of China, Hefei 230026, China

3.
Institutes of Physical Science and Information Technology, Anhui University, Hefei 230601, China

Corresponding author: Du Hai-Feng, duhf@hmfl.ac.cn

Funds: Project supported by the National Key R&D Program of China (Grant No. 2022YFA1403603), the Strategic Priority Research Program of Chinese Academy of Sciences, China (Grant No. XDB33030100), the National Natural Science Foundation of China (Grant No. 12241406), the National Natural Science Fund for Distinguished Young Scholars of China (Grant No. 52325105), the Equipment Development Project of Chinese Academy of Sciences, China (Grant No. YJKYYQ20180012), and the Project for Young Scientists in Basic Research of Chinese Academy of Sciences, China (Grant No. YSBR-084).

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参考文献

[1]	Abanov Ar, Pokrovsky V L 1998 Phys. Rev. B 58 R8889 Google Scholar
[2]	Rößler U K, Bogdanov A N, Pfleiderer C 2006 Nature 442 797 Google Scholar
[3]	Heinze S, von Bergmann K, Menzel M, Brede J, Kubetzka A, Wiesendanger R, Bihlmayer G, Bluegel S 2011 Nat. Phys. 7 713 Google Scholar
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[5]	Ye C, Li L L, Shu Y, Li Q R, Xia J, Hou Z P, Zhou Y, Liu X X, Yang Y Y, Zhao G P 2022 Rare Met. 41 2200 Google Scholar
[6]	Braun H 1994 Phys. Rev. B 50 16485 Google Scholar
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[8]	Rohart S, Thiaville A 2013 Phys. Rev. B 88 184422 Google Scholar
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施引文献

图 1 磁性斯格明子结构　(a) 布洛赫型斯格明子; (b) 奈尔型斯格明子; (c)单磁畴壁

Fig. 1. Magnetic skyrmion structures: (a) Bloch-type skyrmion; (b) Néel-type skyrmion; (c) magnetic domain wall.

下载: 全尺寸图片幻灯片

图 2 符号回归算法中二叉树结构与遗传操作示意图　(a) 用二叉树表示公式形式; (b) 复制操作; (c) 交叉操作; (d) 变异操作

Fig. 2. Binary tree structure and schematic diagrams of genetic operations in symbolic regression: (a) Representing the formula in binary tree form; (b) copy operation; (c) crossover operation; (d) mutation operation.

下载: 全尺寸图片幻灯片

图 3 符号回归方法流程图

Fig. 3. Flowchart of the symbolic regression.

下载: 全尺寸图片幻灯片

图 4 不同A与K值下, (8)式拟合结果与一维单磁畴壁结构模拟数据比较图　(a) A = 1×10^–12 J/m, K = 1×10³ J/m³; (b) A = 5×10^–12 J/m, K = 2×10³ J/m³; (c) A = 13×10^–12 J/m, K = 3×10³ J/m³

Fig. 4. Comparison between the fitting results of Eq. (8) and simulation data of one-dimensional magnetic domain wall under various values of A and K: (a) A = 1×10^–12 J/m, K = 1×10³ J/m³; (b) A = 5×10^–12 J/m, K = 2×10³ J/m³; (c) A = 13×10^–12 J/m, K = 3×10³ J/m³.

下载: 全尺寸图片幻灯片

图 5 $ {\lambda }^{2}/h=0.01 $ 时的帕累托最优　(a) (10a)式; (b) (10b)式

Fig. 5. Pareto optimum when $ {\lambda }^{2}/h=0.01 $: (a) Eq. (10a); (b) Eq. (10b).

下载: 全尺寸图片幻灯片

图 6 不同$ {\lambda }^{2}/h $大小下(10a)式与 (10b)式的拟合情况　(a) 0.01; (b) 0.16; (c) 0.167; (d) 0.9. (e) 不同$ {\lambda }^{2}/h $大小下, 更高适应度解析式统计图(1代表(10a)式, 2代表(10b)式)

Fig. 6. Fitting results of equations (10a) and (10b) under various $ {\lambda }^{2}/h $ values: (a) 0.01; (b) 0.16; (c) 0.167; (d) 0.9$ . $ (e) Statistical chart of equations with higher fitness under various $ {\lambda }^{2}/h $ values (1 represents equation (10a), 2 represents equation (10b)).

下载: 全尺寸图片幻灯片

图 7 不同$ {\lambda }^{2}/h $大小下 (11a)式或(11b)式的拟合情况　(a) 0.01; (b) 0.16; (c) 0.167; (d) 0.9

Fig. 7. Fitting results of Eq. (11a) or Eq. (11b) under various $ {\lambda }^{2}/h $ values: (a) 0.01; (b) 0.16; (c) 0.167; (d) 0.9.

下载: 全尺寸图片幻灯片

图 8 不同$ {\lambda }^{2}/h $大小下, (11a)式或(11b)式的拟合情况　(a) 0.009; (b) 0.015; (c) 0.139; (d) 0.192; (e) 0.227; (f) 0.769

Fig. 8. Fitting results of Eq. (11a) or (11b) under various $ {\lambda }^{2}/h $ values: (a) 0.009; (b) 0.015; (c) 0.139; (d) 0.192; (e) 0.227; (f) 0.769

下载: 全尺寸图片幻灯片

[1]	Abanov Ar, Pokrovsky V L 1998 Phys. Rev. B 58 R8889 Google Scholar
[2]	Rößler U K, Bogdanov A N, Pfleiderer C 2006 Nature 442 797 Google Scholar
[3]	Heinze S, von Bergmann K, Menzel M, Brede J, Kubetzka A, Wiesendanger R, Bihlmayer G, Bluegel S 2011 Nat. Phys. 7 713 Google Scholar
[4]	Wei W S, He Z D, Qu Z, Du H F 2021 Rare Met. 40 3076 Google Scholar
[5]	Ye C, Li L L, Shu Y, Li Q R, Xia J, Hou Z P, Zhou Y, Liu X X, Yang Y Y, Zhao G P 2022 Rare Met. 41 2200 Google Scholar
[6]	Braun H 1994 Phys. Rev. B 50 16485 Google Scholar
[7]	Romming N, Kubetzka A, Hanneken C, von Bergmann K, Wiesendanger R 2015 Phys. Rev. Lett. 114 177203 Google Scholar
[8]	Rohart S, Thiaville A 2013 Phys. Rev. B 88 184422 Google Scholar
[9]	Zhou Y, Iacocca E, Awad A A, Dumas R K, Zhang F C, Braun H B, Akerman J 2015 Nat. Commun. 6 8193 Google Scholar
[10]	Buttner F, Lemesh I, Beach G S D 2018 Sci. Rep. 8 4464 Google Scholar
[11]	Komineas S, Melcher C, Venakides S 2020 Nonlinearity 33 3395 Google Scholar
[12]	Komineas S, Melcher C, Venakides S 2021 Physica D 418 132842 Google Scholar
[13]	Komineas S, Melcher C, Venakides S 2023 New J. Phys. 25 023013 Google Scholar
[14]	Udrescu S M, Tegmark M 2020 Sci. Adv. 6 eaay2631 Google Scholar
[15]	Kim S, Lu P Y, Mukherjee S, Gilbert M, Jing L, Ceperic V, Soljacic M 2021 IEEE Trans. Neural Networks Learn. Syst. 32 4166 Google Scholar
[16]	Raissi M, Perdikaris P, Karniadakis G E 2019 J. Comput. Phys. 378 686 Google Scholar
[17]	Sun S, Ouyang R, Zhang B, Zhang T Y 2019 MRS Bull. 44 559 Google Scholar
[18]	Koksbang S M 2023 Phys. Rev. D 107 103522 Google Scholar
[19]	Hernandez A, Balasubramanian A, Yuan F, Mason S A M, Mueller T 2019 NPJ Comput. Mater. 5 112 Google Scholar
[20]	Baldi P, Sadowski P, Whiteson D 2014 Nat. Commun. 5 4308 Google Scholar
[21]	Carleo G, Troyer M 2017 Science 355 602 Google Scholar
[22]	Zhao G P, Zhao L, Shen L C, Zou J, Qiu L 2019 Chin. Phys. B 28 77505 Google Scholar
[23]	Jones A 1993 Nature 363 222 Google Scholar
[24]	Cranmer M 2023 arXiv: 10.48550/arXiv.2305.01582 [astro-ph.IM
[25]	Wu H, Hu X, Jing K, Wang X R 2021 Commun. Phys. UK 4 1 Google Scholar

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基于符号回归方法探索磁性斯格明子结构近似解析式