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Systematic Magnetic Structure Generation Based on Oriented Spin Space Groups: Formulation, Applications, and High-Throughput First-Principles Calculations

Takuya Nomoto, Kohei Shinohara, Hikaru Watanabe, Ryotaro Arita

TL;DR

This work introduces an oriented spin space group (SSG) framework to exhaustively generate magnetic structures as spin-symmetry-adapted (SSA) representations and their oriented variants, enforcing fixed moment magnitudes and leveraging SOC-driven energy-scale hierarchies. It provides a rigorous enumeration algorithm to derive SSA structures from a given space group and spin-only group, followed by a systematic reduction to oriented SSA structures that are inequivalent under the parent space group. Through application to MAGNDATA and high-throughput first-principles calculations, the authors demonstrate substantial reproducibility: about $77\%$ of entries are reproducible at the SSG level and $\sim82\%$ of those at the oriented-SSG level, with a clear SOC-induced energy-scale separation (SSA gaps ~$\sim$103 meV per magnetic atom vs oriented-SSA gaps ~$\sim$0.29 meV per atom). The two-step workflow—SCF calculations without SOC for SSA structures, then fixed-charge non-SCF calculations with SOC for oriented SSA structures—yields stable, efficient predictions, suggesting a scalable path for large-scale magnetic-structure discovery and symmetry-guided materials design.

Abstract

We propose a framework for generating magnetic structures, inspired by the concept of oriented spin space groups (SSGs): magnetic structures are first generated as totally symmetric representations of an SSG and are then rotated such that they belong to the maximal magnetic space group of the SSG, which we term spin-symmetry-adapted (SSA) structures and oriented SSA structures, respectively. This is a natural framework to enforce fixed magnetic moment magnitudes on the symmetry-equivalent sites as well as to exploit the spin-orbit coupling (SOC)-induced hierarchy of energy scales. To examine the present scheme, we analyze the MAGNDATA database and find that 77% of the reported structures are reproducible at the SSG level, among which 82% are fully reproduced within the oriented SSG scheme, regardless of their spin-only group types or propagation vectors. To quantitatively assess computational and predictive performance, we perform spin density functional theory calculations for 283 materials, first carrying out self-consistent calculations for SSA structures without SOC, followed by fixed-charge calculations including SOC for the descendant oriented SSA structures. The experimental magnetic structures are reproduced as energetically most stable in 82% of cases at the SSG level without SOC and in 76% of cases at the oriented SSG level with SOC, showing that the fixed-charge scheme enables accurate evaluation of SOC-induced energy differences at low computational cost. The characteristic energy scale among oriented SSA structures is only $\sim$0.29 meV per magnetic atom, about 300 times smaller than that of distinct SSA structures. These results demonstrate that oriented SSG-based enumeration, combined with the two-step calculations for SSA and oriented SSA structures, provides an efficient and robust route for large-scale magnetic-structure prediction.

Systematic Magnetic Structure Generation Based on Oriented Spin Space Groups: Formulation, Applications, and High-Throughput First-Principles Calculations

TL;DR

This work introduces an oriented spin space group (SSG) framework to exhaustively generate magnetic structures as spin-symmetry-adapted (SSA) representations and their oriented variants, enforcing fixed moment magnitudes and leveraging SOC-driven energy-scale hierarchies. It provides a rigorous enumeration algorithm to derive SSA structures from a given space group and spin-only group, followed by a systematic reduction to oriented SSA structures that are inequivalent under the parent space group. Through application to MAGNDATA and high-throughput first-principles calculations, the authors demonstrate substantial reproducibility: about of entries are reproducible at the SSG level and of those at the oriented-SSG level, with a clear SOC-induced energy-scale separation (SSA gaps ~103 meV per magnetic atom vs oriented-SSA gaps ~0.29 meV per atom). The two-step workflow—SCF calculations without SOC for SSA structures, then fixed-charge non-SCF calculations with SOC for oriented SSA structures—yields stable, efficient predictions, suggesting a scalable path for large-scale magnetic-structure discovery and symmetry-guided materials design.

Abstract

We propose a framework for generating magnetic structures, inspired by the concept of oriented spin space groups (SSGs): magnetic structures are first generated as totally symmetric representations of an SSG and are then rotated such that they belong to the maximal magnetic space group of the SSG, which we term spin-symmetry-adapted (SSA) structures and oriented SSA structures, respectively. This is a natural framework to enforce fixed magnetic moment magnitudes on the symmetry-equivalent sites as well as to exploit the spin-orbit coupling (SOC)-induced hierarchy of energy scales. To examine the present scheme, we analyze the MAGNDATA database and find that 77% of the reported structures are reproducible at the SSG level, among which 82% are fully reproduced within the oriented SSG scheme, regardless of their spin-only group types or propagation vectors. To quantitatively assess computational and predictive performance, we perform spin density functional theory calculations for 283 materials, first carrying out self-consistent calculations for SSA structures without SOC, followed by fixed-charge calculations including SOC for the descendant oriented SSA structures. The experimental magnetic structures are reproduced as energetically most stable in 82% of cases at the SSG level without SOC and in 76% of cases at the oriented SSG level with SOC, showing that the fixed-charge scheme enables accurate evaluation of SOC-induced energy differences at low computational cost. The characteristic energy scale among oriented SSA structures is only 0.29 meV per magnetic atom, about 300 times smaller than that of distinct SSA structures. These results demonstrate that oriented SSG-based enumeration, combined with the two-step calculations for SSA and oriented SSA structures, provides an efficient and robust route for large-scale magnetic-structure prediction.
Paper Structure (37 sections, 56 equations, 6 figures, 3 tables)

This paper contains 37 sections, 56 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Enumerating spin-symmetry-adapted structures
  3. Group structure of spin space group
  4. Enumeration of spin space groups from family space group and spin-only group
  5. Generation of spin-symmetry-adapted structures from spin space group
  6. Practical choices of family space group
  7. Enumerating symmetry-adapted magnetic structures by subduing SSG to MSG
  8. Noncoplanar oriented spin space group
  9. Collinear oriented spin space group
  10. Coplanar oriented spin space group
  11. Chirality of spin space group
  12. Representative examples
  13. Collinear MnTe
  14. Coplanar Mn3Sn
  15. Noncoplanar CoTa3S6
  16. ...and 22 more sections

Figures (6)

  • Figure 1: Enumeration of collinear magnetic structures of MnTe. (a) SSGs compatible with the nonmagnetic family space group $\mathcal{G} = P 6_{3}/m m c$ (No. 194), collinear, and $N_k = 1$. Two SSGs are obtained: SSG-1 with normal space subgroup $\mathcal{H} = P \overline{3} m 1$ (No. 164), and SSG-2 with $\mathcal{H} = P 6_{3} / m m c$ (No. 194). For each SSG, its SSA structure is shown beneath the corresponding box. (b) Collinear oriented SSA structures derived from SSG-1. Three distinct oriented SSA structures, OSSA-1-1, OSSA-1-2, and OSSA-1-3, are obtained, corresponding to magnetic moments oriented along the crystallographic directions $\langle 0 1 0 \rangle$, $\langle 0 0 1 \rangle$, and $\langle 1 2 0 \rangle$, respectively. Each magnetic space-group type of the oriented SSA structures is indicated by the BNS numbers belov1957neronova. The magnetic moments lying in the $ab$ plane (OSSA-1-1 and OSSA-1-3) are consistent with the experimentally reported one (#0.800 in MAGNDATA).
  • Figure 2: Enumeration of coplanar magnetic structures of Mn$_{3}$Sn. (a) SSGs compatible with the nonmagnetic family space group $\mathcal{G} = P 6_{3}/m m c$ (No. 194), coplanar, and $N_k = 1$. Five SSGs are obtained: SSG-1 with normal space subgroup $\mathcal{H} = P m$ (No. 6), SSG-2 with $\mathcal{H} = P 2_{1}/m$ (No. 11), SSG-3 and SSG-4 with $\mathcal{H} = P \overline{6} m 2$ (No. 187), and SSG-5 with $\mathcal{H} = P 6_{3}/m m c$ (No. 194). For each SSG, SSA structures from linearly independent symmetry-adapted magnetic moment bases are shown beneath the corresponding boxes. (b) Coplanar oriented SSA structures derived from SSG-2 and its enantiomorphic one. For three inequivalent crystallographic orientations of the coplanar spin axis, eight oriented SSA structures (OSSA-2-1 to OSSA-2-8) are obtained in total. For $\langle 0 0 1 \rangle$, four oriented SSA structures appear, among which OSSA-2-3 (BNS No. 63.463) and OSSA-2-4 (BNS No. 63.464) correspond to the experimentally reported magnetic structures of Mn$_{3}$Sn in MAGNDATA entries #0.199 and #0.200, respectively. For $\langle 0 1 0 \rangle$ and $\langle 1 2 0 \rangle$, two oriented SSA structures are generated for each direction.
  • Figure 3: Enumeration of noncoplanar magnetic structures of CoTa$_{3}$S$_{6}$. (a) SSGs compatible with the nonmagnetic family space group $\mathcal{G} = P 6_{3} 2 2$ (No. 182), noncoplanar, and $N_k = 4$. Six SSGs are obtained: SSG-1 with normal space subgroup $\mathcal{H} = P 1$ (No. 1) and $2 \times 2 \times 1$ magnetic unit cell, SSG-2 with $\mathcal{H} = P 2_{1}$ (No. 4) and $2 \times 2 \times 1$ magnetic unit cell, and SSG-3 to SSG-6 with $\mathcal{H} = P 3$ (No. 143) and $1 \times 1 \times 4$ magnetic unit cell. For each SSG, SSA structures from linearly independent symmetry-adapted magnetic moment bases are shown beneath the corresponding boxes. (b) Noncoplanar oriented SSA structures generated from SSG-2. Three distinct oriented SSA structures (OSSA-2-1 to OSSA-2-3) are obtained, among which OSSA-2-2 (BNS No. 150.27) corresponds to the experimentally reported magnetic structure of CoTa$_{3}$S$_{6}$ in MAGNDATA entry #3.25. The nonmagnetic atoms of sulfur are not shown for clarity.
  • Figure 4: Magnetic structures of DyB$_4$. (a) Magnetic structures of DyB$_4$ reported in MAGNDATA entry #0.22. (b)-(d) Three SSA structures with $N_k=1$, whose family space group $\mathcal{G}=P4/mbm$ (No. 127) is identical to the space group. (e) and (f) Two SSA structures with the family space group $\mathcal{G}=Pbam$ (No. 55).
  • Figure 5: Magnetic structures of CoSO$_4$. (a) Magnetic structures of CoSO$_4$ reported in MAGNDATA entry #0.96. (b)-(e) Four SSA structures with $N_k=1$, whose family space group $\mathcal{G}=Pnma$ (No. 62) is identical to the space group. (c)-(e) correspond to collinear AFMs characterized by modulation directions $[100]$, $[010]$, and $[110]$, respectively.
  • ...and 1 more figures