Symmetry-guided prediction of magnetic-ordered ground states

Technical Summary: Symmetry-guided prediction of magnetic-ordered ground states

Problem Statement
The reliable prediction of magnetic ground states remains a significant challenge in materials science due to the immense complexity of the magnetic potential energy landscape and the scarcity of experimentally confirmed magnetic structures. While crystallographic databases contain hundreds of thousands of structures, experimentally reported magnetic structures are limited to a few thousand. Current computational strategies face two primary bottlenecks:

1. Energy-driven approaches (e.g., effective spin Hamiltonians, machine learning) often require a priori assumptions about the Hamiltonian form or demand extensive high-throughput calculations across thousands of random configurations to train models.
2. Conventional symmetry-guided approaches (e.g., magnetic representation analysis using Magnetic Space Groups, MSGs) treat magnetic moments as axial vectors coupled to the lattice. This coupling limits the resolution needed to differentiate distinct magnetic phases and results in too many undetermined continuous free parameters for efficient sampling.

Methodology
The authors propose a symmetry-guided framework that systematically generates realistic magnetic configurations without requiring experimental input, predefined propagation vectors, or prior assumptions. The workflow operates within a hierarchical symmetry-breaking scenario, decoupling the effects of magnetic ordering and Spin-Orbit Coupling (SOC):

1. Group-Subgroup Chain Construction (SG-SSG-MSG):

   • The framework constructs chains from the parent crystallographic Space Group (SG) to Spin Space Groups (SSGs) and finally to Magnetic Space Groups (MSGs) or Oriented Spin Space Groups (OSSGs).
   • SSG Level: In the absence of SOC, the system possesses $G_{SG} \times O(3)$ symmetry. Magnetic ordering breaks this down to an SSG subgroup ($G_{SSG}$), which captures the relative spin arrangement (magnetic geometry) independent of global spin orientation.
   • OSSG Level: The introduction of SOC lifts the global $O(3)$ rotational freedom, locking the spin space orientation relative to the lattice. This is described by OSSGs, which serve as parent groups for the final MSGs. The MSG emerges as a subgroup where spin and lattice operations are fully locked.

2. Systematic Generation via Spin Wyckoff Positions:

   • Magnetic geometries are enumerated using Spin Wyckoff positions of SSGs. These positions assign symmetry-allowed magnetic moments to crystallographic orbits, generating all possible relative spin arrangements.
   • To ensure computational tractability, the generation is constrained by statistical analysis of the MAGNDATA database. Constraints include:
     • Effective space group ($G_0$) subgroup-chain level $\le 1$.
     • Magnetic-order-induced Wyckoff-position splitting number $\le 1$.
     • Magnetic cell volume expansion factor $\le 4$.
   • This restricts the search to high-symmetry subgroups and physically equivalent sublattices, avoiding combinatorial explosions.

3. Two-Stage Energy Evaluation:

   • Stage I (Non-relativistic): First-principles calculations (DFT) without SOC are performed on the generated SSG-symmetry-constrained magnetic geometries to determine the dominant isotropic exchange interactions and identify the ground-state magnetic geometry.
   • Stage II (Relativistic): The identified ground-state geometry is further refined using compatible MSG subgroups and OSSG orientation matrices to resolve specific spin orientations. Relativistic DFT calculations (with SOC) are then used to evaluate magneto-crystalline anisotropy and determine the final ground-state magnetic structure.

Key Contributions

• Theoretical Framework: The integration of Oriented Spin Space Groups (OSSGs) into a hierarchical workflow allows for the exhaustive generation of symmetry-allowed magnetic configurations while explicitly accounting for SOC-induced symmetry breaking. This resolves the limitation of conventional MSG analysis in distinguishing distinct magnetic phases.
• Efficient Screening Strategy: By utilizing Spin Wyckoff positions and data-driven constraints derived from MAGNDATA, the method reduces the magnetic configuration space to a computationally tractable size (typically a few dozen configurations per material) without sacrificing physical completeness.
• Decoupling of Scales: The framework separates the strong isotropic exchange interactions (determining geometry) from the weaker SOC-induced anisotropy (determining orientation), enabling a more efficient search for local energy minima.

Results
The framework was validated through three representative examples and large-scale benchmarks:

1. Case Studies:

   • CoNb$_3$S$_6$: Successfully predicted the noncoplanar "all-in-all-out" antiferromagnetic ground state (consistent with recent experiments) and identified a metastable collinear state. The method captured the weak ferromagnetic moment induced by SOC.
   • MnTe and Mn$_3$Sn: Correctly identified the collinear altermagnetic ground state of MnTe and the coplanar triangular antiferromagnetic ground state of Mn$_3$Sn. The energy differences between the ground state and the first excited state were found to be robust (>10 meV/mag. atom).
   • In all three cases, the ground state was identified using only a few dozen non-relativistic DFT calculations, supplemented by a small number of relativistic calculations.

2. MAGNDATA Benchmark (Generation):

   • Tested on 1,178 distinct crystal structures from the MAGNDATA database.
   • The framework successfully reproduced the experimentally reported magnetic geometries for 78% of the materials.
   • When considering the OSSG-constrained spin orientations, the success rate rose to 93%, with the remaining deviations attributed to weak SOC-induced canting (e.g., Dzyaloshinskii-Moriya interactions) or single-ion anisotropy.

3. First-Principles Benchmark (Energy Ranking):

   • A high-throughput test on 305 compounds (excluding f-electron systems) showed that 82% of experimentally reported magnetic structures were captured within an energy tolerance of 5 meV per magnetic atom from the computed ground state.
   • The framework successfully predicted low-energy metastable phases, including altermagnets, spin-orbit magnets, and noncollinear antiferromagnets exhibiting spin splitting or geometric Hall effects.

Significance
The paper establishes a general and efficient route for the large-scale prediction of magnetic structures and unconventional magnetic phases. By leveraging the decoupled symmetries of spin and lattice spaces, the framework offers a systematic alternative to empirical spin models or exhaustive random sampling. It provides a pathway to:

• Predict magnetic ground states prior to costly experimental characterization (e.g., neutron diffraction).
• Identify competing metastable phases with emergent properties (e.g., altermagnetism, anomalous Hall effects) that may be accessible via external field tuning.
• Offer insights into the microscopic origins of magnetic interactions by analyzing the energetic hierarchy of symmetry-allowed configurations.

The authors note that while the framework is robust for commensurate structures and systems where magnetic order is the primary symmetry-breaking mechanism, it currently does not address complex phase transitions driven by multi-order-parameter competition (e.g., strong magnetoelastic coupling) or incommensurate magnetic orders, which remain areas for future development.