Automatic calculation of symmetry-adapted tensors under spin-group symmetry. STENSOR, a new tool of the Bilbao Crystallographic Server

Technical Summary: STENSOR – Automatic Calculation of Symmetry-Adapted Tensors under Spin-Group Symmetry

Problem Statement
Magnetic materials possess symmetries described by magnetic space groups, which are exact symmetries including spin-orbit coupling (SOC). However, in the non-relativistic limit where SOC is absent, these materials are governed by "spin groups" (or spin space groups). While SOC effects are typically weak, they are crucial for distinguishing dominant, robust physical properties from those that are purely relativistic. A significant challenge in the analysis of magnetic materials is the automatic derivation of symmetry-adapted forms for physical property tensors under spin-group symmetry. Unlike magnetic point groups (MPGs), spin point groups (SpPGs) involve a more complex interplay between spatial operations ($R$) and spin operations ($U$), requiring generalized transformation rules that extend beyond standard Neumann's principle. Existing tools, such as MTENSOR, handle magnetic symmetries but do not address the specific constraints imposed by spin groups, particularly the distinction between non-relativistic and SOC-driven tensor coefficients.

Methodology
The authors present STENSOR, a computational tool integrated into the Bilbao Crystallographic Server, designed to automate the calculation of symmetry-adapted tensors under SpPG symmetry. The software is implemented in the Mathematica language and utilizes a combination of projector methods, Gaussian decomposition (row reduction), and SparseArray handling to manage the high-dimensional matrices associated with tensor reduction.

The program accepts input via two distinct options:

1. Option A: Upload of structural data files in scif (spin CIF) or mcif (magnetic CIF) formats. If an mcif file is provided, the program assumes a minimal SpPG, deriving spin-only operations based on the collinearity or coplanarity of the magnetic moments.
2. Option B: Manual entry of the generators of the oriented SpPG. Users provide 6-component vectors (representing lattice and spin bases) or 7-component vectors (including the determinant of the spin operation). The user must also specify the type of spin alignment (collinear, coplanar, or non-coplanar) and the associated direction vector.

The core algorithm proceeds in several steps:

• MPG Identification: The program identifies the magnetic point group as a subgroup of the SpPG by finding operations where the spin and spatial parts are related ($U = \pm R$).
• Tensor Reduction (MPG): Using projector operators, the program calculates the constraints imposed by the MPG on the tensor components. This involves constructing $3^r \times 3^r$ matrices (where $r$ is the tensor rank) representing the Kronecker products of the symmetry operations.
• Tensor Reduction (SpPG): The program extends the reduction to the full SpPG. For collinear and coplanar cases, this involves an additional step to account for the "intrinsic spin-only group" (e.g., the infinite rotation group $\infty_n$ for collinear spins). For the infinite group, the program employs a series expansion of the rotation matrix to derive linear homogeneous equations that further constrain the tensor coefficients.
• Symmetrization: The algorithm incorporates intrinsic symmetries (symmetric or antisymmetric pairs of indices) using additional projector matrices, handling complex Jahn symbols that may include toroidic components.

The output provides the symmetry-adapted tensor forms for both the SpPG and the MPG, expressed in an orthonormal frame.

Key Contributions and Results
The primary contribution is the development of a tool capable of handling the generalized Jahn symbols required for spin groups, which characterize how tensor components transform under combined spatial and spin operations. The paper demonstrates the utility of STENSOR through two examples drawn from the MAGNDATA database:

1. MnF$_2$ (Collinear): The program analyzes the Hall effect resistivity tensor. It reveals that while the magnetic symmetry allows multiple coefficients, the spin symmetry restricts the tensor significantly, leaving only one non-zero coefficient ($c_{63}$). This indicates that coefficients vanishing under spin symmetry (like $c_{41}$) must arise from SOC. The analysis also identifies a non-relativistic $d$-wave spin splitting ($k_x k_y$ dependence) in the electronic bands.
2. Mn$_3$Sn (Coplanar): The tool analyzes tensors for magnetic susceptibility, piezomagnetism, and the anomalous Hall effect (AHE). It shows that the AHE tensor is null under SpPG symmetry but non-zero under MPG symmetry, confirming its SOC origin. Furthermore, the paper illustrates the "spin basis" option, demonstrating how to handle structures that differ only by a rigid rotation of the spin arrangement (e.g., entry 0.199 vs. 0.200 in MAGNDATA), which results in different physical tensor properties despite having the same underlying spin space group symmetry.

Significance and Claims
The authors claim that STENSOR fills a critical gap in the computational toolkit for magnetic materials research by enabling the systematic distinction between non-relativistic and relativistic (SOC) effects. By comparing the tensor forms allowed under SpPG versus MPG, researchers can immediately identify which physical properties are robust (non-relativistic) and which are likely small, SOC-driven corrections.

The paper positions STENSOR as a complement to existing tools like MTENSOR and databases like MAGNDATA. It emphasizes that while the tool is powerful for deriving theoretical constraints, experimental magnetic structures may incorporate SOC features that reduce the actual symmetry below the ideal SOC-free case. Consequently, the authors caution that in such real-world scenarios, the comparison of tensor forms may not perfectly separate relativistic from non-relativistic contributions if the observed symmetry is already reduced by SOC effects. The tool is presented as a means to facilitate systematic studies of magnetic compounds, particularly in the emerging field of altermagnetism and non-relativistic spintronics.