ToMSGKpoint: A user-friendly package for computing symmetry transformation properties of electronic eigenstates of nonmagnetic and magnetic crystalline materials

The calculation of irreducible (co-)representations of energy bands at high-symmetry points (HSPs) is essential for high-throughput research on topological materials based on symmetry-indicators or topological quantum chemistry. However, existing computational packages usually require transforming crystal structures adapted to specific conventions, thus hindering extensive application, especially to materials whose symmetries are yet to be identified. To address this issue, we developed a Mathematica package, ToMSGKpoint, capable of determining the little groups and irreducible (co-)representations of little groups of HSPs, high-symmetry lines (HSLs), and high-symmetry planes (HSPLs) for any nonmagnetic and magnetic crystalline materials in two and three dimensions, with or without considering spin-orbit coupling. To the best of our knowledge, this is the first package to achieve such functionality. The package also provides magnetic space group operations, supports the analysis of irreducible (co-)representations of energy bands at HSPs, HSLs, and HSPLs using electronic wavefunctions obtained from ab initio calculations interfaced with VASP. Designed for user convenience, the package generates results in a few simple steps and presents all relevant information in a clear tabular format. Its versatility is demonstrated through applications to nonmagnetic topological insulator Bi2Se3 and Dirac semimetal Na3Bi, as well as the antiferromagnetic topological material MnBi2Te4. Suitable for any crystal structure, this package can be conveniently applied in a streamlined study once magnetic space group varies with various symmetry-breakings caused by phase transitions.针对基于对称指示符或拓扑量子化学的拓扑材料高通量研究而言，计算高对称点（high-symmetry points，HSPs）处能带的不可约（共）表示是至关重要的核心步骤。然而，现有计算软件包通常要求晶体结构适配特定的标准化约定，这极大阻碍了其广泛应用，尤其是针对对称性尚未被明确鉴定的材料。为解决这一痛点，我们开发了一款Mathematica软件包ToMSGKpoint，其可针对二维及三维的非磁性与磁性晶体材料，在考虑或不考虑自旋轨道耦合（spin-orbit coupling）的情况下，准确确定高对称点、高对称线（high-symmetry lines，HSLs）与高对称面（high-symmetry planes，HSPLs）对应的小群及其不可约（共）表示。据我们所知，这是首款实现该全功能的计算软件包。该软件包还内置磁性空间群操作功能，支持结合与VASP对接的从头算（ab initio calculations）得到的电子波函数，分析高对称点、高对称线与高对称面处能带的不可约（共）表示。为提升用户使用便利性，该软件包仅需数步简单操作即可生成结果，并以清晰规整的表格形式呈现所有相关信息。我们通过将其应用于非磁性拓扑绝缘体Bi₂Se₃、狄拉克半金属Na₃Bi以及反铁磁性拓扑材料MnBi₂Te₄，验证了该软件包的通用性。该软件包适配任意晶体结构，可便捷应用于因相变引发各类对称性破缺而导致磁性空间群发生变化的流程化研究。