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.

Program summary

Program Title: ToMSGKpoint

Developer's repository link: https://github.com/FengTang1990/ToMSGKpoint

Licensing provisions: GPLv3

Programming language: Wolfram

Nature of problem: The package ToMSGKpoint provides magnetic space group operations for any crystal structure, along with the little groups of high-symmetry points, lines, and planes, and their corresponding irreducible (co-)representations. It also facilitates the transformation from a customized crystal structure to the Bradley-Cracknell convention. Furthermore, based on electronic wavefunctions obtained from VASP calculations, the package computes the irreducible (co-)representations of energy bands at high-symmetry points, lines, and planes.

Solution method: In order to calculate the irreducible (co-)representations of the little groups at high-symmetry points, lines, and planes, we first obtain the transformation from the customized crystal structure convention to the Bradley-Cracknell convention. Using this transformation, we derive the irreducible (co-)representations for the little groups in the customized crystal structure convention based on those in the Bradley-Cracknell convention.

Additional comments including restrictions and unusual features: The current program that we are utilizing is specifically designed to read electronic wave functions that have been meticulously calculated by VASP. Unfortunately, this means that it is not compatible with other first-principles computational software, such as Quantum ESPRESSO or Gaussian. Additionally, it's important to note that this program does not have the capability to directly compute the irreducible (co-)representations of the energy bands in phonon materials.