SpaceGroupIrep: A package for irreducible representations of space group

Abstract

We have developed a Mathematica program package SpaceGroupIrep which is a database and tool set for irreducible representations (IRs) of space group in BC convention, i.e. the convention used in the famous book “The mathematical theory of symmetry in solids” by C.J. Bradley & A.P. Cracknell. Using this package, elements of any space group, little group, Herring little group, or central extension of little co-group can be easily obtained. This package can give not only little-group (LG) IRs for any k-point but also space-group (SG) IRs for any k-stars in intuitive table form, and both single-valued and double-valued IRs are supported. This package can calculate the decomposition of the direct product of SG IRs for any two k-stars. This package can determine the LG IRs of Bloch states in energy bands in BC convention and this works for any input primitive cell thanks to its ability to convert any input cell to a cell in BC convention. This package can also provide the correspondence of k-points and LG IR labels between BCS (Bilbao Crystallographic Server) and BC conventions. In a word, the package SpaceGroupIrep is very useful for both study and research, e.g. for analyzing band topology or determining selection rules. Program summaryProgram title:SpaceGroupIrepCPC Library link to program files:https://doi.org/10.17632/3vm4g32t4d.1Developer's repository link:https://github.com/goodluck1982/SpaceGroupIrepLicensing provisions: GNU General Public Licence 3.0Programming language: MathematicaExternal routines/libraries used:spglib (http://spglib.github.io/spglib)Nature of problem: Space groups and their representations are important mathematical language to describe symmetry in crystals. The book—“The mathematical theory of symmetry in solids” by C.J. Bradley & A.P. Cracknell (called the BC book)—is highly influential because it contains not only systematic theory but also detailed complete data of space groups and their representations. The package SpaceGroupIrep digitizes these data in the BC book and provides tens of functions to manipulate them, such as obtaining group elements and calculating their multiplications, identifying k-points, showing the character table of any little group, determining the little-group (LG) irreducible representations (IRs) of energy bands, and calculating the direct product of space-group (SG) IRs. This package is a useful database and tool set for space groups and their representations in BC convention.Solution method: The direct data in the BC book is used to calculate the LG IRs for standard k-points defined in the book. For a non-standard k-point, we first relate it to a standard k-point by an element which makes the space group self-conjugate and then calculate the LG IRs through the element. SG IRs are obtained by calculating the induced representations of the corresponding LG IRs. The full-group method based on double coset is used to calculate the direct products of SG IRs. In addition, an external package spglib is utilized to help convert any input cell to a cell in BC convention.