Even though the $k·p$ effective Hamiltonian has been widely exploited to predict a large variety of phenomena in condensed matter systems, the general method to construct a $k·p$ Hamiltonian is case-by-case and can even be complicated. Moreover, deriving a $k·p$ model in magnetic materials usually requires representation matrices of a magnetic space group (MSG), however, those for type III/IV MSGs are not constructed well compared with 230 type-I/II MSGs. In this work, we explicitly and exhaustively tabulate the representation matrices for all single-valued/double-valued irreducible representations (irreps) and co-irreps for the little groups of all special $k$ points in 1651 MSGs. Then through analysis by group theory, we obtain a large database composed of 4 857 832 elementary $k·p$ matrix blocks. Directly using these matrix blocks, one can obtain any $k·p$ Hamiltonian for any periodic system, for both nonmagnetic and magnetic materials. In light of the huge successes achieved by $k·p$ models, our work could aid studies in various fields, such as semiconductors, topological physics, spintronics, valleytronics, twistronics, etc. The exhaustive effective models, each attached to an MSG, would facilitate identification and realization of all possible emergent massless/massive excitations. More importantly, through comparing the theoretical model with novel property and those in our database, one can find the MSGs of materials realizations of the model. We also expect our exhaustive results on $k·p$ models to be utilized to connect other fields and condensed matter physics.

• Received 18 May 2021
• Revised 1 August 2021
• Accepted 3 August 2021

DOI:https://doi.org/10.1103/PhysRevB.104.085137