Exhaustive constructions of effective models in 1651 magnetic space groups.

The $k\cdot p$ effective Hamiltonians have been widely applied to predict a large variety of phenomena in condensed matter systems. Currently, the popular way to construct a $k\cdot p$ Hamiltonian is in a case-by-case manner, which significantly limits its applications especially for magnetic systems. In this work, we first explicitly tabulate all the representation matrices for all single-valued and double-valued irreducible representations (irreps) and co-irreps for the little groups of all special $k$ points in 1651 magnetic space groups (including nonmagnetic 230 space groups). Then through group theory analysis, we obtain 4 857 832 elementary $k\cdot p$ matrix blocks, and directly using these matrix blocks given in this work one can obtain any $k\cdot p$ Hamiltonian for any periodic system, including bulk or boundary. We believe our work will accelerate the studies in various fields in condensed matter physics, such as semiconductors, topological physics, spintronics, etc. We also expect our exhaustive results on $k\cdot p$ models will play vital roles in connecting other fields with condensed matter physics and promote realizations of diverse theoretical models which possess exotic properties but lack practical materials.