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The $k\cdot p$ effective Hamiltonians have been widely applied to predict alarge variety of phenomena in condensed matter systems. Currently, the popularway to construct a $k\cdot p$ Hamiltonian is in a case-by-case manner, whichsignificantly limits its applications especially for magnetic systems. In thiswork, we first explicitly tabulate all the representation matrices for allsingle-valued and double-valued irreducible representations (irreps) andco-irreps for the little groups of all special $k$ points in 1651 magneticspace groups (including nonmagnetic 230 space groups). Then through grouptheory analysis, we obtain 4 857 832 elementary $k\cdot p$ matrix blocks, anddirectly using these matrix blocks given in this work one can obtain any$k\cdot p$ Hamiltonian for any periodic system, including bulk or boundary. Webelieve our work will accelerate the studies in various fields in condensedmatter physics, such as semiconductors, topological physics, spintronics, etc.We also expect our exhaustive results on $k\cdot p$ models will play vitalroles in connecting other fields with condensed matter physics and promoterealizations of diverse theoretical models which possess exotic properties butlack practical materials.

Exhaustive constructions of effective models in 1651 magnetic space groups