A programmable $k\cdot p$ Hamiltonian method and application to magnetic topological insulator MnBi$_2$Te$_4$

In the band theory, first-principles calculations, the tight-binding method and the effective $k\cdot p$ model are usually employed to investigate the electronic structure of condensed matters. The effective $k\cdot p$ model has a compact form with a clear physical picture, and first-principles calculations can give more accurate results. Nowadays, it has been widely recognized to combine the $k\cdot p$ model and first-principles calculations to explore topological materials. However, the traditional method to derive the $k\cdot p$ Hamiltonian is complicated and time-consuming by hand. In this work, we independently develop a programmable algorithm to construct effective $k\cdot p$ Hamiltonians. Symmetries and orbitals are used as the input information to produce the one-/two-/three-dimension $k\cdot p$ Hamiltonian in our method, and the open-source code can be directly downloaded online. At last, we also demonstrate the application to MnBi$_2$Te$_4$-family magnetic topological materials.