  The recent discovery of antiferromagnetic (AFM) topological insulator (TI)
MnBi$_2$Te$_4$ has triggered great research efforts on exploring novel magnetic
topological physics. Based on first-principles calculations, we find that the
manipulation of magnetic orientation and order not only significantly affects
material symmetries and orbital hybridizations, but also results in variant new
magnetic topological phases in MnBi$_2$Te$_4$. We thus predict a series of
unusual topological quantum phase transitions that are magnetically
controllable in the material, including phase transitions from AFM TI to AFM
mirror topological crystalline insulator, from type-II to type-I topological
Weyl semimetal, and from axion insulator to Chern insulator. The findings open
new opportunities for future research and applications of magnetic topological
materials.
