  Topological states of quantum matter have attracted great attention in
condensed matter physics and materials science. The study of
time-reversal-invariant (TRI) topological states in quantum materials has made
tremendous progress in both theories and experiments. As a great success,
thousands of TRI topological materials are predicted through sweeping search.
Richer exotic phenomena are expected to appear in magnetic topological
materials because of varied magnetic configurations, but this study falls much
behind due to the complex magnetic structures and transitions. Here, we predict
the tetradymite-type compound MnBi$_2$Te$_4$ and its related materials host
interesting magnetic topological states. The magnetic ground state of
MnBi$_2$Te$_4$ is an antiferromagnetic phase which leads to an antiferromagetic
topological insulator state with a large topologically non-trivial energy gap
($\sim$0.2~eV). It is the parent state for the axion state, which has gapped
bulk and surface states, and quantized topological magnetoelectric effect. The
ferromagnetic phase of MnBi$_2$Te$_4$ leads to an ideal minimal type-II Weyl
semimetal with two Weyl points accompanied by one hole-type and one
electron-type Fermi pocket at the Fermi level, which has never been discovered
elsewhere. We further present a simple and unified continuum model to capture
the salient topological features of this kind of materials.
