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Topological states of quantum matter have attracted great attention incondensed matter physics and materials science. The study oftime-reversal-invariant (TRI) topological states in quantum materials has madetremendous 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 topologicalmaterials because of varied magnetic configurations, but this study falls muchbehind due to the complex magnetic structures and transitions. Here, we predictthe tetradymite-type compound MnBi$_2$Te$_4$ and its related materials hostinteresting magnetic topological states. The magnetic ground state ofMnBi$_2$Te$_4$ is an antiferromagnetic phase which leads to an antiferromagetictopological insulator state with a large topologically non-trivial energy gap($\sim$0.2~eV). It is the parent state for the axion state, which has gappedbulk and surface states, and quantized topological magnetoelectric effect. Theferromagnetic phase of MnBi$_2$Te$_4$ leads to an ideal minimal type-II Weylsemimetal with two Weyl points accompanied by one hole-type and oneelectron-type Fermi pocket at the Fermi level, which has never been discoveredelsewhere. We further present a simple and unified continuum model to capturethe salient topological features of this kind of materials.

Topological axion states in magnetic insulator MnBi$_2$Te$_4$ with the quantized magnetoelectric effect