  Here, we report a new intrinsic magnetic topological insulator FeBi$_2$Te$_4$
based on first-principles calculations and it can achieve a rich topological
phase under pressure modulation. Without pressure, we predict that both
FeBi$_2$Te$_4$ ferromagnetic and antiferromagnetic orders are non-trivial
topological insulators. Furthermore, FeBi$_2$Te$_4$ of FM-z order will undergo
a series of phase transitions from topological insulator to semimetals and then
to trivial insulator under pressure. Finally, we further clarify and verify
topological phase transitions with low-energy effective model calculations.
This topological phase transition process is attributed to the synergy of the
magnetic moment and the spin-orbit coupling. The unique topological properties
of FeBi$_2$Te$_4$ will be of great interest in driving the development of
quantum effects.
