  The concept of electronic topology and the associated topological protection
brings excellent opportunities for developing next-generation devices. Ideally,
magnetic topological materials (MTM) should have their Dirac/Weyl points and/or
associated mass gaps at the Fermi energy (EF) or be readily tunable such that
they can be placed at EF via external perturbations such as electric field
gating, chemical substitutions, or doping. Three-dimensional antiferromagnetic
(AFM) materials like Mn$Bi_{2}$$X_{4}$ (X=Se, Te) that have strong spin-orbit
coupling (SOC) and broken time-reversal symmetry (TRS) due to magnetic ordering
have been the subject of enormous interest.. In this work, using density
functional theory (DFT), we have studied the electronic properties and
topological phases of the first intrinsic magnetic topological insulator family
Mn$Bi_{2}$$Te_{4}$ (MBT) in the presence of an external magnetic field. Our
calculations reveal that the topological phase of bulk rhombohedral
(R$\overline{3}$m) Mn$A_{2}$$X_{4}$ (A = Sb, Bi; X=Se, Te) depends on the spin
direction and the chemistry. The antiferromagnetic (AFM) ground state of
Mn$Sb_{2}$$Se_{4}$ (MSS) is a trivial insulator, whereas the AFM ground state
of Mn$Bi_{2}$$Se_{4}$(MBS) is an Axion insulator. Both materials become nodal
point or nodal line Weyl semimetals in the presence of a sufficiently strong
external magnetic field. The AFM ground state of Mn$Sb_{2}$$Te_{4}$ (MST) is an
Axion insulator. MST is a type-II Weyl semimetal with spins aligned in the
$\hat{Z}$-direction, but becomes insulating with an inverted band gap for spins
in-plane. Similarly, the AFM phase of MBT is an Axion insulator, but remains
insulating with an inverted gap in the ferromagnetic phase. Additionally, we
demonstrated the evolution of the topological phase of Mn$Bi_{2}$$Te_{4}$ (MBT)
by substituting the Bi atoms with the Sb atoms.
