  Recently, the interplay between the Dirac/Weyl fermion and various bulk
properties, such as magnetism, has attracted considerable attention, since
unconventional transport and optical phenomena were discovered. However, the
design principles for such materials have not been established well. Here, we
propose that the layered material $A$Mn$X_2$ ($A$: alkaline and rare-earth
ions, $X$: Sb, Bi) is a promising platform for systematically exploring
strongly correlated Dirac metals, which consists of the alternative stack of
the $X^-$ square net layer hosting a 2D Dirac fermion and the
$A^{2+}$-Mn$^{2+}$-$X^{3-}$ magnetic block layer. In this article, we shall
review recent high-field studies on this series of materials to demonstrate
that various types of Dirac fermions are realized by designing the block layer.
First, we give an overview of the Dirac fermion coupled with the magnetic order
in EuMnBi$_2$ ($A$=Eu). This material exhibits large magnetoresistance by the
field-induced change in the magnetic order of Eu layers, which is associated
with the strong exchange interaction between the Dirac fermion and the local Eu
moment. Second, we review the Dirac fermion coupled with the lattice
polarization in BaMn$X_2$ ($A$=Ba). There, spin-valley coupling manifests
itself owing to the Zeeman-type spin-orbit interaction, which is experimentally
evidenced by the bulk quantum Hall effect observed at high fields.
