  Recent discovery of the time reversal symmetry breaking magnetic Weyl
semimetals has created a huge surge of activities in the field of quantum
topological materials. In this work, we have studied systematically the ground
state magnetic order, electronic structure and the interplay between the
magnetic order and band topology in one such materials, EuMg$_2$Bi$_2$ (EMB)
and its Ca doped variant using first principles method within the framework of
density functional theory (DFT). The detailed investigation unravels the
existence of different topological phases in this single material which can be
tuned by an external probe such as magnetic field or chemical substitution. Our
DFT calculations including Coulomb correlation (U) and spin-orbit (SO)
interaction within GGA+U+SO approximation confirms that the magnetic ground
state of EMB is A-type Antiferromagnetic (A-AFM) with Eu magnetic moments
aligned along the crystallographic $a$ or $b$ direction. Although the ground
state of EMB is A-AFM, the Ferromagnetic (FM) state lies very close in energy.
We observe a single pair of Weyl points connecting valence and conduction band
very close to the Fermi level (FL) along $\Gamma$-A direction in the FM state
of EuMg$_2$Bi$_2$ with Eu moments aligned along crystallographic $c$ direction.
On doping 50\% Ca at Eu sites, we observe single pair of Weyl points moving
closer to the FL which is highly desirable for application purposes. Further we
observe that the separation between the Weyl points in the pair decreases in
doped compound compared to that in the parent compound which has direct
consequence on anomalous Hall conductivity (AHC). Our first principles
calculation of AHC shows high peak values exactly at these Weyl points and the
peak height decreases when we dope the system with Ca. Therefore, Ca doping can
be a good external handle to tune AHC in this system.
