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Recent discovery of the time reversal symmetry breaking magnetic Weylsemimetals has created a huge surge of activities in the field of quantumtopological materials. In this work, we have studied systematically the groundstate magnetic order, electronic structure and the interplay between themagnetic 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 ofdensity functional theory (DFT). The detailed investigation unravels theexistence of different topological phases in this single material which can betuned by an external probe such as magnetic field or chemical substitution. OurDFT calculations including Coulomb correlation (U) and spin-orbit (SO)interaction within GGA+U+SO approximation confirms that the magnetic groundstate of EMB is A-type Antiferromagnetic (A-AFM) with Eu magnetic momentsaligned along the crystallographic $a$ or $b$ direction. Although the groundstate 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 bandvery close to the Fermi level (FL) along $\Gamma$-A direction in the FM stateof 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 movingcloser to the FL which is highly desirable for application purposes. Further weobserve that the separation between the Weyl points in the pair decreases indoped compound compared to that in the parent compound which has directconsequence on anomalous Hall conductivity (AHC). Our first principlescalculation of AHC shows high peak values exactly at these Weyl points and thepeak height decreases when we dope the system with Ca. Therefore, Ca doping canbe a good external handle to tune AHC in this system.

Interplay of magnetism and band topology in Eu$_{1-x}$Ca$_x$Mg$_2$Bi$_2$ (x=0, 0.5) from first principles study