  New materials with magnetic order driven topological phases are hugely sought
after for their immense application potential. In this work, we propose a new
compound EuMn$_2$Bi$_2$ from our first principles density functional theory
calculations to host novel topological phases such as Dirac/Weyl semimetal and
topological insulator in its different magnetic states which are energetically
close to one another. We started with an isostructural compound EuMn$_2$As$_2$
where the magnetic structure has been studied experimentally. From our
calculations we could explain the nature of two magnetic transitions observed
experimentally in this system and could also establish the correct magnetic
ground state. Our electronic structure calculations reveal the insulating
nature of the ground state consistent with the experiments. By replacing all As
by Bi in EuMn$_2$As$_2$ and by optimizing the new structure, we obtained the
new compound EuMn$_2$Bi$_2$. We observe this compound to be dynamically stable
from our phonon calculations supporting its experimental preparation in future.
By comparing the total energies of various possible magnetic structures we
identified the ground state. Though the magnetic ground state is found to be
insulating in nature with tiny band gap which is an order of magnitude less
than the same in EuMn$_2$As$_2$, there were other magnetic states energetically
very close to the ground state which display remarkable non-trivial band
topology such as Dirac/Weyl points close to the Fermi level and topological
insulator state. The energetic proximity of these magnetic order driven
topological phases makes them tunable via external handle which indicates that
the proposed new material EuMn$_2$Bi$_2$ would be a very versatile magnetic
topological material.
