  Flat band systems in condensed matter physics are intriguing because they can
exhibit exotic phases and unconventional properties. In this work, we studied
three correlated magnetic systems, Na$_2$BaX(PO$_4$)$_2$ (X = Mn, Co, Ni), and
revealed their unusual electronic structure and magnetic properties. Despite
their different effective angular momentum, our first-principles calculations
showed a similar electronic structure among them. However, their different
valence configurations led to different responses to electronic correlations in
the high-temperature paramagnetic phase. Using the dynamical mean-field method,
we found that all systems can be understood as a multi-band Hubbard model with
Hund'ss coupling. Our calculations of spin susceptibility and the {\it
ab-initio} estimation of magnetic exchange coupling indicated strong
intra-plane antiferromagnetic coupling and weak inter-plane coupling in all
systems. The ground states of these systems are largely degenerate. It is
likely that none of these magnetic states would dominate over the others,
leading to the possibility of quantum spin liquid states in these systems. Our
work unifies the understanding of these three structurally similar systems and
opens new avenues for exploring correlated flat bands with distinct electronic
and magnetic responses.
