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

Correlated flat bands in the paramagnetic phase of triangular antiferromagnets Na$_2$BaX(PO$_4$)$_2$ (X = Mn, Co, Ni)