  We performed susceptibility, magnetization, specific heat, and single crystal
neutron diffraction measurements on single crystalline BaMn$_2$Si$_2$O$_7$.
Based on the results, we revisited its spin structure with a more accurate
solution and constructed a magnetic phase diagram with applied field along the
$b$-axis, which contains a spin flop transition around 6 T. We also used
susceptibility, magnetization, and specific heat results confirmed the
ferrimagnetic-like magnetism in polycrystalline BaCo$_2$Si$_2$O$_7$.
Furthermore, we performed LSDA + U calculations for the BaM$_2$Si$_2$O$_7$ (M =
Cu, Co, and Mn) system. Our discussions based on the comparison among the
obtained magnetic exchange interactions suggest the different structures and
electronic configurations are the reasons for the different magnetic properties
among the system members.
