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Magnetic properties of AMoOPO$_4$Cl (A = K, Rb) with Mo$^{5+}$ ions in the4$d^1$ electronic configuration are investigated by magnetization, heatcapacity and NMR measurements on single crystals, combined with powder neutrondiffraction experiments. The magnetization measurements reveal that they aregood model compounds for the spin-1/2 $J_1-J_2$ square lattice magnet with thefirst and second nearest-neighbor interactions. Magnetic transitions areobserved at around 6 and 8 K in the K and Rb compounds, respectively. Incontrast to the normal N\'eel-type antiferromagnetic order, the NMR and neutrondiffraction experiments find a columnar antiferromagnetic order for eachcompound, which is stabilized by a dominant antiferromagnetic $J_2$. Bothcompounds realize the unusual case of two interpenetrating $J_2$ squarelattices weakly coupled to each other by $J_1$.

$J_1-J_2$ Square-Lattice Heisenberg Antiferromagnets with 4$d^1$ spins: AMoOPO$_4$Cl (A = K, Rb)