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We study the phase diagram of a square lattice Hubbard model with a perfectvacancy superstructure. The model can be also defined on a new bipartitelattice with each building blocks consisting of a minimal square. Thenon-interacting model is exactly solved and a mid-band gap opens at the Fermienergy in the weak inter-block hopping regime. Increasing the Coulombinteraction will develop the N\'eel antiferromagnetic order with varying blockspin moments. The metal-insulator transition with $U_{\rm MI}$ smaller than theone without vacancies occurs above the magnetic instability $U_{\rm M}$. Theemergent intermediate magnetic metal phase develops substantially in themoderate inter-block hopping regime. Drastic increases in the ordered momentand gap magnitude on the verge of non-interacting band insulator signal apossible distinction between the magnetic semi-conductor and theMott-insulator. The implications of these results for the recent discovered(A,Tl)$_{y}$Fe$_{2-x}$Se$_2$ compounds are discussed.

Block spin magnetism and metal-insulator transition in a two-dimensional Hubbard model with perfect vacancy superstructure