Although lattice gases composed of particles preventing up to their $k\mathrm{th}$ nearest neighbors from being occupied (the $k\mathrm{NN}$ models) have been widely investigated in the literature, location and universality class fluid-columnar transition 2NN model on square are still a topic debate. Here, we present grand-canonical solutions this Husimi lattices built with diagonal lattices, $...

The experimental observation of a spin-gap in the antiferromagnetic spin2 compound CaV4O9 has been attributed to the formation of a plaquette resonating-valence-bond (PRVB) state in the underlying 1/5-depleted square lattice. We construct a spin2 model on the 1/5-depleted lattice for which the PRVB state can be shown to be the exact ground state in a particular parameter regime. In a subspace o...