Phases of the hard-plate lattice gas on a three-dimensional cubic lattice

We study the phase diagram of a lattice gas $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}1$ hard plates on three-dimensional cubic lattice. Each plate covers an elementary plaquette lattice, with constraint that site can belong to utmost one plate. focus isotropic system, equal fugacities for three orientations plates. show, using grand canonical Monte Carlo simulations, system undergoes two transitions when density is increased: first from disordered fluid layered phase, and second sublattice-ordered phase. In breaks up into disjoint slabs thickness along spontaneously chosen Cartesian direction, corresponding twofold (${Z}_{2}$) symmetry breaking translation layering direction. Plates normals perpendicular this direction are preferentially contained entirely within these slabs, while straddling adjacent have lower density, thus between types show exhibit two-dimensional power-law columnar order even in presence nonzero vacancies. contrast, interslab correlations parameter decay exponentially separation slabs. there all directions. present numerical evidence transition continuous consistent universality class $O(3)$ model anisotropy, sublattice first-order nature.