  We study the phase diagram of a system of $2\times 2\times 1$ hard plates on
the three dimensional cubic lattice, {\em i.e.} a lattice gas of plates that
each cover an elementary plaquette of the cubic lattice and occupy its four
vertices, with the constraint that no two plates occupy the same site of the
cubic lattice. We focus on the isotropic system, with equal fugacities for the
three orientations of plates. We show, using grand canonical Monte Carlo
simulations, that the system undergoes two density-driven phase transitions
with increasing density of plates: the first from a disordered fluid to a
layered phase, and the second from the layered phase to a sublattice-ordered
phase. In the layered phase, the system breaks up into disjoint slabs of
thickness two along one spontaneously chosen cartesian direction. Plates with
normals perpendicular to this layering direction are preferentially contained
entirely within these slabs, while plates straddling two successive slabs have
a lower density. Additionally the symmetry between the three types of plates is
spontaneously broken, as plates with normal along the layering direction have a
lower density than the other two types of plates. Intriguingly, the occupied
slabs exhibit two-dimensional power-law columnar order even in the presence of
a nonzero density of vacancies. In contrast, inter-slab correlations of the
two-dimensional columnar order parameter decay exponentially with the
separation between the slabs. In the sublattice-ordered phase, there is
two-fold ($Z_2$) breaking of lattice translation symmetry along all three
cartesian directions. We present numerical evidence that the disordered to
layered transition is continuous and consistent with the three-dimensional
$O(3)$ universality class, while the layered to sublattice transition is
first-order in nature.

Statistical Mechanics