arXiv2022
We obtain the phase diagram of fully-packed hard plates on a cubic lattice. Each plate covers an elementary plaquette of the cubic lattice and occupies its four vertices, with each vertex of the cubic lattice occupied by exactly one such plate. We consider the general case with fugacities $s_μ$ for `$μ$ plates', whose normal is the $μ$ direction ($μ= x,y,z$). At and close to the isotropic point, we find, consistent with previous work, a phase with long-range sublattice order. When two of the fugacities $s_{\rm μ_1}$ and $s_{μ_2}$ are comparable, and the third fugacity $s_{μ_{3}}$ is much smaller, we find a spontaneously-layered phase. In this phase, the system breaks up into disjoint slabs of width two stacked along the $μ_3$ axis. $μ_1$ and $μ_2$ plates are preferentially contained entirely within these slabs, while plates straddling two successive slabs have a lower density. In the opposite limit, with $μ_3 \gg μ_1 \sim μ_2$, we find a phase with long-range columnar order, corresponding to simultaneous $Z_2$ symmetry breaking of lattice translation symmetry in directions $μ_1$ and $μ_2$. The spontaneously-layered phases display critical behaviour, with power-law decay of correlations in the $μ_1$ and $μ_2$ directions when the slabs are stacked in the $μ_3$ direction, and represent examples of `floating phases' discussed earlier in the context of coupled Luttinger liquids and quasi-two-dimensional classical systems. We ascribe this remarkable behaviour to the constrained motion of defects in this phase, and develop a coarse-grained effective field theoretical understanding of the stability of power-law order in this unusual three-dimensional floating phase.