  Motivated by efforts to create thin nanoscale metamaterials and understand
atomically thin binary monolayers, we study the finite temperature statistical
mechanics of arrays of bistable buckled dilations embedded in free-standing
two-dimensional crystalline membranes that are allowed to fluctuate in three
dimensions. The buckled nodes behave like discrete, but highly compressible,
Ising spins, leading to a phase transition at $T_c$ with singularities in the
staggered "magnetization," susceptibility, and specific heat, studied via
molecular dynamics simulations. Unlike conventional Ising models, we observe a
striking divergence and sign change of the coefficient of thermal expansion
near $T_c$ caused by the coupling of flexural phonons to the buckled spin
texture. We argue that a phenomenological model coupling Ising degrees of
freedom to the flexural phonons in a thin elastic sheet can explain this
unusual response.
