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.
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