Statistical mechanics of the flexural Ising model

Inspired by both nanoscale and macroscale metamaterials, thermalized membranes with embedded arrays of buckled bistable nodes have been studied with simulations and theory. Buckled nodes interact with their neighbors via the elastic energy of the host membrane such that they prefer to be antialigned at zero temperature, similar to an Ising antiferromagnet. Molecular dynamics simulations of this system reveal a phase transition in the ordering of buckled nodes for which critical exponents can be measured (Hanakata, Plummer, Nelson, PRL 2022). Many, but not all, of these critical exponents are consistent with the Ising universality class. In order to understand this finding, renormalization group methods are applied to a model that couples an Ising order parameter to a thin elastic sheet with both in-plane and out-of-plane fluctuations. The analysis finds that the coupling is relevant for two-dimensional sheets embedded in three-dimensional space but does not identify a new fixed point towards which the Hamiltonian is expected to flow.