Statistical Mechanics of Puckered Membranes

A triangular lattice of points connected by springs that resists bending and stretching provides a discrete model of a thin elastic plate. We consider such a surface with a superlattice of `impurities'-- sites that have longer springs connecting them to the rest of the lattice. In the continuum limit, this corresponds to a preferred metric with periodic dilations. These impurities tend to pucker either above or below the lattice. We regard this as an Ising-like degree of freedom, and characterize interactions between neighboring puckers. We find we can tune these puckered membranes from a `ferromagnetic' state to an `antiferromagnetic' state using elastic constants and superlattice structure, and investigate these states theoretically and numerically.