Spontaneous corrugation of dipolar membranes

We present a simple model for dipolar elastic membranes that gives lattice-bound point dipoles complete orientational freedom as well as translational freedom along one coordinate (out of the plane of the membrane). There is an additional harmonic term which binds each of the dipoles to the six nearest neighbors on either triangular or distorted lattices. The translational freedom of the dipoles allows triangular lattices to find states that break out of the normal orientational disorder of frustrated configurations and which are stabilized by long-range antiferroelectric ordering. In order to break out of the frustrated states, the dipolar membranes form corrugated or "rippled" phases that make the lattices effectively nontriangular. We observe three common features of the corrugated dipolar membranes: (1) the corrugated phases develop easily when hosted on triangular lattices, (2) the wave vectors for the surface ripples are always found to be perpendicular to the dipole director axis, and (3) on triangular lattices, the dipole director axis is found to be parallel to any of the three equivalent lattice directions.