Geometrical Folding Transitions of the Triangular Lattice in the Face-Centred Cubic Lattice

We study the folding of the regular two-dimensional triangular lattice embedded in the regular three-dimensional Face-Centred Cubic lattice, a discrete model for the crumpling of membranes. Possible folds are complete planar folds, folds with the angle of a regular tetrahedron (71 degrees) or with that of a regular octahedron (109 degrees). We study this model in the presence of a negative bending rigidity K, which favours the folding process. We use both a cluster variation method (CVM) approximation and a transfer matrix approach. The system is shown to undergo two separate geometrical transitions with increasing |K|: a first discontinuous transition separates a phase where the triangular lattice is preferentially wrapped around octahedra from a phase where it is preferentially wrapped around tetrahedra. A second continuous transition separates this latter phase from a phase of complete folding of the lattice on top of a single triangle.