Folding transitions in three-dimensional space with defects

A model describing the three-dimensional folding of the triangular lattice on the face-centered cubic lattice is generalized allowing the presence of defects, which are related to cuts in the two-dimensional network. The model can be expressed in terms of Ising-like variables with nearest-neighbor and plaquette interactions in the hexagonal lattice; its phase diagram is determined by means of the cluster variation method. The results found by varying the curvature and defect energy show that the introduction of defects turns the first-order crumpling transitions of the model without defects into continuous transitions. New phases also appear by decreasing the energy cost of defects and the behavior of their densities has been analyzed.