Folding transitions of the square-diagonal lattice

We address the problem of "phantom" folding of the tethered membrane modeled by the two-dimensional square lattice, with bonds on the edges and diagonals of each face. Introducing bending rigidities K₁ and K₂ for respectively long and short bonds, we derive the complete phase diagram of the model, using transfer matrix calculations. The latter displays two transition curves, one corresponding to a first-order (ferromagnetic) folding transition, and the other to a continuous (anti-ferromagnetic) unfolding transition.