The crumpling transition of crystalline random surfaces

We investigate the crumpling transition in two different models of crystalline random surfaces with extrinsic curvature which have recently caused some confusion and find that many of the results previously obtained are erroneous. Using the Fourier acceleration technique to ameliorate critical slowing down problems we have made numerical simulations of surfaces of up to 128² points embedded in three dimensions. The first model, which has a non-compact lattice version of the extrinsic curvature, suffers from a sickness in the non-crumpled phase which is a lattice artefact; it is smooth in one intrinsic direction and folded up on the scale of the lattice spacing in the other, so we call this phase corrugated. The crumpling transition is continuous, having a diverging persistence length with critical exponent v = 1.15 ± 0.15 and a cusp in the specific heat indicating that 0. The second model, in which the extrinsic curvature depends upon the cosine of the angle between normals of adjacent triangles, also has a continuous transition with v = 0.94 ± 0.20 and = 0.53 ± 0.15. Just beyond the crumpling transition, the smooth phase is found to have Hausdorff dimension d_H < 2.14 at two standard deviations and so we conclude that d_H = 2 throughout this phase. A study of the correlation functions shows that, in the crumpled phase, the system is apparently described by a very simple gaussian action. If true, this result could have important implications which we discuss briefly.