On the transition from crystalline to dynamically triangulated random surfaces

We consider methods of interpolating between the crystalline and dynamically triangulated random surface models. We argue that actions based on the deviation from six of the coordination number at a site are inadequate and propose an alternative based on Alexander moves. Two simplified models, one of which has a phase transition and the other of which does not, are discussed. Theoretical Physics preprint OUTP-93/17P 6th August 1993 1 e-mail: [email protected] 2 e-mail: [email protected] 1 Many apparently disparate physical systems, such as non-critical string theory, 2D quantum gravity, and polymer membranes, are described by the statistical mechanics of random surfaces (for recent reviews see [1]). This letter is concerned with understanding the differences, if any, in the critical behaviour of crystalline surfaces and dynamically triangulated surfaces. Crystalline surfaces are described by a fixed triangulation T (in continuum terms a fixed intrinsic metric), with the partition function formed by summing over all the possible embeddings of this surface in a D-dimensional euclidean space [2]; usually the fixed triangulation is a regular one but this is not essential. If the lattice sites i of T have coordinates Xi in the D-dimensional embedding space the action is given by ST (κ,D) = 1 2 ∑ ∈T (Xi − Xj) + κ ∑ △△∈T 1− n̂△.n̂△′ (1) where < ij > denotes the link fron i to j and n̂△,△′ are the unit normal vectors of the triangles on either side of the link. The canonical partition function is