- Published 1990

Connection between discrete and continuum models of polymerized (tethered) surfaces has been investigated by applying a transfer matrix method to a discrete rigid-bond triangular lattice, which is allowed to fold on itself along its bonds in a two-dimensional embedding space. As its continuum counterpart, the model has an extensive entropy and the mean squared distance between two sites of a folded lattice increases logarithmically with the linear distance between the sites in the unfolded state. The model lattice with bending rigidity remains unfolded at any finite temperature, unlike real polymerized surfaces. Properties of polymerized or tethered membranes (surfaces) have been an object of numerous recent studies [l-61. We can view the polymerized surfaces as a generalization [l] of linear polymers [7]. This analogy has been used [3] to investigate the properties of selfavoiding tethered surfaces. However, unlike in the linear polymers, the long length-scale behavior of tethered surfaces strongly depends on the details of the Hamiltonian. In particular, very rigid surfaces exhibit a nontrivial [6] flat one. As the rigidity of a surface changes it undergoes a second-order phase transition [l, 4,5] from a crumpled (linearpolymerlike) phase to a flat phase. Neither the flat phase nor the crumpling transition has an analogy in linear polymers. The theoretical treatment of long length-scale properties of linear polymers rests on a fkm foundation, since in the absence of self-avoiding (excluded volume, steric) interactions their properties can be calculated exactly. In particular, it can be shown that the end-to-end distance (both in continuum and on a discrete lattice) of a long polymer described by any local Hamiltonian obeys the Gaussian probability distribution. Thus on sufficiently long length-scales the polymer can be described by an effective Hamiltonian H I = = KkB T$dx(dr/dx)2, where r is the position of a monomer in the d-dimensional embedding space, while x is the i n t e m l coordinate (label) of a monomer. A straightforward generalization of H I assumes that a t long length-scale two-dimensional (2d) polymerized membranes without self-avoiding interactions will be described by H 2 = KkB T$d2x(Vr)2, where x is the internal coordinate of a monomer, i.e. its position in the 2d network, while (Vr)2 (W&1)2 + ( W ~ X ~ ) ~ . This expression has indeed been confirmed 158 EUROPHYSICS LETTERS by Monte Carlo simulations and by an approximate Migdal-Kadanoff renormalization group treatments for a 2d network embedded in a continuous three-dimensional space for several types of intermonomer potentials [l], and has been used as a starting point in the theoretical treatment of self-avoiding membranes [3]. The validity of H2 also has been demonstrated in the limit of the infinite embedding dimension for quite a large class of models [8]. Obviously this cannot be a general result for an arbitrary microscopic Hamiltonian in an arbitrary embedding space dimension, because the mere presence of the crumpling transition and of the flat phase indicates that the longh length-scale limit does depends on the details of the microscopic Hamiltonian. One may also question the equivalence of discrete and continuum models, and wonder whether a situation resembling roughening transition in solid interfacesf91 may also be present in tethered surfaces. Actually, one can easily find a discrete example of a different behavior [4]: consider a two-dimensional square lattice embedded in two dimensions and free to fold on itself along its bonds. The folds of the lattice can be only along infinite straight lines, thus, configurations of such a lattice can essentially be represented by an external product of two one-dimensional random walks. The number of distinct folded configurations of a finite L x L lattice is 4L, meaning that the entropy of the system increases as its linear size and therefore the entropy per unit area vanishes as L + CQ. The mean-squared distance ( r 2 ) between two points of the lattice increases linearly with the internal separation w between the points, as opposed to the prediction (r ') In w following from the continuum Hamiltonian H2. If one introduces a bending rigidity by assigning energy x per unit length of a fold, one can easily see that the lattice remains flat at any finite temperature T, because the energetic cost of a single fold xL cannot be offset by the gain in the entropy. One may wonder, whether such a pathological behavior of a square lattice characterizes all discrete systems. We consider all possible foldings of a w x L parallelogram excised from a triangular lattice in the limit L+ CQ. Figure la) depicts such a lattice before the folding, while fig. l b )

@inproceedings{Kant1990TriangularLF,
title={Triangular Lattice Foldings - a Transfer Matrix Study},
author={Yogesh Kant and Marko Vukobrat Jari{\'c}},
year={1990}
}