The flat phase of fixed-connectivity membranes

The statistical mechanics of flexible two-dimensional surfaces (membranes) appears in a wide variety of physical settings. In this talk we discuss the simplest case of fixed-connectivity surfaces. We first review the current theoretical understanding of the remarkable flat phase of such membranes. We then summarize the results of a recent large scale Monte Carlo simulation of the simplest conceivable discrete realization of this system \cite{BCFTA}. We verify the existence of long-range order, determine the associated critical exponents of the flat phase and compare the results to the predictions of various theoretical models.Comment: 7 pages, 5 figures, 3 tables. LaTeX w/epscrc2.sty, combined contribution of M. Falcioni and M. Bowick to LATTICE96(gravity), to appear in Nucl. Phys. B (proc. suppl.

Unitary One Matrix Models: String Equations and Flows

We review the Symmetric Unitary One Matrix Models. In particular we discuss the string equation in the operator formalism, the mKdV flows and the Virasoro Constraints. We focus on the \t-function formalism for the flows and we describe its connection to the (big cell of the) Sato Grassmannian \Gr via the Plucker embedding of \Gr into a fermionic Fock space. Then the space of solutions to the string equation is an explicitly computable subspace of \Gr\times\Gr which is invariant under the flows.Comment: 20 pages (Invited talk delivered by M. J. Bowick at the Vth Regional Conference on Mathematical Physics, Edirne Turkey: December 15-22, 1991.

Folding transition of the triangular lattice in a discrete three--dimensional space

A vertex model introduced by M. Bowick, P. Di Francesco, O. Golinelli, and E. Guitter (cond-mat/9502063) describing the folding of the triangular lattice onto the face centered cubic lattice has been studied in the hexagon approximation of the cluster variation method. The model describes the behaviour of a polymerized membrane in a discrete three--dimensional space. We have introduced a curvature energy and a symmetry breaking field and studied the phase diagram of the resulting model. By varying the curvature energy parameter, a first-order transition has been found between a flat and a folded phase for any value of the symmetry breaking field.Comment: 11 pages, latex file, 2 postscript figure

Anisotropic Membranes

We describe the statistical behavior of anisotropic crystalline membranes. In particular we give the phase diagram and critical exponents for phantom membranes and discuss the generalization to self-avoiding membranes.Comment: LATTICE98(surfaces) 5 pages, 4 Postscript figure

Fixed-Connectivity Membranes

The statistical mechanics of flexible surfaces with internal elasticity and shape fluctuations is summarized. Phantom and self-avoiding isotropic and anisotropic membranes are discussed, with emphasis on the universal negative Poisson ratio common to the low-temperature phase of phantom membranes and all strictly self-avoiding membranes in the absence of attractive interactions. The study of crystalline order on the frozen surface of spherical membranes is also treated.Comment: Chapter 11 in "Statistical mechanics of Membranes and Surfaces", ed. by D.R. Nelson, T. Piran and S. Weinberg (World Scientific, Singapore, 2004); 25 pages with 26 figures (high resolution figures available from author

First-order phase transition of the tethered membrane model on spherical surfaces

We found that three types of tethered surface model undergo a first-order phase transition between the smooth and the crumpled phase. The first and the third are discrete models of Helfrich, Polyakov, and Kleinert, and the second is that of Nambu and Goto. These are curvature models for biological membranes including artificial vesicles. The results obtained in this paper indicate that the first-order phase transition is universal in the sense that the order of the transition is independent of discretization of the Hamiltonian for the tethered surface model.Comment: 22 pages with 14 figure

First-order phase transition in the tethered surface model on a sphere

We show that the tethered surface model of Helfrich and Polyakov-Kleinert undergoes a first-order phase transition separating the smooth phase from the crumpled one. The model is investigated by the canonical Monte Carlo simulations on spherical and fixed connectivity surfaces of size up to N=15212. The first-order transition is observed when N>7000, which is larger than those in previous numerical studies, and a continuous transition can also be observed on small-sized surfaces. Our results are, therefore, consistent with those obtained in previous studies on the phase structure of the model.Comment: 6 pages with 7 figure