New Analytical Results on Anisotropic Membranes

We report on recent progress in understanding the tubular phase of self-avoiding anisotropic membranes. After an introduction to the problem, we sketch the renormalization group arguments and symmetry considerations that lead us to the most plausible fixed point structure of the model. We then employ an epsilon-expansion about the upper critical dimension to extrapolate to the physical interesting 3-dimensional case. The results are $\nu=0.62$ for the Flory exponent and $\zeta=0.80$ for the roughness exponent. Finally we comment on the importance that numerical tests may have to test these predictions.Comment: LATTICE98(surfaces), 3 pages, 2 eps 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

Phase transitions of a tethered membrane model on a torus with intrinsic curvature

A tethered surface model is investigated by using the canonical Monte Carlo simulation technique on a torus with an intrinsic curvature. We find that the model undergoes a first-order phase transition between the smooth phase and the crumpled one.Comment: 12 pages with 8 figure

Phase transition of an extrinsic curvature model on tori

We show a numerical evidence that a tethered surface model with extrinsic curvature undergoes a first-order crumpling transition between the smooth phase and a non-smooth phase on triangulated tori. The results obtained in this Letter together with the previous ones on spherical surfaces lead us to conclude that the tethered surface model undergoes a first-order transition on compact surfaces.Comment: 13 pages with 10 figure

Monte Carlo simulations of a tethered membrane model on a disk with intrinsic curvature

A first-order phase transition separating the smooth phase from the crumpled one is found in a fixed connectivity surface model defined on a disk. The Hamiltonian contains the Gaussian term and an intrinsic curvature term.Comment: 10 pages with 6 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

Grand Canonical simulations of string tension in elastic surface model

We report a numerical evidence that the string tension \sigma can be viewed as an order parameter of the phase transition, which separates the smooth phase from the crumpled one, in the fluid surface model of Helfrich and Polyakov-Kleinert. The model is defined on spherical surfaces with two fixed vertices of distance L. The string tension \sigma is calculated by regarding the surface as a string connecting the two points. We find that the phase transition strengthens as L is increased, and that \sigma vanishes in the crumpled phase and non-vanishes in the smooth phase.Comment: 7 pages with 7 figure

Nonvanishing string tension of elastic membrane models

By using the grand canonical Monte Carlo simulations on spherical surfaces with two fixed vertices separated by the distance L, we find that the second-order phase transition changes to the first-order one when L is sufficiently large. We find that string tension \sigma \not= 0 in the smooth phase while \sigma \to 0 in the wrinkled phase.Comment: 10 pages with 6 figure