Crumpling self-avoiding surfaces

Self-avoiding plaquette surfaces with a folding or bending fugacity are believed to undergo a “crumpling” transition from a flaccid phase with branched polymer characteristics (corresponding to surfaces with a high degree of folding), to a “smooth” phase (corresponding to surfaces faith a low degree of folding). I develop rigorous techniques in order to bound the free energy of this model. In particular, the limiting free energy is proven to be positive for all positive values of the folding fugacity. In addition, the existence of a nonanalyticity in the limiting free energy of a (nontrivial) subclass of surfaces is proven. This implies the existence of a phase transition in this model, which I conjecture to be from a “flaccid” to a “smooth” phase.     

New renormalization group results for scaling of self-avoiding tethered membranes

The scaling properties of self-avoiding polymerized two-dimensional membranes are studied via renormalization group methods based on a multilocal operator product expansion. The renormalization group functions are calculated to second order. This yields the scaling exponent ν to order ε² Our extrapolations for ν agree with the Gaussian variational estimate for large space dimension d and are close to the Flory estimate for d = 3. The interplay between self-avoidance and rigidity at small d is briefly discussed.     

Crumpling of a stiff tethered membrane

A first-principles numerical model for crumpling of a stiff tethered membrane is introduced. This model displays wrinkles, ridge formation, ridge collapse, and initiation of stiffness divergence. The amplitude and wavelength of the wrinkles and the scaling exponent of the stiffness divergence are consistent with both theory and experiment. Close to the stiffness divergence further buckling is hindered by the nonzero thickness of the membrane, and its elastic behavior becomes similar to that of dry granular media. No change in the distribution of contact forces can be observed at the crossover, implying that the network of ridges is then simultaneously a granular force-chain network.     

Universality classes of self-avoiding fixed-connectivity membranes

We present an analysis of extensive large-scale Monte Carlo simulations of self-avoiding fixed-connectivity membranes for sizes (number of faces) ranging from 512 to 17672 (triangular) plaquettes. Self-avoidance is implemented via impenetrable plaquettes. We simulate the impenetrable plaquette model in both three and four bulk dimensions. In both cases we find the membrane to be flat for all temperatures: the size exponent in three dimensions is ν = 0.95(5) (Hausdorff dimension d _H = 2.1(1)). The single flat phase appears, furthermore, to be equivalent to the large bending rigidity phase of non-self-avoiding fixed-connectivity membranes --the roughness exponent in three dimensions is ξ = 0.63(4). This suggests that there is a unique universality class for flat fixed-connectivity membranes without attractive interactions. Finally, we address some theoretical and experimental implications of our work. Received 23 June 2000 and Received in final form 25 October 2000     

Large order behavior for self-avoiding membranes

We derive the large order behavior of the perturbative expansion for the continuous model of tethered self-avoiding membranes. It is controlled by a classical configuration for an effective potential in bulk space, which is the analog of the Lipatov instanton, solution of a highly non-local equation. The nth order is shown to have factorial growth as (−cst)ⁿ (n!)^(1−ϵ/D), where D is the “internal” dimension of the membrane and ϵ the engineering dimension of the coupling constant for self-avoidance. The instanton is calculated within a variational approximation, which is shown to become exact in the limit of large dimension d of bulk space. This is the starting point of a systematic 1/d expansion. As a consequence, the ϵ-expansion of self-avoidinng membranes has a factorial growth, like the ϵ-expansion of polymers and standard critical phenomena, suggesting Borel summability. Consequences for the applicability of the 2-loop calculations are examined.     

Universal negative poisson ratio of self-avoiding fixed-connectivity membranes

We determine the Poisson ratio of self-avoiding fixed-connectivity membranes, modeled as impenetrable plaquettes, to be sigma = -0.37(6), in statistical agreement with the Poisson ratio of phantom fixed-connectivity membranes sigma = -0.32(4). Together with the equality of critical exponents, this result implies a unique universality class for fixed-connectivity membranes. Our findings thus establish that physical fixed-connectivity membranes provide a wide class of auxetic (negative Poisson ratio) materials with significant potential applications in materials science.     

Self-avoiding tethered membranes at the tricritical point

The scaling properties of self-avoiding tethered membranes at the tricritical point (Θ-point) are studied by perturbative renormalization group methods. To treat the 3-body repulsive interaction (known to be relevant for polymers), new analytical and numerical tools are developed and applied to 1-loop calculations. These techniques are a prerequisite to higher-order calculations for self-avoiding membranes. The crossover between the 3-body interaction and the modified 2-body interaction, attractive at long range, is studied through a new double ε-expansion. It is shown that the latter interaction is relevant for 2-dimensional membranes at the Θ-point.     

Polymer-decorated tethered membranes under good- and poor-solvent conditions

We study tethered membranes grafted by polymer chains on one side. Mean-field and scaling arguments predicting a spontaneous curvature are compared to the results of lattice-based Monte Carlo simulations using the Bond Fluctuation Model, which are carried out for various grafting densities and chain lengths. We show that already slightly overlapping chains bend the membrane significantly. This proves the entropic origin for the bending stiffness, which is of order kT . To understand the membrane curvature under conditions of very small bending stiffness we apply a geometrical model which takes into account the state of chains at the overlap threshold. Applying a thermal solvent model for the grafted chains, we demonstrate that the bending direction of the membrane can be triggered by variation of the solvent quality. This indicates that polymer-decorated membranes may serve as switchable nanoscale devices.     

Inverse Anderson transition caused by flatbands

We propose a new disorder-induced insulator-metal transition of one-electron states, which may be called the "inverse Anderson transition." We first make a highly degenerated localized states by constructing a three-dimensional periodic system possessing only flat dispersion relations. When we introduce a disorder into it, a finite-size scaling of the level statistics shows two clear (localization-delocalization and delocalization-localization) transitions for a wide range of the energy, with increasing the degree of disorder. These transitions are confirmed also by finding the system-size-independent characteristic of the wave function.     

Polymers with self-avoiding interaction in random medium: a localization-delocalization transition

In this paper we investigate the problem of a long self-avoiding polymer chain immersed in a random medium. We find that in the limit of a very long chain and when the self-avoiding interaction is weak, the conformation of the chain consists of many “blobs” with connecting segments. The blobs are sections of the molecule curled up in regions of low potential in the case of a Gaussian distributed random potential or in regions of relatively low density of obstacles in the case of randomly distributed hard obstacles. We find that as the strength of the self-avoiding interaction is increased the chain undergoes a delocalization transition in the sense that the appropriate free energy per monomer is no longer negative. The chain is then no longer bound to a particular location in the medium but can easily wander around under the influence of a small perturbation. For a localized chain we estimate quantitatively the expected number of monomers in the “blobs” and in the connecting segments. Received 13 November 2002 Published online 14 March 2003     

Renormalization of self-avoiding walks by non-linear transformations

Renormalizations of self-avoiding lattice walks by non-linear transformations are discussed. A procedure to obtain approximate renormalization group equations, together with the value of the critical index ν, is carried through for the triangular and the square lattice.     

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.