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.

The Phase Diagram of Crystalline Surfaces

We report the status of a high-statistics Monte Carlo simulation of non-self-avoiding crystalline surfaces with extrinsic curvature on lattices of size up to $128^2$ nodes. We impose free boundary conditions. The free energy is a gaussian spring tethering potential together with a normal-normal bending energy. Particular emphasis is given to the behavior of the model in the cold phase where we measure the decay of the normal-normal correlation function.Comment: 9 pages latex (epsf), 4 EPS figures, uuencoded and compressed. Contribution to Lattice '9

A New Phase of Tethered Membranes: Tubules

We show that fluctuating tethered membranes with {\it any} intrinsic anisotropy unavoidably exhibit a new phase between the previously predicted ``flat'' and ``crumpled'' phases, in high spatial dimensions $d$ where the crumpled phase exists. In this new "tubule" phase, the membrane is crumpled in one direction but extended nearly straight in the other. Its average thickness is $R_G\sim L^{\nu_t}$ with $L$ the intrinsic size of the membrane. This phase is more likely to persist down to $d=3$ than the crumpled phase. In Flory theory, the universal exponent $\nu_t=3/4$, which we conjecture is an exact result. We study the elasticity and fluctuations of the tubule state, and the transitions into it.Comment: 4 pages, self-unpacking uuencoded compressed postscript file with figures already inside text; unpacking instructions are at the top of file. To appear in Phys. Rev. Lett. November (1995

The shape of invasion perclation clusters in random and correlated media

The shape of two-dimensional invasion percolation clusters are studied numerically for both non-trapping (NTIP) and trapping (TIP) invasion percolation processes. Two different anisotropy quantifiers, the anisotropy parameter and the asphericity are used for probing the degree of anisotropy of clusters. We observe that in spite of the difference in scaling properties of NTIP and TIP, there is no difference in the values of anisotropy quantifiers of these processes. Furthermore, we find that in completely random media, the invasion percolation clusters are on average slightly less isotropic than standard percolation clusters. Introducing isotropic long-range correlations into the media reduces the isotropy of the invasion percolation clusters. The effect is more pronounced for the case of persisting long-range correlations. The implication of boundary conditions on the shape of clusters is another subject of interest. Compared to the case of free boundary conditions, IP clusters of conventional rectangular geometry turn out to be more isotropic. Moreover, we see that in conventional rectangular geometry the NTIP clusters are more isotropic than TIP clusters

Effects of Self-Avoidance on the Tubular Phase of Anisotropic Membranes

We study the tubular phase of self-avoiding anisotropic membranes. We discuss the renormalizability of the model Hamiltonian describing this phase and derive from a renormalization group equation some general scaling relations for the exponents of the model. We show how particular choices of renormalization factors reproduce the Gaussian result, the Flory theory and the Gaussian Variational treatment of the problem. We then study the perturbative renormalization to one loop in the self-avoiding parameter using dimensional regularization and an epsilon-expansion about the upper critical dimension, and determine the critical exponents to first order in epsilon.Comment: 19 pages, TeX, uses Harvmac. Revised Title and updated references: to appear in Phys. Rev.

Polymer Shape Anisotropy and the Depletion Interaction

We calculate the second and third virial coefficients of the effective sphere-sphere interaction due to polymer depletion. By utilizing the anisotropy of a typical polymer conformation, we can consider polymers that are roughly the same size as the spherical inclusions. We argue that recent experiments can confirm this anisotropy.Comment: 4 pages, 4 eps figures, RevTe

Fluctuating Nematic Elastomer Membranes: a New Universality Class

We study the flat phase of nematic elastomer membranes with rotational symmetry spontaneously broken by in-plane nematic order. Such state is characterized by a vanishing elastic modulus for simple shear and soft transverse phonons. At harmonic level, in-plane orientational (nematic) order is stable to thermal fluctuations, that lead to short-range in-plane translational (phonon) correlations. To treat thermal fluctuations and relevant elastic nonlinearities, we introduce two generalizations of two-dimensional membranes in a three dimensional space to arbitrary D-dimensional membranes embedded in a d-dimensional space, and analyze their anomalous elasticities in an expansion about D=4. We find a new stable fixed point, that controls long-scale properties of nematic elastomer membranes. It is characterized by singular in-plane elastic moduli that vanish as a power-law eta_lambda=4-D of a relevant inverse length scale (e.g., wavevector) and a finite bending rigidity. Our predictions are asymptotically exact near 4 dimensions.Comment: 18 pages, 4 eps figures. submitted to PR

Folding of the Triangular Lattice with Quenched Random Bending Rigidity

We study the problem of folding of the regular triangular lattice in the presence of a quenched random bending rigidity + or - K and a magnetic field h (conjugate to the local normal vectors to the triangles). The randomness in the bending energy can be understood as arising from a prior marking of the lattice with quenched creases on which folds are favored. We consider three types of quenched randomness: (1) a ``physical'' randomness where the creases arise from some prior random folding; (2) a Mattis-like randomness where creases are domain walls of some quenched spin system; (3) an Edwards-Anderson-like randomness where the bending energy is + or - K at random independently on each bond. The corresponding (K,h) phase diagrams are determined in the hexagon approximation of the cluster variation method. Depending on the type of randomness, the system shows essentially different behaviors.Comment: uses harvmac (l), epsf, 17 figs included, uuencoded, tar compresse

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