Marco Falcioni

The Flat Phase of Crystalline Membranes

We present the results of a high-statistics Monte Carlo simulation of a phantorn crystalline (fixed-connectivity) membrane with free boundary. We verify trie existence of a fiai phase by exarnining lattices of size up to 128~. The Harniltonian of the rnodel is trie sum of a simple spnng pair potential, with no hard-core repulsion, and beuding energy. The only free pararneter is the bending ngidity ~. In-plane elastic constants are non explicitly introduced. We obtain the rernarkable result thon this simple model dynamically generates the elastic constants required to stabilize the fiai phase. We present measurements of the size (Flory) exportent v and the roughness exportent (. We also determine the critical exponents ~ and ~u descnbing the scale dependence of the bending rigidity (~(q) mJ q~~) and trie mduced elastic constants (A(q)mJ ~1(q) ~J q~~ ). Ai bending rigidity ~ = l-1, we find v = 0.95(5) (Hausdorlf dimension dH = 2 Iv = 2,1(1)), ( = 0.64(2 and ~u = 0.50(1). These results are consistent with the scaling relation ( = (2 + ~u)/4. The additional scaling relation ~ = 2(1- () irnplies ~ = 0.72(4). A direct measurernent of ~ from the power-law decay of the normal-normal correlation functiou yields ~ m 0.6 on the 128~ lattice.

Fluid random surfaces with extrinsic curvature. II

Abstract We present the results of an extension of our previous work on large-scale simulations of dynamically triangulated toroidal random surfaces embedded in R 3 with extrinsic curvature. We find that the extrinsic-curvature specific heat peak ceases to grow on lattices with more than 576 nodes and that the location of the peak λ c also stabilizes. The evidence for a true crumpling transition is still weak. If we assume it exists we can say that the finite-size scaling exponent α / ηd is very close to zero or negative. On the other hand our new data does rule out the observed peak as being a finite-size artifact of the persistence length becoming comparable to the extent of the lattice.

Numerical Observation of a Tubular Phase in Anisotropic Membranes

We provide the first numerical evidence for the existence of a tubular phase, predicted by Radzihovsky and Toner (RT), for anisotropic tethered membranes without self-avoidance. Incorporating anisotropy into the bending rigidity of a simple model of a tethered membrane with free boundary conditions, we show that the model indeed has two phase transitions corresponding to the flat-to-tubular and tubular-to-crumpled transitions. For the tubular phase we measure the Flory exponent {nu}{sub F} and the roughness exponent {zeta} . We find {nu}{sub F}=0.305(14) and {zeta}=0.895(60) , which are in reasonable agreement with the theoretical predictions of RT; {nu}{sub F}=1/4 and {zeta}=1 . {copyright} {ital 1997} {ital The American Physical Society}

The flat phase of fixed-connectivity membranes

Abstract 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 [1]. 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.

Two Ising models coupled to two-dimensional gravity

Abstract To investigate the properties of c = 1 matter coupled to 2d-gravity we have performed large-scale simulations ot two copies of the Ising model on a dynamical lattice. We measure spin susceptibility and percolation critical exponents using finite-size scaling. We show explicity how algorithmic corrections are needed for a proper comparison with theoretical exponents. We also exhibit correlations, mediated by gravity, between the energy and magnetic properties of the two Ising species. The prospects for extending this work beyond c = 1 are dressed.

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.

Improved algorithms for simulating crystalline membranes

Abstract The physics of crystalline membranes, i.e. fixed-connectivity surfaces embedded in three dimensions and with an extrinsic curvature term, is very rich and of great theoretical interest. Numerical simulations are commonly used to study this class of models. Unfortunately, traditional Monte Carlo algorithms suffer from very long auto-correlation times, especially near critical points. In this paper we study the performance of improved Monte Carlo algorithms for simulating crystalline membrane, such as hybrid overrelaxation and unigrid methods, and compare their performance to the more traditional Metropolis algorithm. We find that although the overrelaxation algorithm does not reduce the critical slowing down, it gives an overall gain of a factor 15 over the Metropolis algorithm. The unigrid algorithm does, on the other hand, reduce the critical slowing down exponent to z ≈ 1.7.

Critical slowing down of cluster algorithms for Ising models coupled to 2-d gravity

Abstract We simulate single and multiple Ising models coupled to 2-d gravity using both the Swendsen-Wang and Wolff algorithms to update the spins. We study the integrated autocorrelation time and find that there is considerable critical slowing down, particularly in the magnetization. We argue that this is primarily due to the local nature of the dynamical triangulation algorithm and to the generation of a distribution of baby universes which inhibits cluster growth.

Multiple Ising models coupled to 2-d gravity: a CSD analysis

Abstract We simulate single and multiple Ising models coupled to 2-d gravity and we measure critical slowing down (CSD) with the standard methods. We find that the Swendsen-Wang and Wolff cluster algorithms do not eliminate CSD. We interpret the result as an effect of the mesh dynamics.