Fourier Monte Carlo Simulation of Elastic Membranes

Based on our recently developed Fourier Monte Carlo algorithm [A. Tröster, Phys. Rev. B 76, 012402 (2007)] we propose a novel approach to study the elasticity of solid and hexatic membranes. In detail, we offer three different but related approaches to study the low-temperature ``flat`` phase, based on (i) Wilson`s momentum shell renormalization group scheme for finite shell thickness, (ii) a computational version of field-theoretic renormalization methods, and (iii) a direct evaluation of the correlation function of the membrane`s Fourier-transformed unit normals in Fourier space using a small wave vector cutoff. These calculations are expected to be highly relevant for understanding the important problem of the formation of intrinsic ripples in graphene sheets. Moreover, our pioneer approaches (i) and (ii) should also be extremely interesting from a theoretical point of view, as they both constitute exciting new nonperturbative computational renormalization group schemes.

The project at hand Fourier Monte Carlo simulation of elastic membranes has aimed to investigate the elastic properties of crystalline and so-called hexatic membranes, whose structure roughly represents an intermediate state during the melting of a crystalline membrane into a fluid phase, using a novel computer simulation approach known as the Fourier Monte Carlo (FMC) algorithm that was developed by the project's principal investigator. A quantitative understanding of the spatial fluctuation behavior of such membranes is of vital importance in a biological context (cell membranes) as well as for the understanding of the physical properties of novel nano-membranes like graphene. In the limit of long wavelengths their elastic fluctuations should obey a so-called scaling law. The considerable dispersion in the values for the corresponding exponent eta of a crystalline membrane published in the literature up to the present date may serve to illustrate the technical and conceptual difficulties encountered in an effort to determine this value precisely. As to computer simulation approaches, the main obstacles that need to be overcome arise from the combined presence of long-ranged interactions and the so-called critical slowing down that seriously affects the efficiency of conventional simulations. In our project we were able to overcome both difficulties by implementing an FMC simulation in combination with a new algorithmic optimization. An important factor for the success of our approach was the design of a parallel version of the algorithm that allows to exploit the possibilities offered by modern supercomputer facilities. Armed with these tools, we succeeded not only in computing highly precise estimates of the numerical value of eta, but for the first time even to predict the leading corrections to the corresponding scaling law. Our results also provide clear evidence of the absence of the so-called intrinsic ripples in graphene, whose existence has been debated lively in the literature. This success also rests on the combination of FMC with two distinct but closely related techniques. While finite size scaling (systematic extrapolation of results obtained for finite systems for infinite system size) is an indispensable concept for computer simulations, the paradigm of the renormalization group (RG), i.e. the mathematical analysis of the dependence of the coupling constants of the theory on the underlying length scale, plays a paramount role in a number of different branches of modern physics. Our results show that the combination of RG and FMC can deliver solutions to problems for which all other approaches have failed so far. For the so-called crinkled phase of hexatic membranes our results suggest a logarithmic behavior of the corresponding elastic fluctuations, but definitive conclusions concerning the numerical value of eta must be postponed to future investigations.