Fourier Monte Carlo simulation of crystalline membranes in the flat phase

• Published in: Journal of physics. Conference series 2013-01, Vol.454 (1), p.12032-17
• Main Author: Troster, Andreas
• Format: Article
• Language: English
• Subjects: Brackets; Computer simulation; Correlation; Crystal structure; Crystallinity; Elasticity; Fourier analysis; Graphene; Heat treatment; Mathematical analysis; Mathematical models; Membranes; Monte Carlo methods; Monte Carlo simulation; Numerical analysis; Physics; Scaling; Simulation; Size effects
• ISSN: 1742-6596; 1742-6588; 1742-6596
• DOI: 10.1088/1742-6596/454/1/012032

Stimulated by the recent interest in graphene, the elastic behavior of crystalline membranes continues to be under debate. In their flat phase, one observes scaling of the correlation functions of in-plane and out-of-plane deformations u(x) and f(x) at long wavelengths with respect to a given reference plane governed by a single universal exponent η. The purpose of the present article is to explain the ideas and techniques underlying our Fourier Monte Carlo simulation approach to the numerical determination of η in much greater detail than was possible in a recent letter that is currently under review. Our simulations are based on an effective Hamiltonian first derived by Nelson and Peliti formulated exclusively in terms of the Fourier amplitudes (q) of the field f(x), and we calculate the out-of-plane correlation function ⟨| (q)|2⟩ (q) and their related mean squared displacement ⟨(Δf)2⟩. The key to the progress reported in this work is the observation that on tuning the Monte Carlo acceptance rates separately for each wave vector, we are able to eliminate critical slowing down and thus achieve unprecedented statistical accuracy. A finite size scaling analysis for ⟨(Δf)2⟩ gives η = 0.795(10). In the alternative approach, where we study the scaling of (q), we observe an unexpected anisotropic finite size effect at small wave vectors which hampers a similarly accurate numerical analysis.