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arXiv:1303.3726 · Physical Review B

High-precision Fourier Monte Carlo simulation of crystalline membranes

A. Tröster

Published Mar 25, 2013• 39 citations
arXivHTMLPDF

Abstract

We report an essential improvement of the plain Fourier Monte Carlo algorithm that promises to be a powerful tool for investigating critical behavior in a large class of lattice models, in particular those containing microscopic or effective long-ranged interactions. On tuning the Monte Carlo acceptance rates separately for each wave vector, we are able to drastically reduce critical slowing down. We illustrate the resulting efficiency and unprecedented accuracy of our algorithm with a calculation of the universal elastic properties of crystalline membranes in the flat phase and derive a numerical estimate $\ensuremath{\eta}=0.795(10)$ for the critical exponent $\ensuremath{\eta}$ that challenges those derived from other recent simulations. The large system sizes accessible to our present algorithm also allow us to demonstrate that insufficiently taking into account corrections to scaling may severely hamper a finite size scaling analysis. This observation may also help to clarify the apparent disagreement of published numerical estimates of $\ensuremath{\eta}$ in the existing literature.

DOI
10.1103/physrevb.87.104112

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We report an essential improvement of the plain Fourier Monte Carlo algorithm that promises to be a powerful tool for investigating critical behavior in a large class of lattice models, in particular those containing microscopic or effective long-ranged interactions. On tuning the Monte Carlo acceptance rates separately for each wave vector, we are able to drastically reduce critical slowing down. We illustrate the resulting efficiency and unprecedented accuracy of our algorithm with a calculation of the universal elastic properties of crystalline membranes in the flat phase and derive a numerical estimate \ensuremathη=0.795(10)\ensuremath{\eta}=0.795(10)\ensuremathη=0.795(10) for the critical exponent \ensuremathη\ensuremath{\eta}\ensuremathη that challenges those derived from other recent simulations. The large system sizes accessible to our present algorithm also allow us to demonstrate that insufficiently taking into account corrections to scaling may severely hamper a finite size scaling analysis. This observation may also help to clarify the apparent disagreement of published numerical estimates of \ensuremathη\ensuremath{\eta}\ensuremathη in the existing literature.
The computation of the critical exponent η characterizing the universal elastic behavior of crystalline membranes in the flat phase continues to represent challenges to theorists as well as computer simulators that manifest themselves in a considerable spread of numerical results for η published in the literature. We present additional insight into this problem that results from combining Wilson's momentum shell renormalization-group method with the power of modern computer simulations based on the Fourier Monte Carlo algorithm. After discussing the ideas and difficulties underlying this combined scheme, we present a calculation of the renormalization-group flow of the effective two-dimensional Young modulus for momentum shells of different thickness. Extrapolation to infinite shell thickness allows us to produce results in reasonable agreement with those obtained by functional renormalization group or by Fourier Monte Carlo simulations in combination with finite-size scaling. Moreover, our method allows us to obtain a decent estimate for the value of the Wegner exponent ω that determines the leading correction to scaling, which in turn allows us to refine our numerical estimate for η previously obtained from precise finite-size scaling data.
We present a Monte Carlo study of the two-component 4 model on the simple cubic lattice in three dimensions. By suitable tuning of the coupling constant, , we eliminate leading-order corrections to scaling. High-statistics simulations using finite-size scaling techniques yield = 0.6723(3)[8] and = 0.0381(2)[2], where the statistical and systematical errors are given in the first and second bracket, respectively. These results are more precise than any previous theoretical estimate of the critical exponents for the 3D XY universality class.
We present a finite size scaling technique to calculate the central charge and some critical exponents of two-dimensional critical models with a Monte Carlo simulation. We use systems with dimensions $L\ifmmode\times\else\texttimes\fi{}M,$ and focus on the scaling behavior in M/L.M/L.M/L. The finite size scaling relation that we use is the universal expression for the stress tensor on the torus. The stress tensor is the operator that governs the anisotropy of the system, and stems for the theory of conformal invariance. We show that a lattice representation of the stress tensor can easily be constructed, such that its expectation value on the torus can be calculated in a Monte Carlo simulation. In doing so, we observe that the stress tensor turns out to be remarkably insensitive to critical slowing down. We show that the typical simulation time scales with the linear system dimension LLL roughly as L4,{L}^{4},L4, and that this scaling holds for the ordinary Metropolis algorithm as well as for more sophisticated cluster algorithms, such that it is fruitless to invoke the latter. We test the method for the Ising model (with central charge c=c=c=12\frac{1}{2}21​), the Ashkin-Teller model (c=1),(c=1),(c=1), and the F model (also c=1c=1c=1).
Corrections to scaling in the 3D Ising model are studied based on Monte Carlo (MC) simulation results for very large lattices with linear lattice sizes up to L=3456. Our estimated values of the correction-to-scaling exponent omega tend to decrease below the usually accepted value about 0.83 when the smallest lattice sizes are discarded from the fits. This behavior apparently confirms some of the known estimates of the Monte Carlo renormalization group (MCRG) method, i.e., omega about 0.7 and omega = 0.75(5). We discuss the possibilities that omega is either really smaller than usually expected or these values of omega describe some transient behavior. We propose refining MCRG simulations and analysis to resolve this issue. In distinction from omega, our actual MC estimations of the critical exponents eta and nu provide stable values eta=0.03632(13) and nu=0.63017(31), which well agree with those of the conformal bootstrap method, i.e., eta=0.0362978(20) and nu=0.6299709(40).
Our recently developed Fourier Monte Carlo algorithm permits a nonperturbative calculation of momentum-shell renormalization-group flows by simulation which despite its apparent simplicity is illustrative both numerically as well as conceptually interesting. We study the example of a varphi(4) model with long-range lattice interaction. For this model we show that the topology of the renormalization flow is globally accessible in a particularly convenient way. The nontrivial fixed point of Wilson-Fisher type is observed its accompanying critical exponents are numerically determined from fitting its surrounding flow pattern to a linearized renormalization-group transformation. The results are compared to those obtained from perturbation theory, -expansion and earlier Monte Carlo simulations. Application of our method is also expected to be rewarding in other models with long-range interactions.