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Physics Procedia

Critical phenomena without “hyper scaling”: How is the finite-size scaling analysis of Monte Carlo data affected?

Kurt Binder

Published Jan 1, 2010• 1 citation
Publisher

Abstract

The finite size scaling analysis of Monte Carlo data is discussed for two models for which hyperscaling is violated: (i) the random field Ising model (using a model for a colloid-polymer mixture in a random matrix as a representative) (ii) The Ising bi-pyramid in computing surface fields.
DOI
10.1016/j.phpro.2010.09.041

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In chapter one, we explain briefly the continuum limit, scaling, and high temperature expansion of critical phenomena, Monte Carlo algorithms and fitting. In chapter two, different continuum limits of the Ising model in dimensions (D) 2, 3 and 4 are investigated numerically. The data indicate that triviality occurs for D = 4 and fails for D < 4 in each limit. In chapter three, a relation between the critical exponents of the leading and confluent scaling terms is derived using the finite size scaling argument. We also determine the new scaling variable of the 4D Ising model based on a new Monte Carlo simulation data. In chapter four, a Monte Carlo study of two dimensional diluted Ising systems is reported. It is shown that regular dilution does not affect critical exponents, while a random one does, with critical exponents varying continuously with impurity concentration. The importance of fluctuations in producing such effects is emphasized. In chapter five, a different point of view regarding the critical exponent of the specific heat of the 3D Ising model is presented. Based on the analysis of high temperature expansion, finite size scaling and Monte Carlo data in the symmetric phase of the 3D Ising model, it is shown that logarithmic scaling behavior of specific heat is more consistent than power scaling behavior.
The implications of the scaling theory for the crossover from one universality class of critical phenomena to another one are worked out for finite-size scaling analyses of numerical simulation data. Near a multicritical point finite-size scaling critical amplitudes have a singular dependence on the distance from the multicritical point. Attention is also paid to the crossover to Landau-like behaviour when the interaction range diverges, and as an application Monte Carlo results for polymer mixtures are discussed.
Using single cluster flip Monte Carlo simulations we accurately determine new finite size scaling functions which are expressed only in terms of the variable x=${\ensuremath{\xi}}_{\mathit{L}}$/L, where ${\ensuremath{\xi}}_{\mathit{L}}$ is the correlation length in a finite system of size L. Data for the d=2 and d=3 Ising models, taken at different temperatures and for different size lattices, show excellent data collapse over the entire range of scaling variable for susceptibility and correlation length. From these finite size scaling functions we can estimate critical temperatures and exponents with rather high accuracy even though data are not obtained extremely close to the critical point. The bulk values of the renormalized four-point coupling constant are accurately measured and show strong evidence for hyperscaling. \textcopyright{} 1996 The American Physical Society.
In systems belonging to the universality class of the random field Ising model, the standard hyperscaling relation between critical exponents does not hold, but is replaced with a modified hyperscaling relation. As a result, standard formulations of finite-size scaling near critical points break down. In this work, the consequences of modified hyperscaling are analyzed in detail. The most striking outcome is that the free-energy cost ΔF of interface formation at the critical point is no longer a universal constant, but instead increases as a power law with system size, ΔF∝L(θ), with θ as the violation of hyperscaling critical exponent and L as the linear extension of the system. This modified behavior facilitates a number of numerical approaches that can be used to locate critical points in random field systems from finite-size simulation data. We test and confirm the approaches on two random field systems in three dimensions, namely, the random field Ising model and the demixing transition in the Widom-Rowlinson fluid with quenched obstacles.
We present a Monte Carlo simulation of the d = 3 Ising model in a random magnetic field with values ±H on each lattice site. Calculations have been performed on cubic lattices with linear dimension L = 6, 8, 10 and 12, and several values of H. A finite-size scaling analysis of the data for H = 0.5, 0.75 and 1.0 is consistent with hyperscaling, and a correlation exponent v = 0.7 ± 0.1. For H = 1.50, 1.75 and 2.0 some of the L = 12 data show a discontinuity as a function of the inverse temperature β. The implications of these results are discussed.