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arXiv:cond-mat/9807415 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Nature of crossover from classical to Ising-like critical behavior

Erik Luijten, Kurt Binder

Published Oct 1, 1998• 57 citations
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Abstract

We present an accurate numerical determination of the crossover from classical to Ising-like critical behavior upon approach of the critical point in three-dimensional systems. The possibility of varying the Ginzburg number in our simulations allows us to cover the entire crossover region. We employ these results to scrutinize several semiphenomenological crossover scaling functions that are widely used for the analysis of experimental results. In addition, we present strong evidence that the exponent relations do not hold between effective exponents.

DOI
10.1103/physreve.58.r4060

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We present an accurate numerical determination of the crossover from classical to Ising-like critical behavior upon approach of the critical point in three-dimensional systems. The possibility of varying the Ginzburg number in our simulations allows us to cover the entire crossover region. We employ these results to scrutinize several semiphenomenological crossover scaling functions that are widely used for the analysis of experimental results. In addition, we present strong evidence that the exponent relations do not hold between effective exponents.
We determine the scaling functions describing the crossover from Ising-like critical behavior to classical critical behavior in two-dimensional systems with a variable interaction range. Since this crossover spans several decades in the reduced temperature as well as in the finite-size crossover variable, it has up to now largely evaded a satisfactory numerical determination. Using a dedicated Monte Carlo method, we could obtain accurate results for sufficiently large interaction ranges. Our data cover the full crossover region both above and below the critical temperature and support the hypothesis that the crossover functions are universal. Also the so-called effective exponents are discussed and we show that these can vary nonmonotonically in the crossover region.
We study the crossover between classical and nonclassical critical behaviors. The critical crossover limit is driven by the Ginzburg number G. The corresponding scaling functions are universal with respect to any possible microscopic mechanism which can vary G, such as changing the range or the strength of the interactions. The critical crossover describes the unique flow from the unstable Gaussian to the stable nonclassical fixed point. The scaling functions are related to the continuum renormalization-group functions. We show these features explicitly in the large-NNN limit of the $\mathrm{O}(N) {\ensuremath{\varphi}}^{4}$ model. We also show that the effective susceptibility exponent is nonmonotonic in the low-temperature phase of the three-dimensional Ising model.
Accurate numerical results are presented for the three-dimensional equivalent-neighbor model on a cubic lattice, for 12 different interaction ranges (coordination number between 18 and 250). These results allow the determination of the range dependences of the critical temperature and various critical amplitudes, which are compared to renormalization-group predictions. In addition, the analysis yields an estimate for the interaction range at which the leading corrections to scaling vanish for the spin-1 / 2 model, and confirms earlier conclusions that the leading Wegner correction must be negative for the three-dimensional (nearest-neighbor) Ising model. By complementing these results with Monte Carlo data for systems with coordination numbers as large as 52 514, the full finite-size crossover curves between classical and Ising-like behavior are obtained as a function of a generalized Ginzburg parameter. Also, the crossover function for the effective magnetic exponent is determined.
A semi-microscopic scaling-field theory is developed for crossover phenomena near critical and tricritical points. The theory is based on a renormalization-group description of a model with two competing fixed points (such as a critical and a tricritical fixed point) in terms of scaling fields. The coupled nonlinear differential equations for scaling fields are truncated such as to preserve the physics essential for crossover phenomena. The approach allows the explicit calculation of thermodynamic functions for (i) tricritical systems and (ii) critical systems with an irrelevant scaling field. We obtain, for example, an explicit expression for the scaling function of the susceptibility, which describes the crossover from the tricritical to the critical region. The idea of "flow diagrams" in the scaling-field space is used to characterize crossover phenomena globally in the whole critical region. The concept of asymptotic critical exponents is generalized and effective critical exponents are introduced as logarithmic derivatives of thermodynamic quantities with respect to experimental fields and scaling fields, respectively. By using the method of effective exponents the size of the crossover region between regions of different asymptotic critical behavior is estimated. For the susceptibility, the width of the crossover region in decades of the effective temperature variable is roughly equal to the inverse of the crossover exponent. In the case of a critical system with a slow transient the asymptotic critical exponent is only reached extremely close to the critical point (unless the amplitude of the transient vanishes). It might then be impossible to determine the asymptotic exponent experimentally or by conventional series-expansion techniques, and an analysis of the data in terms of effective exponents is the alternative. The scaling-field approach is applied to three systems with crossover phenomena: (i) the model for tricritical systems with molecular-field tricritical exponents, (ii) the Ashkin-Teller model in three dimensions, and (iii) a model for phase transitions with Fisher exponent renormalization due to a constraint.
We have numerically determined the behavior of the magnetic susceptibility upon approach of the critical point in two-dimensional spin systems with an interaction range that was varied over nearly 2 orders of magnitude. The full crossover from classical to Ising-like critical behavior, spanning several decades in the reduced temperature, could be observed. Our results convincingly show that the effective susceptibility exponent ${\ensuremath{\gamma}}_{\mathrm{eff}}$ changes nonmonotonically from its classical to its Ising value when approaching the critical point in the ordered phase. In the disordered phase the behavior is monotonic. Furthermore, the hypothesis that the crossover function is universal is supported.
Motivated by recent, apparently conflicting, Monte Carlo simulations of three-dimensional critical wetting in Ising systems and interface displacement models, we employ a linear functional renormalization group to calculate explicitly the effective surface magnetization exponent in fixed-temperature, field-driven wetting. Our results indicate, for some values of the parameters, a quick exit from the classical regime, followed by a slow, very broad crossover to pure unadulterated nonuniversal criticality that requires a many-decade decrease of the magnetic field.