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arXiv:cond-mat/9804264 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Crossover scaling from classical to nonclassical critical behavior

Andrea Pelissetto C, Paolo Rossi, Ettore Vicari

Published Dec 1, 1998• 34 citations
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Abstract

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-$N$ 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.

DOI
10.1103/physreve.58.7146

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Physical review. B, Condensed matter•Apr 1, 1986•84 citations
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 O(N)\ensuremathφ4\mathrm{O}(N) {\ensuremath{\varphi}}^{4}O(N)\ensuremathφ4 model. We also show that the effective susceptibility exponent is nonmonotonic in the low-temperature phase of the three-dimensional Ising model.
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.
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 show that the critical behavior of disordered ferromagnets can be understood in terms of an effective, translational invariant Hamiltonian with renormalized coupling parameters which exhibit a tricritical as well as a critical point. The stable critical fixed point describes the critical behavior of weakly disordered systems, whereas the tricritical fixed point is responsible for the critical behavior of strongly disordered systems at least in the experimentally accessible range of temperature. This necessitates a crossover theory to describe the crossover between these two fixed points and to calculate critical properties as a function of concentration and temperature. In this paper we study the symmetric phase. Susceptibility and specific heat are calculated in scaling form using nonlinear scaling fields to verify that disordered ferromagnets exhibit universal critical behavior. The corresponding effective exponents exhibit nonmonotonous concentration- and temperature-dependent behavior. According to our theory, Fisher-renormalized tricritical behavior dominates in the experimentally accessible temperature range which leads to extreme values of the effective exponents ${\ensuremath{\gamma}}_{\mathrm{eff}}$=2 and ${\ensuremath{\alpha}}_{\mathrm{eff}}$=-1 in the limit of strong dilution. However, in accord with the Harris criterion we verify that the asymptotic behavior (T\ensuremath{\rightarrow}Tc{T}_{c}Tc​) is not changed by dilution if the critical exponent of the specific heat of the pure system ${\ensuremath{\alpha}}_{\mathrm{pure}<0}$, independent of the value of the concentration.
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.
The crossover from ideal-Bose-gas behavior to nonclassical, XY-like criticality in a dilute, interacting Bose fluid is considered in detail with emphasis on the superfluid density, ${\ensuremath{\rho}}_{s}$(T), below the transition. A general discussion of crossover scaling in constrained systems shows that ${\ensuremath{\rho}}_{s}$ and other temperature-dependent thermodynamic functions at controlled overall density can be expressed in terms of universal scaling functions. The finite-temperature Green's function formalism is used to calculate the associated linear scaling fields and the crossover exponent exactly for all dimensionalities d>2, scaling predictions, including logarithmic singularities just at d=3, being checked via second-order computations. A mapping of the Bose problem onto the classical s4{s}^{4}s4 spin model is achieved which is asymptotically exact in the critical region. In this way, previously known renormalization-group results are used to calculate the scaling functions for ${\ensuremath{\rho}}_{s}$, for the off-diagonal susceptibility, etc., to first order in \ensuremath{\epsilon}=4-d. Good agreement is found with the experimental measurements by Reppy and co-workers of ${\ensuremath{\rho}}_{s}$(T) for helium absorbed at low coverages in Vycor glass (of pore size 50--80 A\r{}). An effective-mass ratio m∗{m}^{\mathrm{*}}m∗/m\ensuremath{\simeq}1.5, and an effective-scattering-length ratio a∗{a}^{\mathrm{*}}a∗/a\ensuremath{\simeq}1.3 are indicated for mobile helium in the Vycor (although the irregular character of Vycor has not, at this point, been fully allowed for in the theory). Other approaches in the literature to computing the critical behavior of quantum-mechanical systems are reviewed and related to the present analysis.