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Ferroelectrics

Finite size effects on phase transitions

Kurt Binder

Published Jun 1, 1987• 250 citations
Publisher

Abstract

Both first- and second-order transitions get smeared and shifted due to the finite size of a sample. Recent theoretical developments on this finite size rounding of singularities associated with phase transitions are briefly reviewed, and evidence from Monte Carlo simulations for the validity of these concepts will be discussed. It is also shown how finite size scaling theories in conjunction with simulation data from finite systems are a powerful tool for the quantitative study of bulk critical phenomena. We shall also mention extensions such as finite size tests of hyperscaling, finite size scaling of interfacial properties, finite size effects on critical relaxation, etc. A brief discussion of the experimental situation concludes this review.

DOI
10.1080/00150198708227908

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Both first- and second-order transitions get smeared and shifted due to the finite size of a sample. Recent theoretical developments on this finite size rounding of singularities associated with phase transitions are briefly reviewed, and evidence from Monte Carlo simulations for the validity of these concepts will be discussed. It is also shown how finite size scaling theories in conjunction with simulation data from finite systems are a powerful tool for the quantitative study of bulk critical phenomena. We shall also mention extensions such as finite size tests of hyperscaling, finite size scaling of interfacial properties, finite size effects on critical relaxation, etc. A brief discussion of the experimental situation concludes this review.
Abstract In finite systems singularities, such as discontinuities or divergences, associated with phase transitions are rounded. The extrapolation of finite size calculations to the singular thermodynamic limit makes use of scaling laws, introduced by Fisher, which involve the size of the system. The use and the theoretical basis of these laws are briefly reviewed here; some difficulties associated with the analytic calculation of the scaling functions are discussed. The correct treatment of zero‐momentum modes leads to a modified renormalization group picture; above the upper critical dimension (u.c.d.) finite size scaling does not hold but the properties of the finite system are easily calculable; below the u.c.d. the scaling functions are obtained in a systematic expansion in fractional powers of ϵ = 4–d. The rounding of first order transitions is also briefly considered; in spite of the lack of universality of the bulk transition itself, the rounding is universal because the correlation length of the finite system is large.
The current state of Monte Carlo work on phase transitions and critical phenomena is reviewed. Both classical and quantum Monte Carlo results are discussed with emphasis on statistical lattice-model studies involving the highly accurate calculation of critical exponents. It is shown that finite-size scaling is an effective method whereby not only simple lattice models but also those of complex practical magnetic materials can be treated. For models allowing crossover phenomena, it is shown that both finite-size scaling theory and conventional power-law functions must be used in carrying out a Monte Carlo analysis.
The theory of Finite Size Scaling describes a build-up of the bulk properties when a small system is increased in size. This description is particularly important in strongly correlated systems where critical fluctuations develop with increasing system size, including phase transition points, polymer conformations. Since numerical computer simulations are always done with finite samples, they rely on the Finite Size Scaling theory for data extrapolation and analysis. With the advent of large scale computing in recent years, the use of the size-scaling methods has become increasingly important.
First-order phase transitions are ubiquitous in nature but their presence is often uncertain because of the effects which finite size has on all transitions. In this article we consider a general treatment of size effects on lattice systems with discrete degrees of freedom and which undergo a first-order transition in the thermodynamic limit. We review recent work involving studies of the distribution functions of the magnetization and energy at a first-order transition in a finite sample of size N connected to a bath of size N′. Two cases: N′ = ∞ and N′ = finite are considered. In the former (canonical ensemble) case, the distributions are approximated by a superposition of Gaussians corresponding to the different phases; all finite-size effects then vary as N or 1/N. The latter case involves the Gaussian ensemble where the entropy of the bath has a convenient form; for small N′, first-order transitions are characterized by van der Waals' loops in (for example) the energy vs. temperature curves. Results from extensive Monte Carlo simulations of Ising and Potts models in d = 2 are presented to confirm the predictions. Comparison is made with data from second-order transitions to show that the order of a transition can be distinguished through such studies, although problems still occur for first-order transitions very close to critical points.
We develop a detailed finite-size-scaling theory at a general, asymmetric, temperature-driven, strongly first-order phase transition in a system with periodic boundary conditions. We compute scaling functions for various cumulants of energy in the form U(L,t)=U0{\mathit{U}}_{0}U0​(tLd{\mathit{tL}}^{\mathit{d}}tLd)+${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$U1{\mathit{U}}_{1}U1​(tLd{\mathit{tL}}^{\mathit{d}}tLd) with t=1-Tc{\mathit{T}}_{\mathit{c}}Tc​/T. In particular, we consider the specific heat and Binder's fourth cumulant and show this has a minimum value of 2/3-(e1{\mathit{e}}_{1}e1​/e2{\mathit{e}}_{2}e2​-e2{\mathit{e}}_{2}e2​/e1{\mathit{e}}_{1}e1​)2{)}^{2})2/12+O(${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$) at a temperature Tc(2){\mathit{T}}_{\mathit{c}}^{(2)}Tc(2)​(L)-Tc{\mathit{T}}_{\mathit{c}}Tc​=O(${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$). Various other pseudocritical temperatures corresponding to extrema of other cumulants are evaluated. We compare these theoretical predictions with extensive Monte Carlo simulations of the nominally strong first-order transitions in the eight- and ten-state Potts models in two dimensions for system sizes L\ensuremath{\le}50. The ten-state simulations agree with theory in all details in contrast to the eight-state data, and we give estimates for the bulk specific heats at Tc{\mathit{T}}_{\mathit{c}}Tc​ using all exactly known analytic results. A criterion is developed to estimate numerically whether or not system sizes used in a simulation of a first-order transition are in the finite-size-scaling regime.
We study the finite-size effects at a temperature-driven first-order transition by analyzing various moments of the energy distribution. The distribution function for the energy is approximated by the superposition of two weighted Gaussian functions yielding quantitative estimates for various quantities and scaling form for the specific heat. The rounding of the singularities and the shifts in the location of the specific-heat maximum are analyzed and the characteristic features of a first-order transition are identified. The predictions are tested on the ten-state Potts model in two dimensions by carrying out extensive Monte Carlo calculations. The results are found to be in good agreement with theory. Comparison is made with the second-order transitions in the two- and three-state Potts models.