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Physical review. B, Condensed matter

Finite-size scaling at first-order phase transitions

Kurt Binder, D. P. Landau

Published Aug 1, 1984• 634 citations
Publisher

Abstract

Using thermodynamic fluctuation theory, we study the finite-size rounding of anomalies occurring at first-order phase transitions of the corresponding infinite system. Explicit expressions for thermodynamic functions are derived both for "symmetric transitions" (such as the jump of the spontaneous magnetization in the Ising model from $+{M}_{\mathrm{sp}}$ to $\ensuremath{-}{M}_{\mathrm{sp}}$ as the field changes from ${0}^{+}$ to ${0}^{\ensuremath{-}}$) as well as for asymmetric cases, but restricting attention to (hyper)cubic system shapes. As an explicit example for the usefulness of these considerations in Monte Carlo simulations where it may be a problem to (i) locate a phase transition and (ii) distinguish first-order from second-order transitions, we present numerical results for the two-dimensional nearest-neighbor Ising ferromagnet in a field, both below the critical temperature ${T}_{c}$ and at ${T}_{c}$. The numerical results are found to be in very good agreement with the phenomenological theory and it is shown that one may extract the magnitudes of jumps occurring at first-order phase transitions in a well-defined and accurate way.

DOI
10.1103/physrevb.30.1477

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Physical review. B, Condensed matter•Feb 1, 1991•316 citations
Using thermodynamic fluctuation theory, we study the finite-size rounding of anomalies occurring at first-order phase transitions of the corresponding infinite system. Explicit expressions for thermodynamic functions are derived both for "symmetric transitions" (such as the jump of the spontaneous magnetization in the Ising model from +Msp+{M}_{\mathrm{sp}}+Msp​ to \ensuremath−Msp\ensuremath{-}{M}_{\mathrm{sp}}\ensuremath−Msp​ as the field changes from 0+{0}^{+}0+ to 0\ensuremath−{0}^{\ensuremath{-}}0\ensuremath−) as well as for asymmetric cases, but restricting attention to (hyper)cubic system shapes. As an explicit example for the usefulness of these considerations in Monte Carlo simulations where it may be a problem to (i) locate a phase transition and (ii) distinguish first-order from second-order transitions, we present numerical results for the two-dimensional nearest-neighbor Ising ferromagnet in a field, both below the critical temperature Tc{T}_{c}Tc​ and at Tc{T}_{c}Tc​. The numerical results are found to be in very good agreement with the phenomenological theory and it is shown that one may extract the magnitudes of jumps occurring at first-order phase transitions in a well-defined and accurate way.
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.
First order transitions are rounded in finite cells or in infinite strips of finite width (cylinders). In this work we have considered the rounding of the jump of the magnetization of an Ising system at a fixed temperature T below Tc when the external field h goes from a small positive, to a small negative value. This rounding is well-known in a finite cell (block geometry) but the situation is more complicated, and more interesting, in a cylinder. Indeed it is known from the work of Privman and Fisher that these rounding effects are related to the behaviour of the correlation length along the axis of the cylinder which increases exponentially with the cross-sectional area A of the strip. It then follows that if one studies, for a given magnetization M, the external field h, this field approaches exponentially with A its bulk limit, but the coefficient inside the exponential is different if M is smaller, larger than or equal to the bulk spontaneous magnetization M0. We present numerical data for the two-dimensional case (or equivalently for the one-dimensional quantum model in a transverse field) which are in satisfactory agreement with the theory.
Finite-size effects on the first-order phase transition in a three-dimensional classical Heisenberg model with nearest-neighbor ferromagnetic interactions and periodic boundary conditions are studied theoretically and numerically in this paper. We find that the current Fisher-Privman theory fails at temperatures close to criticality and/or for small systems. We rederive explicit scaling functions for the longitudinal magnetization and susceptibility under another phenomenological assumption. The results are in very good agreement with Monte Carlo simulations.
The finite-size rounding of a first-order transition is studied in systems representable as n-vector ferromagnets, so that O(n) symmetry (n\ensuremath{\ge}2) is broken at the bulk transition point. Both ``block,'' V=Ld{L}^{d}Ld, and ``cylinder,'' ${L}^{d\mathrm{\ensuremath{-}}1}$\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}, geometries are considered for general dimensionality d. Explicit expressions are obtained for the scaling functions describing the rounded transitions and the crossover in shape. Spin-wave effects are shown to be of relative order 1/${L}^{d\mathrm{\ensuremath{-}}2}$, and are calculated in detail in the block case. For n=3 (and d=3) this provides an extension of N\'eel's phenomenological theory of superparamagnetism. The analysis for cylinders involves the formulation of a ``degeneracy kernel'' to describe the asymptotic rounding of first-order transitions and establishes a general relation between the helicity modulus (or ``spin-wave stiffness'' or ``superfluid density'') and the transfer operator spectrum. The relationship to finite-size scaling in the critical region is examined with emphasis on the extra scaling combination, ${\mathrm{Vt}}^{2\mathrm{\ensuremath{-}}\ensuremath{\alpha}}$, that is needed for d>d>{d}_{>}d>​=4. All the results found can be checked in the limit n\ensuremath{\rightarrow}\ensuremath{\infty} against exact results for spherical models (described elsewhere).
The magnetic critical exponent of the two-dimensional axial next-nearest-neighbor Ising model is estimated by means of the finite-size scaling of the order parameter. For a finite system without ordering field, this magnitude is computed from the two leading eigenvectors of the transfer matrix. The ferromagnetic-paramagnetic transition temperatures estimated with this method exhibit a small lattice-size dependence. Moreover, they are in good agreement with those obtained using phenomenological renormalization, and also with analytical approximations. The correlation-length critical exponent is calculated by means of phenomenological renormalization. The results found with these numerical techniques indicate that the ferromagnetic-paramagnetic transition of the model seems to be continuous, in contradiction with a previous result obtained using the persistence-length criterion, which indicated that the transition is always first order. We found that the phase transition is first order only when the ratio of the next-nearest-neighbor and nearest-neighbor interactions equals one-half.
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.