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International Journal of Modern Physics C

EQUAL WEIGHT VERSUS EQUAL HEIGHT —A CONTROVERSY IN FINITE-SIZE SCALING THEORY

Stefan Kappler, Christian Borgs

Published Oct 1, 1992• 2 citations
Publisher

Abstract

Describing asymmetric first-order phase transitions by an order parameter distribution, there is a controversy in the literature concerning the relative normalization of the two occurring peaks. There are two rules for normalizing the distribution, called the “equal weight”-rule and the “equal height”-rule, which lead to different predictions of the finite-size scaling. We tested these predictions for an asymmetric model with two co-existing phases by means of a Monte Carlo simulation. We find overwhelming numerical evidence in favour of the “equal weight”-rule, showing at the same time that the shift of the susceptibility maximum with respect to the infinite volume transition point ht is proportional to L−2d, and not to L−d. In addition we tested a new method to determine the transition point ht which was recently proposed by Borgs and Janke.

DOI
10.1142/s0129183192000725

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Describing asymmetric first-order phase transitions by an order parameter distribution, there is a controversy in the literature concerning the relative normalization of the two occurring peaks. There are two rules for normalizing the distribution, called the “equal weight”-rule and the “equal height”-rule, which lead to different predictions of the finite-size scaling. We tested these predictions for an asymmetric model with two co-existing phases by means of a Monte Carlo simulation. We find overwhelming numerical evidence in favour of the “equal weight”-rule, showing at the same time that the shift of the susceptibility maximum with respect to the infinite volume transition point ht is proportional to L−2d, and not to L−d. In addition we tested a new method to determine the transition point ht which was recently proposed by Borgs and Janke.
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We present a rigorous description of finite-size effects for a large class of models with an asymmetric first-order transition, assuming that all phases contributing to the transition have a finite correlation length. If the model describes the coexistence of two phases, it is shown that, at sufficiently low temperatures, the shift of the transition point due to finite-size effects in a volume Ld{\mathit{L}}^{\mathit{d}}Ld with periodic boundary conditions is O(${\mathit{L}}^{\mathrm{\ensuremath{-}}2\mathit{d}}$), in contrast to certain claims in the literature. We also discuss different ways to determine the transition point from finite-size data, which involve only exponentially small systematic errors in L.
We consider a temperature-driven first-order phase transition describing the coexistence of q ordered low-temperature phases and one disordered high-temperature phase at the infinite-volume transition temperature T0{\mathit{T}}_{0}T0​. Analyzing the exponential corrections to the periodic partition function in a box of volume V, Zper{\mathit{Z}}_{\mathrm{per}}Zper​=tsumm{\mathit{tsum}}_{\mathit{m}}tsumm​ exp(-\ensuremath{\beta}fm{\mathit{f}}_{\mathit{m}}fm​V), where fm{\mathit{f}}_{\mathit{m}}fm​ is the (metastable) free energy of the phase m, we propose several definitions of a finite-volume transition temperature T0{\mathit{T}}_{0}T0​(V) which involve only exponential corrections with respect to T0{\mathit{T}}_{0}T0​. We test our propositions in the d=2 Potts model for q=5, 8, and 10 by means of Monte Carlo simulations, using the single-cluster update procedure.
In this paper, I give an overview over a recently developed rigorous theory of finite-size scaling near first order transitions. Leaving out details of the mathematical proofs, the main emphasis is put on the underlying physical ideas and the discussion of the validity of the results for regions which are, in the sense of mathematical rigour, not covered by the original papers. I present both the finite-size scaling for cubic systems and for long cylinders, discussing also a recent controversy which stems from our work on the finite-size scaling of the mass gap in long cylinders.
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 $\ensuremath{-}{M}_{\mathrm{sp}}$ as the field changes from 0+{0}^{+}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 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.
We consider a particular case of the two dimensional Blume-Emery-Griffiths model to study the finite-size scaling for a field driven first-order phase transition with two coexisting phases not related by a symmetry. For low temperatures we verify the asymptotic (large volume) predictions of the rigorous theory of Borgs and Kotecky. Near the critical temperature we show that all data fit onto a unique curve, even when the correlation length ξ becomes comparable to or larger than the size of the system, provided the linear dimension L of the system is rescaled by ξ.