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Physical Review A

Monte Carlo evaluations of interfacial tension and universal amplitude ratios of the three-dimensional Ising model

K. K. Mon, David Jasnow

Published Jun 1, 1985• 30 citations
Publisher

Abstract

The interfacial tension of the simple-cubic Ising model near the transition temperature has been calculated with use of an extension of a recently proposed novel Monte Carlo method. Finite-size scaling theory was used to analyze the results and to obtain the surface tension amplitude. Universal ratios involving the surface tension amplitude are evaluated. Previous ambiguities in the theoretical value of the ratio involving the surface tension and specific-heat amplitudes have been clarified.

DOI
10.1103/physreva.31.4008

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The interfacial tension of the simple-cubic Ising model near the transition temperature has been calculated with use of an extension of a recently proposed novel Monte Carlo method. Finite-size scaling theory was used to analyze the results and to obtain the surface tension amplitude. Universal ratios involving the surface tension amplitude are evaluated. Previous ambiguities in the theoretical value of the ratio involving the surface tension and specific-heat amplitudes have been clarified.
Critical interfacial tension amplitudes can be obtained without the usual difficult task of calculating the related excess free energy. This permits for the first time practical Monte Carlo calculations with very large lattices. By use of up to 128 \ifmmode\times\else\texttimes\fi{} 128 \ifmmode\times\else\texttimes\fi{} 128 for the simple-cubic Ising model, a new amplitude is obtained and is larger than previous estimates with smaller lattices. This resolves the long-standing large disagreements between experiments and theory on the universality of the related amplitude ratios.
The study of the universal finite-size scaling amplitudes of interfacial free energies in Monte Carlo simulations provides an accurate method of determining the critical temperature and the universality class of phase transitions. These universal amplitudes can replace the critical exponents to identify universality classes. The authors use Bennett's optimisation scheme (1976) for the calculation of the interfacial free energies in Monte Carlo simulations. Tests for the Ising and three-state Potts model confirm the effectiveness of this method.
Estimates for the critical interfacial tension amplitudes of the three-state Potts model in two dimensions via direct Monte Carlo sampling are presented for the first time. Both the square and triangular lattices are analyzed with use of finite-size scaling considerations. Also presented for the first time are estimates for the correlation-length amplitudes as derived from high-temperature series. These numerical results allow investigation of the ratio U=\ensuremath{\tau}${\ensuremath{\xi}}^{d\mathrm{\ensuremath{-}}1}$/kB{k}_{B}kB​T as T\ensuremath{\rightarrow}Tc{T}_{c}Tc​. The results confirm the universality of U and are in agreement with exact results for hard hexagons. The finite-size scaling amplitude for the interfacial free energy at criticality is also shown to be universal within the available precision.
We use a high-precision Monte Carlo simulation to determine the universal specific-heat amplitude ratio A + / A − in the three-dimensional Ising model via the impact angle ϕ of complex temperature zeros. We also measure the correlation-length critical exponent ν from finite-size scaling and the specific-heat exponent α through hyperscaling. Extrapolations to the thermodynamic limit yield ϕ = 59.2(1.0)°, A + / A − = 0.56(3), ν = 0.630 48(32) and α = 0.1086(10). These results are compatible with some previous estimates from a variety of sources and rule out recently conjectured exact values.
We review Monte Carlo simulations of the Ising model and similar models in three dimensions that were performed in the last decade. Only recently, Monte Carlo simulations provide more accurate results for critical exponents than field theoretic methods, such as the ∊-expansion. These results were obtained with finite size scaling and "improved actions". In addition, we summarize Monte Carlo results for universal amplitude ratios, the interface tension, and the dimensional crossover from three to two dimensions.
We present a high precision Monte Carlo study of various universal amplitude ratios of the three-dimensional Ising spin model. Using state of the art simulation techniques we study the model close to criticality in both phases. Great care is taken to control systematic errors due to finite size effects and correction to scaling terms. We obtain , and . Our results are compatible with those obtained by field theoretic methods applied to the theory and high- and low-temperature series expansions of the Ising model. The mismatch with a previous Monte Carlo study by Ruge et al remains to be understood.