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Physical Review Letters

New Monte Carlo Estimates of critical interfacial amplitudes and the universality of amplitude ratios

K. K. Mon

Published Jun 27, 1988• 51 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.60.2749

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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.
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.
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.
We present estimates for the 3D Ising model on the cubic lattice, both regarding interface and bulk properties. We have results for the interface tension, in particular the amplitude sigma0 in the critical law sigma = sigma0*t**mu, and for the universal combination R_ = sigma*xi**2. Concerning the bulk properties, we estimate the specific heat universal amplitude ratio A+/A_, together with the exponent alpha, the nonsingular background of energy and specific heat at criticality, together with the exponent nu. There are also results for the universal combination fs*xi**3, where fs is the singular part of the free energy.
We compute a number of universal amplitude ratios in the three-dimensional Ising universality class. To this end, we perform Monte Carlo simulations of the improved Blume-Capel model on the simple cubic lattice. For example, we obtain ${A}_{+}/{A}_{\ensuremath{-}}=0.536(2)$ and ${C}_{+}/{C}_{\ensuremath{-}}=4.713(7)$, where ${A}_{\ifmmode\pm\else\textpm\fi{}}$ and ${C}_{\ifmmode\pm\else\textpm\fi{}}$ are the amplitudes of the specific heat and the magnetic susceptibility, respectively. The subscripts +++ and $\ensuremath{-}$ indicate the high- and the low-temperature phase, respectively. We compare our results with those obtained from previous Monte Carlo simulations, high- and low-temperature series expansions, field theoretic methods, and experiments.