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Physical review. B, Condensed matter

Interfacial tension and correlation lengths for the three-state Potts model in two dimensions

David Jasnow, K. K. Mon, Martin V. Ferer

Published Mar 1, 1986• 9 citations
Publisher

Abstract

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}${\ensuremath{\xi}}^{d\mathrm{\ensuremath{-}}1}$/${k}_{B}$T as T\ensuremath{\rightarrow}${T}_{c}$. 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.

DOI
10.1103/physrevb.33.3349

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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}\ensuremathξd\ensuremath−1{\ensuremath{\xi}}^{d\mathrm{\ensuremath{-}}1}\ensuremathξd\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.
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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.
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.
. Finite-size corrections to scaling of critical correlation lengths and free energies of Ising and three-state Potts ferromagnets are analysed by numerical methods, on strips of width N sites of square, triangular and honeycomb lattices. Strong evidence is given that the amplitudes of the `analytical' correction terms, N -2 , are identically zero for triangular and honeycomb Ising systems. For Potts spins, our results are broadly consistent with this lattice-dependent pattern of cancellations, though for correlation lengths non-vanishing (albeit rather small) amplitudes cannot be entirely ruled out. 1. Introduction The systematic study of subdominant terms in scaling provides researchers with guidelines on how to extrapolate, e.g. finite-size results to the thermodynamic limit [1--5]. Also, it may in itself bring out connections to underlying physical properties. These latter may be universal, such as the relationship between critical free-energy finite-width correction and conforma...