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Journal of Physics A Mathematical and General

Universal finite-size scaling amplitudes on a torus for the triangular Ising lattice gas

Hyunggyu Park

Published May 21, 1990• 2 citations

Abstract

Recently the author suggested studying the scaling properties of complex two-dimensional phase transitions by calculating the universal amplitudes of interfacial free energies on a torus geometry employing Monte Carlo simulations. He tests this method at the 3*3 phase transition in the Ising lattice gas on a triangular lattice belonging to the chiral three-state Potts universality class. The interfacial free energies for this lattice gas model can be written in terms of free energy differences between systems with different sizes. The algorithm to calculate these free energy differences is presented. Using lattice sizes up to 18*18 and modest computing effort, he finds that the universal amplitude A=1.220.02, in accordance with the exact analytical value for the non-chiral three-state Potts model, A=1.203291 . . ., which he determined earlier.

DOI: 10.1088/0305-4470/23/10/019

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Recently the author suggested studying the scaling properties of complex two-dimensional phase transitions by calculating the universal amplitudes of interfacial free energies on a torus geometry employing Monte Carlo simulations. He tests this method at the 33 phase transition in the Ising lattice gas on a triangular lattice belonging to the chiral three-state Potts universality class. The interfacial free energies for this lattice gas model can be written in terms of free energy differences between systems with different sizes. The algorithm to calculate these free energy differences is presented. Using lattice sizes up to 1818 and modest computing effort, he finds that the universal amplitude A=1.220.02, in accordance with the exact analytical value for the non-chiral three-state Potts model, A=1.203291 . . ., which he determined earlier.

We derive the exact values of the universal amplitudes associated with the finite-size-scaling behavior of the free energy and interfacial free energies of the Potts model on a torus, by means of the extended-scaling method. These amplitudes vary continuously with the value of the aspect ratio of the torus. Our results make it possible to apply the type of finite-size-scaling techniques that are standard in the context of transfer-matrix calculations to Monte Carlo simulations, and thus enhance the effectiveness of these simulations in establishing the universality class of a phase transition.

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}${\ensuremath{\xi}}^{d\mathrm{\ensuremath{-}}1}$/

{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.

. 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...

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.