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arXiv:hep-lat/9510050 · Journal of Statistical Physics

Monte Carlo simulations of conformal theory predictions for the three-state Potts model

G. T. Barkema, John McCabe

Published Sep 1, 1996• 8 citations
arXivHTMLPDF
DOI
10.1007/bf02174128

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We study the three-point correlation function of the backbone in the two-dimensional Q-state Potts model using the Fortuin-Kasteleyn (FK) representation. The backbone is defined as the biconnected skeleton of an FK cluster after removing all dangling ends and bridges. To circumvent the severe critical slowing down in direct Potts simulations for large Q, we employ large-scale Monte Carlo simulations of the O(n) loop model on the hexagonal lattice, which is regarded to correspond to the Potts model with Q=n^{2}. Using a highly efficient cluster algorithm, we compute the universal three-point amplitude ratios for the backbone (R_{BB}) and FK clusters (R_{FK}). Our computed R_{FK} exhibits excellent agreement with exact conformal field theory predictions, validating the reliability of our numerical approach. In the critical regime, we find that R_{BB} is systematically larger than R_{FK}. Conversely, along the tricritical branch, R_{BB} and R_{FK} coincide within numerical accuracy, strongly suggesting that R_{BB}=R_{FK} holds throughout this regime. This finding mirrors the known equality of the backbone and FK cluster fractal dimensions at tricriticality, jointly indicating that both structures share the same geometric universality.

arXiv:2511.10393•Physical review. E•Feb 12, 2026•0 citations
Monte Carlo simulations and series expansions data for the energy, specific heat, magnetization, and susceptibility of the three-state Potts model on the square lattice are analyzed in the vicinity of the critical point in order to estimate universal combinations of critical amplitudes. We estimate these amplitudes using the correction-to-scaling exponents predicted by conformal field theory. We also form effective ratios of the observables close to the critical point and analyze how they approach the universal critical-amplitude ratios. In particular, using the duality relation, we show analytically that for the Potts model with a number of states $q\ensuremath{\leqslant}4$, the effective ratio of the energy critical amplitudes always approaches unity linearly with respect to the reduced temperature. This fact leads to the prediction of relations among the amplitudes of correction-to-scaling terms of the specific heat in the low- and high-temperature phases. We present numerical and analytical support for the form of the first two correction-to-scaling terms. Our results for the amplitude ratios closely agree with the theoretical predictions and the earlier numerical estimates of the specific heat and the susceptibility amplitude ratios.
We study the three-point correlation function of the backbone in the two-dimensional Q-state Potts model using the Fortuin-Kasteleyn (FK) representation. The backbone is defined as the biconnected skeleton of an FK cluster after removing all dangling ends and bridges. To circumvent the severe critical slowing down in direct Potts simulations for large Q, we employ large-scale Monte Carlo simulations of the O(n) loop model on the hexagonal lattice, which is regarded to correspond to the Potts model with Q=n^{2}. Using a highly efficient cluster algorithm, we compute the universal three-point amplitude ratios for the backbone (R_{BB}) and FK clusters (R_{FK}). Our computed R_{FK} exhibits excellent agreement with exact conformal field theory predictions, validating the reliability of our numerical approach. In the critical regime, we find that R_{BB} is systematically larger than R_{FK}. Conversely, along the tricritical branch, R_{BB} and R_{FK} coincide within numerical accuracy, strongly suggesting that R_{BB}=R_{FK} holds throughout this regime. This finding mirrors the known equality of the backbone and FK cluster fractal dimensions at tricriticality, jointly indicating that both structures share the same geometric universality.