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Universal amplitude ratios in the two-dimensional $q$-state Potts model and percolation from quantum field theory

Delfino, G, Cardy, John L

Published Jan 1, 1998• 47 citations
OpenAlex

Abstract

We consider the scaling limit of the two-dimensional $q$-state Potts model for $q\\leq 4$. We use the exact scattering theory proposed by Chim and Zamolodchikov to determine the one and two-kink form factors of the energy, order and disorder operators in the model. Correlation functions and universal combinations of critical amplitudes are then computed within the two-kink approximation in the form factor approach. Very good agreement is found whenever comparison with exact results is possible. We finally consider the limit $q\\to 1$ which is related to the isotropic percolation problem. Although this case presents a serious technical difficulty, we predict a value close to 74 for the ratio of the mean cluster size amplitudes above and below the percolation threshold. Previous estimates for this quantity range from 14 to 220.

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High-precision determination of universal amplitude ratios for the q=3 Potts model in two dimensions

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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.

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Numerical revision of the universal amplitude ratios for the two-dimensional 4-state Potts model

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Monte Carlo (MC) simulations and series expansion (SE) data for the energy, specific heat, magnetization and susceptibility of the ferromagnetic 4-state Potts model on the square lattice are analyzed in a vicinity of the critical point in order to estimate universal combinations of critical amplitudes. The quality of the fits is improved using predictions of the renormalization group (RG) approach and of conformal invariance, and restricting the data within an appropriate temperature window. The RG predictions on the cancelation of the logarithmic corrections in the universal amplitude ratios are tested. A direct calculation of the effective ratio of the energy amplitudes using duality relations explicitly demonstrates this cancelation of logarithms, thus supporting the predictions of RG. We emphasize the role of corrections and of background terms on the determination of the amplitudes. The ratios of the critical amplitudes of the susceptibilities obtained in our analysis differ significantly from those predicted theoretically and supported by earlier SE and MC analysis. This disagreement might signal that the two-kink approximation used in the analytical estimates is not sufficient to describe with fair accuracy the amplitudes of the 4-state model.

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An exact universal amplitude ratio for percolation

Katherine Seaton

The universal amplitude ratio $\\tilde{R}_{\\xi}$ for percolation in two dimensions is determined exactly using results for the dilute A model in regime 1, by way of a relationship with the q-state Potts model for q<4. The numerical result obtained by Delfino and Cardy by quantum field theory techniques agrees well with the algebraic expression determined here.

•Jan 1, 2001•2 citations

Critical amplitudes for correlated percolation

Bosiljka Tadić, R. Pirc

We consider the q-state Potts model with correlated random interactions such that their correlations decay at large distances as \ensuremath{\sim}${R}^{\mathrm{\ensuremath{-}}(d\mathrm{\ensuremath{-}}a)}$. Applying the renormalization group and a double (\ensuremath{\epsilon},${\ensuremath{\epsilon}}_{a}$) expansion, we derive the scaling form of the equation of state to linear order in \ensuremath{\epsilon}=6-d and ${\ensuremath{\epsilon}}_{a}$=a-2. In the limit q\ensuremath{\rightarrow}1, which describes correlated percolation, the critical amplitudes ${A}^{\ifmmode\pm\else\textpm\fi{}}$ associated with the mean number of clusters F\ensuremath{\sim}${A}^{\ifmmode\pm\else\textpm\fi{}}$\ensuremath{\Vert}t${\ensuremath{\Vert}}^{2\mathrm{\ensuremath{-}}\ensuremath{\alpha}}$ are shown to diverge when \ensuremath{\alpha} approaches a negative integer as a function of ${\ensuremath{\epsilon}}_{a}$.

Physical review. B, Condensed matter•Jul 1, 1987•0 citations
Universal amplitude ratios in the two-dimensional qqq-state Potts model and percolation from quantum field theory
We consider the scaling limit of the two-dimensional qqq-state Potts model for qleq4q\\leq 4qleq4. We use the exact scattering theory proposed by Chim and Zamolodchikov to determine the one and two-kink form factors of the energy, order and disorder operators in the model. Correlation functions and universal combinations of critical amplitudes are then computed within the two-kink approximation in the form factor approach. Very good agreement is found whenever comparison with exact results is possible. We finally consider the limit qto1q\\to 1qto1 which is related to the isotropic percolation problem. Although this case presents a serious technical difficulty, we predict a value close to 74 for the ratio of the mean cluster size amplitudes above and below the percolation threshold. Previous estimates for this quantity range from 14 to 220.
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.
Abstract We use scale invariant scattering theory to exactly determine the renormalization group fixed points of a q-state Potts model coupled to an r-state Potts model in two dimensions. For integer values of q and r the fixed point equations are very constraining and show in particular that scale invariance in coupled Potts ferromagnets is limited to the Ashkin–Teller case ( q = r = 2 ). Since our results extend to continuous values of the number of states, we can access the limit r → 1 corresponding to correlated percolation, and show that the critical properties of Potts spin clusters cannot in general be obtained from those of Fortuin–Kasteleyn clusters by analytical continuation.
Monte Carlo (MC) simulations and series expansion (SE) data for the energy, specific heat, magnetization and susceptibility of the ferromagnetic 4-state Potts model on the square lattice are analyzed in a vicinity of the critical point in order to estimate universal combinations of critical amplitudes. The quality of the fits is improved using predictions of the renormalization group (RG) approach and of conformal invariance, and restricting the data within an appropriate temperature window. The RG predictions on the cancelation of the logarithmic corrections in the universal amplitude ratios are tested. A direct calculation of the effective ratio of the energy amplitudes using duality relations explicitly demonstrates this cancelation of logarithms, thus supporting the predictions of RG. We emphasize the role of corrections and of background terms on the determination of the amplitudes. The ratios of the critical amplitudes of the susceptibilities obtained in our analysis differ significantly from those predicted theoretically and supported by earlier SE and MC analysis. This disagreement might signal that the two-kink approximation used in the analytical estimates is not sufficient to describe with fair accuracy the amplitudes of the 4-state model.
The universal amplitude ratio tildeRxi\\tilde{R}_{\\xi}tildeRxi​ for percolation in two dimensions is determined exactly using results for the dilute A model in regime 1, by way of a relationship with the q-state Potts model for q<4. The numerical result obtained by Delfino and Cardy by quantum field theory techniques agrees well with the algebraic expression determined here.
We consider the q-state Potts model with correlated random interactions such that their correlations decay at large distances as \ensuremath{\sim}${R}^{\mathrm{\ensuremath{-}}(d\mathrm{\ensuremath{-}}a)}$. Applying the renormalization group and a double (\ensuremath{\epsilon},${\ensuremath{\epsilon}}_{a}$) expansion, we derive the scaling form of the equation of state to linear order in \ensuremath{\epsilon}=6-d and ${\ensuremath{\epsilon}}_{a}$=a-2. In the limit q\ensuremath{\rightarrow}1, which describes correlated percolation, the critical amplitudes ${A}^{\ifmmode\pm\else\textpm\fi{}}$ associated with the mean number of clusters F\ensuremath{\sim}${A}^{\ifmmode\pm\else\textpm\fi{}}$\ensuremath{\Vert}t${\ensuremath{\Vert}}^{2\mathrm{\ensuremath{-}}\ensuremath{\alpha}}$ are shown to diverge when \ensuremath{\alpha} approaches a negative integer as a function of ${\ensuremath{\epsilon}}_{a}$.