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arXiv:0707.2430 · Europhysics Letters (EPL)

A study of logarithmic corrections and universal amplitude ratios in the two-dimensional 4-state Potts model

Щур Лев Николаевич, Bertrand Berche, P. Butera

Published Jan 3, 2008• 10 citations
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Abstract

Monte Carlo (MC) and series expansion (SE) data for the energy, specific heat, magnetization and susceptibility of the two-dimensional 4-state Potts model in the vicinity of the critical point are analysed. The role of logarithmic corrections is discussed and an approach is proposed in order to account numerically for these corrections in the determination of critical amplitudes. Accurate estimates of universal amplitude ratios A + / A - , Γ + /Γ - , Γ T /Γ - and R C ± are given, which arouse new questions with respect to previous works.

DOI
10.1209/0295-5075/81/30008

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Monte Carlo (MC) and series expansion (SE) data for the energy, specific heat, magnetization and susceptibility of the two-dimensional 4-state Potts model in the vicinity of the critical point are analysed. The role of logarithmic corrections is discussed and an approach is proposed in order to account numerically for these corrections in the determination of critical amplitudes. Accurate estimates of universal amplitude ratios A + / A - , Γ + /Γ - , Γ T /Γ - and R C ± are given, which arouse new questions with respect to previous works.
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.
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 present a solution of the nonlinear renormalization group equations leading to the dominant and subdominant singular behaviors of physical quantities (free energy density, correlation length, internal energy, specific heat, magnetization, susceptibility and magnetocaloric coefficient) at the critical temperature in a non-vanishing magnetic field. The solutions (i) lead to exact cancellation of logarithmic corrections in universal amplitude ratios and (ii) prove recently proposed relations among logarithmic exponents.
We investigate the three-dimensional O(2) model near the critical point by Monte Carlo simulations and calculate the major universal amplitude ratios of the model. The ratio U_0=A+/A- is determined directly from the specific heat data at zero magnetic field. The data do not, however, allow to extract an accurate estimate for alpha. Instead, we establish a strong correlation of U_0 with the value of alpha used in the fit. This numerical alpha-dependence is given by A+/A- = 1 -4.20(5) alpha + O(alpha^2). For the special alpha-values used in other calculations we find full agreement with the corresponding ratio values, e. g. that of the shuttle experiment with liquid helium. On the critical isochore we obtain the ratio xi+/xi-_T=0.293(9), and on the critical line the ratio xi_T^c/xi_L^c=1.957(10) for the amplitudes of the transverse and longitudinal correlation lengths. These two ratios are independent of the used alpha or nu-values.
We present a different way of probing the universality class of the site-diluted two-dimensional Ising model. We analyze Monte Carlo data for the magnetic susceptibility, introducing a fitting procedure in the critical region applicable even for a single sample with quenched disorder. This gives us the possibility to fit simultaneously the critical exponent, the critical amplitude, and the sample-dependent pseudocritical temperature. The critical amplitude ratio of the magnetic susceptibility is seen to be independent of the concentration q of the empty sites for all investigated values of q < or =0.25. At the same time the average effective exponent gamma(eff) is found to vary with the concentration q, which may be argued to be due to logarithmic corrections to the power law of the pure system. These corrections are canceled in the susceptibility amplitude ratio as predicted by theory. The central charge of the corresponding field theory was computed and compared well with the theoretical predictions.