Universal ratios of critical amplitudes in the Potts model universality class

Abstract

Monte Carlo (MC) simulations and series expansions (SE) data for the energy, specific heat, magnetization, and susceptibility of the three-state and four-state Potts model and Baxter-Wu model on the square lattice are analyzed in the vicinity of the critical point in order to estimate universal combinations of critical amplitudes. 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 ≤ 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. It is a common belief that the four-state Potts and the Baxter-Wu model belong to the same universality class. At the same time, the critical behavior of the four-state Potts model is modified by logarithmic corrections while that of the Baxter-Wu model is not. Numerical analysis shows that critical amplitude ratios are very close for both models and, therefore, gives support to the hypothesis that the critical behavior of both systems is described by the same renormalization group fixed point.

Critical amplitude ratios of the Baxter–Wu model

“…[7,9] and the discussion in Ref. [26]), which produces the correction-to-scaling exponent 2/3 in the specific heat (16) and magnetization (21).…”

“…Delfino and Grinza [30] obtained practically the same value R χ = 4.02 using the same approximation for the Ashkin-Teller model with parameters which correspond to the 4-state Potts model universality class. A recent analysis of the MC and SE data for the 4-state Potts model gives an amplitude ratio in the range of about 6.5 (4) [20,26,31]. At the same time, the values for the universal combination of amplitudes…”

“…The ratio of the susceptibility amplitudes R χ =Γ 4p (+)/Γ 4p (−)=Γ bw (+)/Γ bw (−), on the other hand, is not known exactly and different approximations (analytical, series expansions, Monte Carlo data) from different groups are not coherent and differ substantially (for a recent discussion, see the papers [20,26]). It is well known that logarithmic corrections, if they exist, complicate the critical behavior and render the analysis of the critical behavior extremely difficult if possible at all, and the determination of critical exponents from Monte Carlo (MC) and series expansions (SE) became non-trivial.…”

“…[28] and Refs. [20,26,31] are 0.00508 and 0.0052 (2), respectively (A 0 and B 0 are the specific-heat and magnetization amplitudes). This is in perfect agreement with our estimate 0.00517 (7) we present here for the Baxter-Wu model.…”

The two-dimensional 4-state Potts model in a magnetic field

“…[1] for details and Refs. [12,14,13,15,16,17,18] for different levels of approximation). Keeping only the leading logarithmic behavior for the present context, expression (34) simply yields What appears extremely useful in these expressions is that when defining appropriate effective ratios, the dependence on the quantity ζ cancels, due to the scaling relations among the critical exponents.…”