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Progress of Theoretical Physics

Critical Amplitude Ratio 0+/ 0T in 1/n and Expansions

Yutaka Okabe, K. Ideura

Published Dec 1, 1981• 15 citations
Publisher

Abstract

The universal critical amplitude ratio ξ0+/ξ0T is investigated in both 1/n and ε expansions. The term up to O(1/n) is considered in 1/n expansion, and the value of ξ0+/ξ0T is estimated numerically for d = 3. The O(ε2) calculation is performed in ε expansion, and it is shown that the result is consistent with that of 1/n expansion.
DOI
10.1143/ptp.66.1959

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We investigate the three-dimensional 0 (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 us to extract an accurate estimate for a. Instead, we establish a strong correlation of U0 with the value of a used in the fit. This numerical a-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 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 alpha- or v-values used here.
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.
The author introduces a new method of calculating critical amplitude ratios using series, which is both simple and powerful. This method, which gives estimates for the amplitude ratios that are neither biased by the values of the critical points nor by the critical exponents, is applied to several models. It is shown that this method produces results where no reliable estimates from series expansion exist. In particular one finds 0.025+or-0.001 for AT'/B 2 for the 3D Ising model and 220+or-10 for C + /C - for the two-dimensional percolation model in agreement with, and with more accuracy than, values obtained by other methods.
We estimate, at d=3d=3d=3, from ${\ensuremath{\varphi}}^{4}$ field theory the universal ratios of confluent critical correction amplitudes relative to the correlation length, the susceptibility, and the specific heat above Tc{T}_{c}Tc​. Our central values are comparable to these available from analysis of high-temperature series and the $\ensuremath{\epsilon}$ expansion. Furthermore, experiments in progress on a binary mixture are in good agreement with one of our results, the ratio of the confluent corrections for the specific heat and susceptibility which have not been previously estimated. We also present, in detail, a derivation of the universality of these quantities in the framework of field theory.