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Register a new userPower-law singularities and critical exponents in n-vector models
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J. Kaupuzs
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Power-law singularities and critical exponents in n-vector models are considered from different theoretical points of view. It includes a theoretical approach called the GFD (grouping of Feynman diagrams) theory, as well as the perturbative renormalization group (RG) treatment. A non-perturbative proof concerning corrections to scaling in the two-point correlation function of the phi^4 model is provided, showing that predictions of the GFD theory rather than those of the perturbative RG theory can be correct. Critical exponents determined from highly accurate experimental data very close to the lambda-transition point in liquid helium, as well as the Goldstone mode singularities in n-vector spin models, evaluated from Monte Carlo simulation results, are discussed with an aim to test the theoretical predictions. Our analysis shows that in both cases the data can be well interpreted within the GFD theory.
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Correlation functions in the O(n) models below the critical temperature are considered. Based on Monte Carlo (MC) data, we confirm the fact stated earlier by Engels and Vogt, that the transverse two-plane correlation function of the O(4) model for lattice sizes about L=120 and small external fields h is very well described by a Gaussian approximation. However, we show that fits of not lower quality are provided by certain non-Gaussian approximation. We have also tested larger lattice sizes, up to L=512. The Fourier-transformed transverse and longitudinal two-point correlation functions have Goldstone mode singularities in the thermodynamic limit at k --> 0 and h=+0, i.e., G_perp(k) = a k^{-lambda_perp} and G_parallel(k) = b k^{-lambda_parallel}, respectively. Here a and b are the amplitudes, k is the magnitude of the wave vector. The exponents lambda_perp, lambda_parallel and the ratio b M^2/a^2, where M is the spontaneous magnetization, are universal according to the GFD (grouping of Feynman diagrams) approach. Here we find that the universality follows also from the standard (Gaussian) theory, yielding b M^2/a^2 = (n-1)/16. Our MC estimates of this ratio are 0.06 +/- 0.01 for n=2, 0.17 +/- 0.01 for n=4 and 0.498 +/- 0.010 for n=10. According to these and our earlier MC results, the asymptotic behavior and Goldstone mode singularities are not exactly described by the standard theory. This is expected from the GFD theory. We have found appropriate analytic approximations for G_perp(k) and G_parallel(k), well fitting the simulation data for small k. We have used them to test the Patashinski--Pokrovski relation and have found that it holds approximately.
Monte Carlo (MC) analysis of the Goldstone mode singularities for the transverse and the longitudinal correlation functions, behaving as G_{perp}(k) simeq ak^{-lambda_{perp}} and G_{parallel}(k) simeq bk^{-lambda_{parallel}} in the ordered phase at k -> 0, is performed in the three-dimensional O(n) models with n=2, 4, 10. Our aim is to test some challenging theoretical predictions, according to which the exponents lambda_{perp} and lambda_{parallel} are non-trivial (3/2<lambda_{perp}<2 and 0<lambda_{parallel}<1 in three dimensions) and the ratio bM^2/a^2 (where M is a spontaneous magnetization) is universal. The trivial standard-theoretical values are lambda_{perp}=2 and lambda_{parallel}=1. Our earlier MC analysis gives lambda_{perp}=1.955 pm 0.020 and lambda_{parallel} about 0.9 for the O(4) model. A recent MC estimation of lambda_{parallel}, assuming corrections to scaling of the standard theory, yields lambda_{parallel} = 0.69 pm 0.10 for the O(2) model. Currently, we have performed a similar MC estimation for the O(10) model, yielding lambda_{perp} = 1.9723(90). We have observed that the plot of the effective transverse exponent for the O(4) model is systematically shifted down with respect to the same plot for the O(10) model by Delta lambda_{perp} = 0.0121(52). It is consistent with the idea that 2-lambda_{perp} decreases for large $n$ and tends to zero at n -> infty. We have also verified and confirmed the expected universality of bM^2/a^2 for the O(4) model, where simulations at two different temperatures (couplings) have been performed.
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