Corrections to finite-size scaling in the 3D Ising model based on non-perturbative approaches and Monte Carlo simulations

Corrections to scaling in the 3D Ising model are studied based on non-perturbative analytical arguments and Monte Carlo (MC) simulation data for different lattice sizes L. Analytical arguments show the existence of corrections with the exponent (gamma-1)/nu (approximately 0.38), the leading correction-to-scaling exponent being omega =< (gamma-1)/nu. A numerical estimation of omega from the susceptibility data within 40 =< L =< 2048 yields omega=0.25(33). It is consistent with the statement omega =< (gamma-1)/nu, as well as with the value omega = 1/8 of the GFD theory. We reconsider the MC estimation of omega from smaller lattice sizes to show that it does not lead to conclusive results, since the obtained values of omega depend on the particular method chosen. In particular, estimates ranging from omega =1.274(72) to omega=0.18(37) are obtained by four different finite-size scaling methods, using MC data for thermodynamic average quantities, as well as for partition function zeros. We discuss the influence of omega on the estimation of exponents eta and nu.

Corrections to finite--size scaling in the phi^4 model on square lattices

Corrections to scaling in the two-dimensional scalar phi^4 model are studied based on non-perturbative analytical arguments and Monte Carlo (MC) simulation data for different lattice sizes L (from 4 to 1536) and different values of the phi^4 coupling constant lambda, i.~e., lambda = 0.1, 1, 10. According to our analysis, amplitudes of the nontrivial correction terms with the correction-to-scaling exponents omega_l < 1 become small when approaching the Ising limit (lambda --> infinity), but such corrections generally exist in the 2D phi^4 model. Analytical arguments show the existence of corrections with the exponent 3/4. The numerical analysis suggests that there exist also corrections with the exponent 1/2 and, very likely, also corrections with the exponent about 1/4, which are detectable at lambda = 0.1. The numerical tests clearly show that the structure of corrections to scaling in the 2D phi^4 model differs from the usually expected one in the 2D Ising model.

Correlation functions, universal ratios and Goldstone mode singularities in n-vector models

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.

Goldstone mode singularities in O(n) models

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.

Power-law singularities and critical exponents in n-vector models

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.

Studies of critical phenomena and phase transitions in large lattices with Monte-Carlo based non-perturbative approaches

Critical phenomena and Goldstone mode effects in spin models with O(n) rotational symmetry are considered. Starting with the Goldstone mode singularities in the XY and O(4) models, we briefly review different theoretical concepts as well as state-of-the art Monte Carlo simulation results. They support recent results of the GFD (grouping of Feynman diagrams) theory, stating that these singularities are described by certain nontrivial exponents, which differ from those predicted earlier by perturbative treatments. Furthermore, we present the recent Monte Carlo simulation results of the three-dimensional Ising model for very large lattices with linear sizes up to L=1536. These results are obtained, using a parallel OpenMP implementation of the Wolff single cluster algorithm. The finite-size scaling analysis of the critical exponent eta, assuming the usually accepted correction-to-scaling exponent omega=0.8, shows that eta is likely to be somewhat larger than the value 0.0335 +/- 0.0025 of the perturbative renormalization group (RG) theory. Moreover, we have found that the actual data can be well described by different critical exponents: eta=omega=1/8 and nu=2/3, found within the GFD theory.

Some aspects of the nonperturbative renormalization of the phi^4 model

A nonperturbative renormalization of the phi^4 model is considered. First we integrate out only a single pair of conjugated modes with wave vectors +/- q. Then we are looking for the RG equation which would describe the transformation of the Hamiltonian under the integration over a shell Lambda - d Lambda < k < Lambda, where d Lambda -> 0. We show that the known Wegner--Houghton equation is consistent with the assumption of a simple superposition of the integration results for +/- q. The renormalized action can be expanded in powers of the phi^4 coupling constant u in the high temperature phase at u -> 0. We compare the expansion coefficients with those exactly calculated by the diagrammatic perturbative method, and find some inconsistency. It causes a question in which sense the Wegner-Houghton equation is really exact.