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 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.
The explicit dependence of correlation length as function of temperature and external field is found for the Ising model on a simple cubic lattice near the phase transition point. , where and h c are the renormalized values of the external field h and reduced temperature τ. The critical amplitude ξ0 is proportional to the lattice constant through the coefficient, which is determined by the parameter of the interaction potential. The size of the critical region is established by the temperature and field. It is shown that the essential increasing of the system correlation length takes place at () with h→0 and at h′<h* (, h′=βh) with τ→0.
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⊥(k)
≃ak−λ⊥ and
G∣∣(k
)≃bk−λ∣∣, respectively. Here
a and
b are the amplitudes,
k=|k| is the magnitude of the wave vector k. The exponents
λ⊥,
λ∣∣ and the ratio
bM2/a2, 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
bM2/a2=
(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⊥(k) and
G∣∣(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 ⊥ (k) ≃ ak -λ ⊥ and G ∥ (k) ≃ bk -λ ∥ 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 λ ⊥ and λ ∥ are non-trivial (3/2 < λ ⊥ < 2 and 0 < λ ∥ < 1 in three dimensions) and the ratio bM 2 /a 2 (where M is a spontaneous magnetization) is universal.The trivial standardtheoretical values are λ ⊥ = 2 and λ ∥ = 1.Our earlier MC analysis gives λ ⊥ = 1.955 ± 0.020 and λ ∥ about 0.9 for the O(4) model.A recent MC estimation of λ ∥ , assuming corrections to scaling of the standard theory, yields λ ∥ = 0.69±0.10for the O(2) model.Currently, we have performed a similar MC estimation for the O(10) model, yielding λ ⊥ = 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 ∆λ ⊥ = 0.0121(52).It is consistent with the idea that 2 -λ ⊥ decreases for large n and tends to zero at n → ∞.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.