Correlation Functions, Universal Ratios and Goldstone Mode Singularities in $n$-Vector Models (Communications in Computational Physics, 2014)
Correlation functions in the $\mathcal{O}$$(n)$ models below the critical temperature areconsidered. Based on Monte Carlo (MC) data, we confirm the fact stated earlier by Engels and Vogt, that the transverse two-plane correlation function of the $\mathcal{O}$$(4)$ model forlattice 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 bycertain non-Gaussian approximation. We have also tested larger lattice sizes, up to $L=512$. The Fourier-transformed transverse and longitudinal two-point correlationfunctions 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 theratio $bM^2/a^2$, where $M$ is the spontaneous magnetization, are universal according tothe GFD (grouping of Feynman diagrams) approach. Here we find that the universality follows also from the standard (Gaussian) theory, yielding $bM^2/a^2$=$(n−1)/16$. OurMC 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 andGoldstone mode singularities are not exactly described by the standard theory. This isexpected from the GFD theory. We have found appropriate analytic approximationsfor $G_⊥$(k) and $G_{||}$(k), well fitting the simulation data for small $k$. We have used themto test the Patashinski-Pokrovski relation and have found that it holds approximately.