Correlation Functions, Universal Ratios and Goldstone Mode Singularities inn-Vector Models

Abstract

AbstractCorrelation functions in the(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(4) model for lattice sizes aboutL= 120 and small external fieldshis 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 toL= 512. The Fourier-transformed transverse and longitudinal two-point correlation functions have Goldstone mode singularities in the thermodynamic limit atk→ 0 andh= +0, i.e.,G⊥(k) ≃ak–λ⊥andG‖(k)≃bk–λ‖, respectively. Hereaandbare the amplitudes,k= |k| is the magnitude of the wave vectork. The exponents λᚆ, λ‖and the ratiobM2/a2, whereMis 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, yieldingbM2/a2=(n−1)/16. Our MC estimates of this ratio are 0.06±0.01 forn=2, 0.17±0.01 forn= 4 and 0.498±0.010 forn= 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 forG⊥(k) andG‖(k), well fitting the simulation data for smallk. We have used them to test the Patashinski-Pokrovski relation and have found that it holds approximately.