Longitudinal and Transverse Correlation Functions in the  4 Model below and near the Critical Point

Abstract

We have extended our method of grouping Feynman diagrams (GFD theory) to study the transverse and longitudinal correlation functions G ⊥ (k) and G (k) in ϕ 4 model below the critical point (T < T c ) in the presence of an infinitesimal external field. Our method allows a qualitative analysis without cutting the perturbation series. The long-wave limit k → 0 has been studied at T < T c , showing that G ⊥ (k) a k −λ ⊥ and G (k) b k −λ with exponents d/2 < λ ⊥ < 2 and λ = 2λ ⊥ − d are the physical solutions of our equations at the spatial dimensionality 2 < d < 4, which coincides with the asymptotic solution at T → T c as well as with a nonperturbative renormalization group (RG) analysis provided in our paper. This has been confirmed also by recent Monte Carlo simulations. The exponents as well as the ratio bM 2 /a 2 (where M is magnetization) are universal. The results of the perturbative RG method are reproduced by formally setting λ ⊥ = 2, although our analysis yields λ ⊥ < 2.Subject Index: 370 §1. IntroductionPhase transitions and critical phenomena are widely investigated fields in physics and natural sciences. 1)-6) This paper is devoted to the further development of our original diagrammatic method introduced in Ref. 7) to study the ϕ 4 phase transition model below the critical point. Our approach is based on a suitable grouping of Feynman diagrams; therefore, we shall call it the GFD theory. Distinct from the conventional perturbative renormalization group (RG) method, 3), 4) it allows a qualitative analysis near and at the critical point, without cutting the perturbation series.The ϕ 4 model exhibits a nontrivial behavior in close vicinity, as well as below the critical temperature T c , if the order parameter is an n-component vector with n > 1. The related long-wave divergence of the longitudinal and transverse correlation functions (in Fourier representation) at T < T c has been studied in Refs. 8)-9) on the basis of the hydrodynamical (Gaussian) approximation. Essentially the same problem has been studied before in Ref. 10) in terms of the Gaussian spin-wave theory. 11), 12) Later perturbative RG studies 13)-21) support the Gaussian approximation. According to conventional belief, 9), 13)-21) the transverse correlation function G ⊥ (k) (where k is the wave vector) diverges like k −λ ⊥ with λ ⊥ = 2 at k → 0 below T c for the systems with O(n ≥ 2) rotational symmetry. It corresponds to the * )

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

“…The exponent λ = 0.736 (12) in Tab. 5 (at minimal h and k max = 0.28) is somewhat smaller than the value 0.858(42), calculated from the scaling relation λ = 2λ ⊥ − d [16] with λ ⊥ = 1.929(21), although it agrees well with the direct estimation λ = 0.69 ± 0.10 in [14]. The discrepancies indicate that corrections to scaling, including non-trivial ones of the GFD theory (discussed in [14,15]), which have not been taken into account in the fitting procedures, are larger for the O(2) model as compared to the O(4) and O(10) models.…”

“…5. For the O(2) model, the agreement between λ ⊥ = 1.929(21), obtained in [12] from (2) via scaling relation ρ = (d/λ ⊥ ) − 1 [16], and the smallest-h estimate λ ⊥ = 1.9763(56) in Tab. 4 is worse.…”

“…for η * = 2 − λ ⊥ > 0 and λ = 2λ ⊥ − 3, corresponding to the relations of the GFD theory at d = 3 [16]. The ratio bM 2 /a 2 and, consequently, also BM 2 /A 2 are universal in this theory.…”

“…The accuracy of the standard theory can be checked by comparing (28)-(29) with Monte Carlo estimates. According to the relation λ = 2λ ⊥ − d, which holds in the GFD theory [16] and also in the standard theory (where λ ⊥ = 2 and λ = 4 − d), in 3D case we have where the quantity…”

“…Let us now consider the approximation (19) in more detail. This approximation does not uniquely follow from the theory in [16], since the letter refers mainly to the case h = +0 in the thermodynamic limit. However, this approximation for non-zero h together with an analogous one for the longitudinal correlation function, i. e., However, according to (32)-(33), the ratio ξ ⊥ /ξ is expected to be a constant at h → 0, but not necessarily two.…”