Critical exponents predicted by grouping of Feynman diagrams inϕ4 model

Kaupužs, J.

doi:10.1002/1521-3889(200104)10:4<299::aid-andp299>3.0.co;2-j

Cited by 26 publications

(103 citation statements)

References 27 publications

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“…The same model, but without the external field h, has been discussed in Ref. 7), representing the ϕ 4 term as…”

Section: §1 Introductionmentioning

confidence: 78%

“…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.…”

Section: §1 Introductionmentioning

confidence: 99%

“…Like in Ref. 7), here, we assume that the field ϕ i (x) does not contain the Fourier components ϕ i (k) with k > Λ.…”

Section: §1 Introductionmentioning

confidence: 99%

“…7), including this symmetry-breaking term. Besides, we will propose an alternative formulation of the diagrammatic equations and will show that our approach is consistent with the variational principle.…”

Section: §1 Introductionmentioning

confidence: 99%

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

Kaupužs¹

2010

Progress of Theoretical Physics

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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 * )

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“…The same model, but without the external field h, has been discussed in Ref. 7), representing the ϕ 4 term as…”

Section: §1 Introductionmentioning

confidence: 78%

Section: §1 Introductionmentioning

confidence: 99%

“…Like in Ref. 7), here, we assume that the field ϕ i (x) does not contain the Fourier components ϕ i (k) with k > Λ.…”

Section: §1 Introductionmentioning

confidence: 99%

Section: §1 Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Kaupužs¹

2010

Progress of Theoretical Physics

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“…apparently is closer to the standard (Gaussian) theoretical value 1/2, as compared to the X Y model considered in [9,10]. Therefore we are looking for sufficiently stable methods, which would provide small enough statistical errors allowing to distinguish between ρ = 1/2 and ρ = 1/2 if a small deviation from the standard value really takes place, as expected from the theoretical treatment in [15][16][17] yielding 1/2 < ρ < 1 and 3/2 < λ ⊥ < 2 in three dimen-…”

Section: Magnetization and Longitudinal Susceptibilitymentioning

confidence: 99%