The Ginzburg-Landau phase transition model is considered in d=4-epsilon dimensions within the renormalization group (RG) approach. The problem of existence of the non-Gaussian fixed point is discussed. An equation is derived from the first principles of the RG theory (under the assumption that the fixed point exists) for calculation of the correction-to-scaling term in the asymptotic expansion of the two-point correlation (Green's) function. It is demonstrated clearly that, within the framework of the standard methods (well justified in the vicinity of the fixed point) used in the perturbative RG theory, this equation leads to an unremovable contradiction with the known RG results. Thus, in its very basics, the RG field theory in 4-epsilon dimensions is contradictory. To avoid the contradiction, we conclude that such a non-Gaussian fixed point, as predicted by the RG field theory, does not exist. Our consideration does not exclude existence of a different fixed point.
The Ginzburg-Landau phase transition model is considered in d=4-epsilon dimensions within the renormalization group (RG) approach. The problem of existence of the non-Gaussian fixed point is discussed. An equation is derived from the first principles of the RG theory (under the assumption that the fixed point exists) for calculation of the correction-to-scaling term in the asymptotic expansion of the two-point correlation (Green's) function. It is demonstrated clearly that, within the framework of the standard methods (well justified in the vicinity of the fixed point) used in the perturbative RG theory, this equation leads to an unremovable contradiction with the known RG results. Thus, in its very basics, the RG field theory in 4-epsilon dimensions is contradictory. To avoid the contradiction, we conclude that such a non-Gaussian fixed point, as predicted by the RG field theory, does not exist. Our consideration does not exclude existence of a different fixed point.
The perturbative renormalization of the Ginzburg-Landau model is reconsidered based on the Feynman diagram technique. We derive renormalization group (RG) flow equations, exactly calculating all vertices appearing in the perturbative renormalization of the phi^4 model up to the epsilon^3 order of the epsilon-expansion. In this case, the phi^2, phi^4, phi^6, and phi^8 vertices appear. All these vertices are relevant. We have tested the expected basic properties of the RG flow, such as the semigroup property. Under repeated RG transformation R_s, appropriately represented RG flow on the critical surface converges to certain s-independent fixed point. The Fourier-transformed two-point correlation function G(k) has been considered. Although the epsilon-expansion of X(k)=1/G(k) is well defined on the critical surface, we have revealed an inconsistency of the perturbative method with the exact rescaling of X(k), represented as an expansion in powers of k at k --> 0. We have discussed also some aspects of the perturbative renormalization of the two-point correlation function slightly above the critical point. Apart from the epsilon-expansion, we have tested and briefly discussed also a modified approach, where the phi^4 coupling constant u is the expansion parameter at a fixed spatial dimensionality d.
Different perturbation theory treatments of the Ginzburg-Landau phase transition model are discussed. This includes a criticism of the perturbative renormalization group (RG) approach and a proposal of a novel method providing critical exponents consistent with the known exact solutions in two dimensions. The usual perturbation theory is reorganized by appropriate grouping of Feynman diagrams of φ4 model with O(n) symmetry. As a result, equations for calculation of the two-point correlation function are obtained which allow to predict possible exact values of critical exponents in two and three dimensions by proving relevant scaling properties of the asymptotic solution at (and near) the criticality. The new values of critical exponents are discussed and compared to the results of numerical simulations and experiments.
The perturbative renormalization group (RG) treatment of the Ginzburg–Landau model is reconsidered based on the Feynman diagram technique. We derive RG flow equations, exactly calculating all vertices appearing in the perturbative RG transformation of the φ4 model up to the ε3 order of the ε-expansion. The Fourier-transformed two-point correlation function G(k) has been considered. Although the ε-expansion of X(k) = 1/G(k) is well defined on the critical surface, we have revealed an inconsistency with the exact rescaling of X(k), represented as an expansion in powers of k at k →0. This new result can serve as a basis to challenge the correctness of the ε-expansion-based perturbative RG method.
Abstract In Chapter 14, the singular behavior of ferromagnetic systems with O(N) symmetry and short-range interactions, near a second order phase transition has been determined in the mean-field approximation, which is also a quasi-Gaussian approximation. The mean-field approximation predicts a set of universal properties, properties independent of the detailed structure of the microscopic Hamiltonian, the dimension of space, and, to a large extent, of the symmetry of systems. However, the leading corrections to the mean-field approximation, in dimensions smaller than or equal to four, diverge at the critical temperature, and the universal predictions of the mean-field approximation cannot be correct. Such a problem originates from the non-decoupling of scales and leads to the question of possible universality. In Chapter 9, the question has been answered in four dimensions using renormalization theory, and related renormalization group (RG) equations. Moreover, below four dimensions, in an expansion around the mean-field, the most singular terms near criticality can be also formally recovered from a continuum, low-mass φ4 field theory. More generally, following Wilson, to understand universality beyond the mean-field approximation, it is necessary to build a general renormalization group in the form of flow equations for effective Hamiltonians and to find fixed points of the flow equations. Near four dimensions, the flow equations can be approximated by the renormalization group of quantum field theory (QFT), and the fixed points and critical behaviours derived within the framework of the Wilson-Fisher ϵ expansion.
The perturbative renormalization of the Ginzburg–Landau model is reconsidered based on the Feynman diagram technique. We derive renormalization group (RG) flow equations, exactly calculating all vertices appearing in the perturbative renormalization of the ' 4 model up to the 3 order of the -expansion. In this case, the ' 2 , ' 4 , ' 6 , and ' 8 vertices appear. All these are relevant in the sense that they do not vanish at the fixed point. We have tested the expected basic properties of the RG flow, such as the semigroup property. Under repeated RG transformation Rs, appropriately represented RG flow on the critical surface converges to certain s–independent fixed point. The Fourier–transformed two–point correlation function G(k) has been considered. Although the -expansion of X(k) = 1=G(k) is well defined on the critical surface, we have revealed a fundamental inconsistency of the perturbative method with the exact rescaling of X(k), represented as an expansion in powers of k at k ! 0. We discuss also certain paradox, related to the -expansion of X(k) above the critical point. Apart from the –expansion, we have tested also a modified approach, where the ' 4 coupling constant u is the expansion parameter at a fixed spatial dimensionality d. Our tests point to some internal inconsistency of such a method.
Different perturbation theory treatments of the Ginzburg-Landau phase transition model are discussed. This includes a criticism of the perturbative renormalization group (RG) approach and a proposal of a novel method providing critical exponents consistent with the known exact solutions in two dimensions. The usual perturbation theory is reorganized by appropriate grouping of Feynman diagrams of φ4 model with O(n) symmetry. As a result, equations for calculation of the two-point correlation function are obtained which allow to predict possible exact values of critical exponents in two and three dimensions by proving relevant scaling properties of the asymptotic solution at (and near) the criticality. The new values of critical exponents are discussed and compared to the results of numerical simulations and experiments.