Second order phase transitions are investigated with theoretical methods of the phi^4 field theory. Thermodynamic quantities like the correlation length, the susceptibility or the specific heat have singularities in the critical point. The goal is the precise temperature dependence near these singularities. A cooperative many particel effect generates critical behavior. Differences in the short range microscopic interaction of the investigated systems are insignificant. The dimension d and the number of components of the order parameter determine the critical behavior. The renormalization group theory identifies universality classes of critical behavior and predicts that critical exponents and certain amplitude ratios are equal for all systems of the same universality class. The case d=3, n=2 is of special interest. Experimental results of the superfluid transition of 4^He in space are compared to MC simulations of spin systems and analytical calculations in the framework of the phi^4 theory. This is a high precision test of the RG prediction of universality. RG theory has a wide range of applications. Hence, this quantitative test of RG is of interest in statistical physics, condensed matter physics and elementary particle physics. A short summary of two main results: A significant improvement of former Borel summations of critical exponents gamma and alpha is achieved with the same information of perturbation series. This improvement stems from an enhancement of the method of resummation. Amplitude functions at the fixpoint are resummed. Analytical results for certain universal amplitude ratios are achieved. So far only numerical or experimental quantitative results for these ratios are known.
Second order phase transitions are investigated with theoretical methods of the phi^4 field theory. Thermodynamic quantities like the correlation length, the susceptibility or the specific heat have singularities in the critical point. The goal is the precise temperature dependence near these singularities. A cooperative many particel effect generates critical behavior. Differences in the short range microscopic interaction of the investigated systems are insignificant. The dimension d and the number of components of the order parameter determine the critical behavior. The renormalization group theory identifies universality classes of critical behavior and predicts that critical exponents and certain amplitude ratios are equal for all systems of the same universality class. The case d=3, n=2 is of special interest. Experimental results of the superfluid transition of 4^He in space are compared to MC simulations of spin systems and analytical calculations in the framework of the phi^4 theory. This is a high precision test of the RG prediction of universality. RG theory has a wide range of applications. Hence, this quantitative test of RG is of interest in statistical physics, condensed matter physics and elementary particle physics. A short summary of two main results: A significant improvement of former Borel summations of critical exponents gamma and alpha is achieved with the same information of perturbation series. This improvement stems from an enhancement of the method of resummation. Amplitude functions at the fixpoint are resummed. Analytical results for certain universal amplitude ratios are achieved. So far only numerical or experimental quantitative results for these ratios are known.
Abstract Universal quantities near the phase transition of O(N) symmetric vector models, can be determined, in the framework of the (f2 )2 field theory, and the corresponding renormalization group (RG), in the form of perturbative series. The O(N) symmetric field theories describe, in particular for N = 0, the universal properties of the statistics of long polymers, for N = 1, the liquid–vapour transition, for N = 2, superfluid helium transition, and so on. Universal quantities have been calculated within two different schemes, the Wilson-Fisher ϵ = 4 − d expansion, and perturbative expansion at fixed dimensions 2 and 3 (as suggested by Parisi). In both cases, the series are divergent, and the expansion parameters are not small. In fixed dimensions smaller than 4, the series are proven to be Borel summable. For the ϵ expansion, there are reasons that the property is equally true, but a proof is lacking. With this assumption, in both cases, although the series are divergent, they define unique functions. Since the expansion parameters are not small, summation methods are then required to determine these functions. A specific summation method, based on a parametric Borel transformation and mapping, in which the knowledge of the large order behaviour has been incorporated, has been successfully applied to the series, and has led to a precise evaluation of critical exponents and other universal quantities.
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.
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 scaling form of the free energy near a critical point allows for the definition of various thermodynamical amplitudes and the determination of their dependence on the microscopic nonuniversal scales. Universal quantities can be obtained by considering special combinations of the amplitudes. Together with the critical exponents they characterize the universality classes and may be useful quantities for their experimental identification. We compute the universal amplitude ratios for the tricritical Ising model in two dimensions by using several theoretical methods from perturbed conformal field theory and scattering integrable quantum field theory. The theoretical approaches are further supported and integrated by results coming from a numerical determination of the energy eigenvalues and eigenvectors of off-critical systems in an infinite cylinder.
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.
Abstract 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.