Parametric representation of the dilute polymer system to O(ϵ2): crossover from Wilson-Fisher to gaussian fixed point as the Flory temperature is approached
It is well known that the properties of scaling and universality exhibited by the polymer system have a very natural description in terms of the critical behaviour of an analogous magnetic system. This analogy is used to derive renormalisation-group equations directly for the polymer system. Solving these renormalisation-group equations a compact parametric description is obtained of the crossover from the Wilson-Fisher to gaussian fixed point displayed by the mean square size of a polymer (R 2 ) and the second A 2 and third A 3 virial coefficients correct to O( epsilon 2 ) in d=4- epsilon spatial dimensions.
It is well known that the properties of scaling and universality exhibited by the polymer system have a very natural description in terms of the critical behaviour of an analogous magnetic system. This analogy is used to derive renormalisation-group equations directly for the polymer system. Solving these renormalisation-group equations a compact parametric description is obtained of the crossover from the Wilson-Fisher to gaussian fixed point displayed by the mean square size of a polymer (R 2 ) and the second A 2 and third A 3 virial coefficients correct to O( epsilon 2 ) in d=4- epsilon spatial dimensions.
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.
Although Flory's approximation describes very well the radius r of a polymer, it disagrees sharply with an exact ϵ expansion (derived from an analogy to n = 0 spins). Here the polymer is described explicitly as a critical cluster of n = 0 spins. While ordinary clusters resemble trees bifurcating into b branches, a cluster of n = 0 spins constitutes a single line, weighted by bphantom bifurcations. Analogous description of the polymer leads to a modified Flory-like equation, which describes r and b, jointly, and to two more equations for r and b in separate. These give very accurate exponents ν and γ, correct to O(ϵ2). In addition, the approach can be extended to a many polymer system, and to other critical clusters.
The large distance behaviour, near the critical temperature, of continuous phase transitions with short range interactions can be reproduced by an effective local field theory. Within this framework, the universal properties of the large distance can be determined by finding fixed points of general, functional renormalization group (RG) equations (Wilson, Wegner). For dimension D > 4, the large scale behavior is governed by the Gaussian fixed point (related to mean field theory), while for D < 4 the Gaussian fixed point is unstable. Near D = 4, the RG equations can be solved in the form of an ǫ = 4 − D expansion, a new stable fixed point is found and this leads to the famous Wilson-Fisher expansion for critical exponents and other universal quantities. Moreover, one realizes that, in this framework, the effective field theory can be reduced to a renormalizable field theory and, at leading order in the critical domain, the much simpler quantum field theory renormalization group can be used. A number of physicists, including the Saclay group (Brezin, Le Guillou, Zinn-Justin) have developed sophisticated field theory techniques, to prove general scaling laws and to calculate exponents to higher orders [1].
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.
For pt.I see ibid., vol.10, p.2985 (1977). The crossover function for the specific heat of a disordered material is constructed to order epsilon by solving the renormalisation group equations for the m n model (m to O,n>1). From this an effective exponent is obtained which provides a local measure of the degree of singularity of the specific heat in the critical region. It is seen that as the critical region is approached from above the critical temperature the effective exponent behaves initially as if no disorder were present. But as the critical temperature is more closely approached the effective exponent crosses over to the value predicted for the disordered material.
Monte Carlo simulations of the critical behaviour of disordered Ising systems have led to concentration-dependent critical exponents which seem to violate universality. We apply the renormalization group to investigate the crossover effects which cause the observed behaviour. We improve the three-loop expansion of the Callan-Symanzik and Wilson functions by a Pade-Borel approximation and solve the flow equations for various initial points. The exponents found in the simulations can be related to regions in the space of coupling coefficients away from the fixed point.