Recent applications of the renormalization group to critical phenomena in magnetic systems have been based mainly on local linear arguments. It has been implicitly assumed that the global nonlinear effects are important only in crossover effects and that the behavior asymptotically close to the critical point is determined by the stablest fixed point alone. We have given a nonlinear analysis which incorporates the crossover between the Wilson–Fisher and mean field behavior. We point out in this paper that this competition expresses itself in globally valid solutions which can upset the dominance presumed from the linear stability analysis unless certain regularity conditions are imposed.
Recent applications of the renormalization group to critical phenomena in magnetic systems have been based mainly on local linear arguments. It has been implicitly assumed that the global nonlinear effects are important only in crossover effects and that the behavior asymptotically close to the critical point is determined by the stablest fixed point alone. We have given a nonlinear analysis which incorporates the crossover between the Wilson–Fisher and mean field behavior. We point out in this paper that this competition expresses itself in globally valid solutions which can upset the dominance presumed from the linear stability analysis unless certain regularity conditions are imposed.
The analysis of selected nonlinear problems in the renormalization group is found to show striking contrasts between the usual local linearized fixed-point analysis and the properties of global solutions of nonlinear equations derived from an approximation of the Wegner-Houghton differential formulation. The competition between various fixed points that is incorporated in general global solutions can upset the asymptotically valid critical behavior deduced from the local analysis. In general, the critical-point exponents of such a solution will not satisfy equalities, but rather the corresponding inequalities. However, these nonscaling solutions have extraneous singularities that are not related to the thermodynamic singularities of the system. If singularities of this type are excluded, then the global solution has the same critical-point exponents as the local solution derived by linearizing around the stablest fixed point. It is shown that in this case the critical surface in the Hamiltonian space is closely related to the surface of order-2 critical points in a thermodynamic field space. The boundaries of this surface are correspondingly related to the critical points of higher order in this thermodynamic space. The nonlinear global solution predicts multiple power scaling behavior from a single scaling equation deduced from the renormalization group. Previously such behavior was obtained by postulating the simultaneous validity of two of more "linear" scaling hypotheses.
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 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].
We show that the critical behavior of disordered ferromagnets can be understood in terms of an effective, translational invariant Hamiltonian with renormalized coupling parameters which exhibit a tricritical as well as a critical point. The stable critical fixed point describes the critical behavior of weakly disordered systems, whereas the tricritical fixed point is responsible for the critical behavior of strongly disordered systems at least in the experimentally accessible range of temperature. This necessitates a crossover theory to describe the crossover between these two fixed points and to calculate critical properties as a function of concentration and temperature. In this paper we study the symmetric phase. Susceptibility and specific heat are calculated in scaling form using nonlinear scaling fields to verify that disordered ferromagnets exhibit universal critical behavior. The corresponding effective exponents exhibit nonmonotonous concentration- and temperature-dependent behavior. According to our theory, Fisher-renormalized tricritical behavior dominates in the experimentally accessible temperature range which leads to extreme values of the effective exponents ${\ensuremath{\gamma}}_{\mathrm{eff}}$=2 and ${\ensuremath{\alpha}}_{\mathrm{eff}}$=-1 in the limit of strong dilution. However, in accord with the Harris criterion we verify that the asymptotic behavior (T\ensuremath{\rightarrow}Tc) is not changed by dilution if the critical exponent of the specific heat of the pure system ${\ensuremath{\alpha}}_{\mathrm{pure}<0}$, independent of the value of the concentration.
The effect of the long-range correlated random magnetic fields which behave like 〈h(x)h(y)〉 $\ensuremath{\sim}\frac{1}{{|x\ensuremath{-}y|}^{d\ensuremath{-}\ensuremath{\sigma}}}$ on the critical phenomena is discussed with the use of the nonlinear $\ensuremath{\sigma}$ model. The crossover between the two critical behaviors dominated by the long-range and short-range disorder fixed points is shown to occur at $(m\ensuremath{-}2)\ensuremath{\sigma}\ensuremath{-}d+4=0$ (m is the spin dimensionality). The critical exponents calculated at the long-range disorder fixed point around the lower critical dimensionality $4+\ensuremath{\sigma}$ are not the same as those in $d\ensuremath{-}2$ expansions in the pure system.
Most renormalization groups studied heretofore are linear, in that the block-spin variables are linearly related to the old spin variables. These renormalization groups all require an adjustable parameter that must be properly fixed in order that the renormalization-group transformation have a useful fixed point. Niemeijer and van Leeuwen, however, have investigated the critical behavior of the two-dimensional Ising model with a nonlinear renormalization group, and find a fixed point without introducing such a parameter. We introduce here another nonlinear renormalization group and study it in detail for the Gaussian model. We find that (i) within certain limits, no such parameter need be adjusted in order to reach a fixed point; (ii) an eigenvalue of the linearized transformation with no physical significance depends on the nonlinearity of the transformation; and (iii) a physically significant eigenvalue is unchanged (to the order examined).