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Physical Review A

Exact renormalization-group equations in one dimension

R. E. Prange

Published Apr 1, 1974• 9 citations
Publisher

Abstract

A renormalization-group method of recursion relations is developed and applied to the general Landau-Ginzburg problem in one-dimensional statistical mechanics. In special cases, simple closed-form expressions are obtained. In one case, the group recursion relations completely determine the free energy, whereas in another the relations are barely stronger than dimensional analysis.

DOI
10.1103/physreva.9.1711

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A renormalization-group method of recursion relations is developed and applied to the general Landau-Ginzburg problem in one-dimensional statistical mechanics. In special cases, simple closed-form expressions are obtained. In one case, the group recursion relations completely determine the free energy, whereas in another the relations are barely stronger than dimensional analysis.
The renormalization-group method for studying critical phenomena is generalized to a class of dynamical systems---the time-dependent Ginzburg-Landau models. The effects of conservation laws on the critical dynamics are investigated through the study of models with different conservation properties for the energy and the space integral of the order parameter. Dynamic critical exponents near four dimensions ($d\ensuremath{\approx}4$) are obtained from recursion relations, analogous to those of Wilson and Fisher. The physical significance of the time-dependent Ginzburg-Landau models is explored and the applicability of the results to experiments on the NMR linewidth of FeF2{\mathrm{F}}_{2}F2​ is discussed.
By explicit construction we show the existence of an exact renormalization group for triangular Ising models with only nearest-neighbor interactions. The recursion relations take the form of a set of three quasilinear first-order partial differential equations for the interactions. We determine a nontrivial fixed point and study the linearized flow around it. This yields the specific-heat exponent $\ensuremath{\alpha}=0$, in agreement with the Onsager and Houtappel solutions, and demonstrates universality. The free energy is expressed as the trajectory integral of an explicitly given function of the interactions.
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.