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Progress of Theoretical Physics

Generalization of Callan-Symanzik Equations to Those for Long-Rang Interactions

Y. Yamazaki

Published Apr 1, 1976• 5 citations
Publisher

Abstract

Regularizations in systems with long-range interactions decaying as 1/rd+σ (d: dimension of space, 0<σ<2) are studied and the Callan-Symanzik equations for short-range interaction are generalized by means of the differential vertex operations to those convenient for studying critical phenomena of the systems with the long-range interactions. The generalized Callan-Symanzik and renormalization-group equations were obtained in the same forms as the old ones.

DOI
10.1143/ptp.55.1093

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Regularizations in systems with long-range interactions decaying as 1/rd+σ (d: dimension of space, 0<σ<2) are studied and the Callan-Symanzik equations for short-range interaction are generalized by means of the differential vertex operations to those convenient for studying critical phenomena of the systems with the long-range interactions. The generalized Callan-Symanzik and renormalization-group equations were obtained in the same forms as the old ones.
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The present review is devoted to the problems of finite-size scaling due to the presence of long-range interaction decaying at large distance as 1/rd+σ, σ>0. The attention is focused mainly on the renormalization group results in the framework of [Formula: see text]-theory for systems with fully finite (block) geometry under periodic boundary conditions. Some bulk critical properties and Monte Carlo results are also reviewed. The role of the cut-off effects as well as their relation with those originating from the long-range interaction is additionally discussed. Special attention is paid to the description of an adequate mathematical technique that allows for the treatment of the long-range and short-range interactions equally. The review concludes with a short discussion of some open problems.
The influence of long-range interactions decaying in d dimensions as 1/R^{d+σ} on the critical behavior of systems with Fisher's correlation-function exponent for short-range interactions η_{SR}<0, is re-examined. Such systems, typically described by Φ^{3}-field theories, are e.g. the Potts-model in the percolation-limit, the Edwards-Anderson spin-glass, and the Yang-Lee edge singularity. In contrast to preceding studies, it is shown by means of Wilson's momentum-shell renormalization-group recursion relations that the long-range interactions dominate as long as σ<2-η_{SR}. Exponents change continuously to their short-range values at the boundary of this region.
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Continuous field models were originally systematically elaborated (and still are) in the context of particle physics. Most relevant to the study of critical phenomena is the derivation of a renormalization group flow, characterized by a set of Callan–Symanzik equations for Green functions, with regular coefficients derived from perturbation theory. This is supplemented by the idea due to Fisher and Wilson of an expansion in powers of the deviation from the strict renormalizability dimension, i.e. four in the case of the ϕ 4 model, but can also be presumed to work directly in the physical dimension three (and even possibly two). We devote this chapter to a general presentation and a survey of some applications. An appendix gives a short introduction to multicritical phenomena. The Lagrangian and dimensional analysis Introduction We want to investigate universal critical properties. A discrete lattice model appears as a regularizing intermediate stage, which allows a precise meaning to be given to the functional integrals of the field theory. Following the scheme suggested by the mean field approximation, one is tempted to start directly from a continuous model, in which the lattice is replaced by a d -dimensional continuous Euclidean space, and also, even for models with a discrete symmetry, the dynamical variables are replaced by continuous fields. From the original formulation, we just retain the idea of a cutoff factor ∧ large with respect to all momenta, which ensures that some possibly divergent expressions remain finite.