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5 - Dynamic renormalization group

from Part I - Near-equilibrium critical dynamics

Published online by Cambridge University Press:  05 June 2014

Uwe C. Täuber
Uwe C. Täuber
Affiliation:
Virginia Polytechnic Institute and State University
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Summary

In this central and essential chapter, we develop the dynamic renormalization group approach to time-dependent critical phenomena. Again, we base our exposition on the simple O(n)-symmetric relaxational models A and B; the generalization to other dynamical systems is straightforward. We begin with an analysis of the infrared and ultraviolet singularities appearing in the dynamic perturbation expansion. Although we are ultimately interested in the infrared critical region, we first take care of the ultraviolet divergences. Below and at the critical dimension dc = 4, only a finite set of Feynman diagrams carries ultraviolet singularities, which we evaluate by means of the dimensional regularization prescription, and then eliminate via multiplicative as well as additive renormalization (the latter takes into account the fluctuation-induced shift of the critical temperature). The renormalization group equation then permits us to explore the ensuing scaling behavior of the correlation and vertex functions of the renormalized theory upon varying the arbitrary renormalization scale. Of fundamental importance is the identification of an infrared-stable renormalization group fixed point, which describes scale invariance and hence allows the derivation of the critical power laws in the infrared limit from the renormalization constants determined in the ultraviolet regime. This program is explicitly carried through for the relaxational models A and B. For model B, we derive a scaling relation connecting the dynamic exponent to the Fisher exponent η. The critical exponents ν, η, and z are computed to first non-trivial order in an ∊ expansion around the upper critical dimension dc = 4.

Type
Chapter
Information
Critical Dynamics
A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior
 , pp. 171 - 206
Publisher: Cambridge University Press
Print publication year: 2014

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