Dimensional Crossover, the Renormalization Group and Finite Size Scaling

Kubyshin, Yuri, O'Connor, Denjoe and Stephens, C. R. (1991) Dimensional Crossover, the Renormalization Group and Finite Size Scaling. (Preprint)

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Abstract

In this paper we review some of our recent investigations into dimensional crossover using renormaiization group arguments. We consider field theory on E = R^d × B where B is a compact space of “size” L. By employing an L dependent renormalization we obtain renormalization group equations that interpolate between the two limits L = ∞ and L = 0, the latter being equivalent to the absence of B. We consider two distinct cases: one where the interpolation is between two renormalizable theories and another where it is between a non-renormalizable theory and a renormalizable one. In the former we find that the correlation functions depend only on the scaling variable L/ξ_L where ξ_L is the correlation length on E. This is finite size scaling. The scaling variable becomes tL^(1/ν) when L/ξ_L → ∞, ξ_L → ∞, and tL^(1/ν') when ξ_L → ∞ for fixed L. Here ν and ν' are the correlation length exponents of the L = ∞ and L = 0 systems respectively and t is the mass parameter on E. The results of this paper have many applications in statistical mechanics and particle physics, e.g. liquid helium in wafer geometries, finite temperature field theory and Kaluza Klein theories.

Item Type: Article
Divisions: School of Theoretical Physics > Preprints
Date Deposited: 19 Jun 2018 14:18
Last Modified: 17 Dec 2022 20:49
URI: https://dair.dias.ie/id/eprint/759

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