Dimensional Crossover and Effective Exponents

TL;DR

This paper explores how finite-size effects influence the critical behavior of the lambda phi^4 theory on a space with one compact dimension, revealing dimensional crossover phenomena through numerical renormalization group analysis.

Contribution

It introduces a numerical RG approach to study finite-size scaling and demonstrates how the phase transition dimensionality depends on the ratio of the system size to the RG scale.

Findings

Finite-size scaling variable determines phase transition dimensionality.

Finite-size effects are significant at intermediate scales.

Polynomial expansion of the effective potential fails near criticality.

Abstract

We investigate the critical behavior of the lambda phi^4 theory defined on S^1 x R^d having two finite length scales beta, the circumference of S^1, and k^{-1}, the blocking scale introduced by the renormalization group transformation. By numerically solving the coupled differential RG equations for the finite-temperature blocked potential U_{beta,k}(Phi) and the wavefunction renormalization constant Z_{beta,k}(Phi), we demonstrate how the finite-size scaling variable betabar = beta k determines whether the phase transition is (d+1)- or d-dimensional in the limits betabar >> 1 and betabar << 1, respectively. For the intermediate values of betabar, finite-size effects play an important role. We also discuss the failure of the polynomial expansion of the effective potential near criticality.