Predicting the critical density of topological defects inO(N)scalar field theories

Abstract

$O(N)$ symmetric $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ field theories describe many critical phenomena in the laboratory and in the early Universe. Given $N$ and $D<~3,$ the spatial dimension, these models exhibit topological defect classical solutions that in some cases fully determine their critical behavior. For $N=2$ and $D=3,$ it has been observed that the defect density is seemingly a universal quantity at ${T}_{c}.$ We prove this conjecture and show how to predict its value based on the universal critical exponents of the field theory. Analogously, for general N and D we predict the universal critical densities of domain walls and monopoles, for which no detailed thermodynamic study exists, to our knowledge. Remarkably this procedure can be inverted, producing an algorithm for generating typical defect networks at criticality, in contrast with the usual procedure [Vachaspati and Vilenkin, Phys. Rev. D 30, 2036 (1984)], which applies only in the unphysical limit of infinite temperature.