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

TL;DR

This paper proves the universality of defect density at criticality in O(N) scalar field theories and provides a method to predict these densities based on critical exponents, applicable across various N and D.

Contribution

It establishes the universality of defect densities at critical points and introduces a predictive method based on critical exponents for general N and D.

Findings

Proved the universality of defect density at T_c for N=2, D=3.

Predicted universal critical densities for domain walls and monopoles.

Developed an algorithm to generate defect networks at criticality.

Abstract

O(N) symmetric $\lambda \phi^4$ field theories describe many critical phenomena in the laboratory and in the early Universe. Given N and $D\leq 3$, the dimension of space, these models exhibit topological defect classical solutions that in some cases fully determine their critical behavior. For N=2, 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. This procedure can also be inverted, producing an algorithm for generating typical defect networks at criticality, in contrast to the canonical procedure, which applies only in the unphysical limit of infinite temperature.