Field-theory results for three-dimensional transitions with complex symmetries

TL;DR

This paper reviews field-theoretical analyses of three-dimensional critical phenomena with complex symmetries, focusing on stability, critical behaviors, and multicritical points using high-order perturbative series in Landau-Ginzburg-Wilson theories.

Contribution

It provides a comprehensive overview of high-order perturbative results for various 3D critical phenomena with complex symmetries, including stability and multicritical behaviors.

Findings

Stability analysis of the O(N) fixed point

Critical behavior of dilute Ising-like systems

Multicritical behavior with competing order parameters

Abstract

We discuss several examples of three-dimensional critical phenomena that can be described by Landau-Ginzburg-Wilson $\phi^4$ theories. We present an overview of field-theoretical results obtained from the analysis of high-order perturbative series in the frameworks of the $\epsilon$ and of the fixed-dimension d=3 expansions. In particular, we discuss the stability of the O(N)-symmetric fixed point in a generic N-component theory, the critical behaviors of randomly dilute Ising-like systems and frustrated spin systems with noncollinear order, the multicritical behavior arising from the competition of two distinct types of ordering with symmetry O($n_1$) and O($n_2$) respectively.