Three-dimensional Abelian and non-Abelian gauge Higgs theories

Gauge symmetries and Higgs mechanisms are key features of theories describing high-energy particle physics and collective phenomena in statistical and condensed-matter physics. In this review we address the collective behavior of systems of multicomponent scalar fields interacting with gauge fields, which can be already present in the underlying microscopic system or emerge only at criticality. The interplay between local gauge and global symmetries determines the phase diagram, the nature of the Higgs phases, and the nature of phase transitions between the high-temperature disordered and the low-temperature Higgs phases. However, additional crucial features determine the universal properties of the critical behavior at continuous transitions. Specifically, their nature also depends on the role played by the gauge modes at criticality. Effective (Abelian or non-Abelian) gauge Higgs field theories emerge when gauge modes develop critical correlations. On the other hand, a more standard critical behavior, which admits an effective description in terms of Landau-Ginzburg-Wilson $\Phi^4$ theories, occurs when gauge-field modes are short ranged at the transition. In the latter case, gauge fields only prevent non-gauge invariant correlation functions from becoming critical. This review covers the recent progress made in the study of Higgs systems with Abelian and non-Abelian gauge fields. We discuss the equilibrium thermodynamic properties of systems with a classical partition function, focusing mainly on three-dimensional systems, and only briefly discussing two-dimensional models. However, by using the quantum-to-classical mapping, the results on the critical behavior for classical systems in $D=d+1$ dimensions can be extended to quantum transitions in $d$ dimensions.