Spontaneous Symmetry Breaking in Compactified $\lambda\phi^4$ Theory

TL;DR

This paper investigates how temperature and spatial confinement influence phase transitions in a massive vector $ ext{(}\lambda ext{)} ext{phi}^4$ theory using an extended Matsubara formalism, deriving formulas for critical behavior in confined systems.

Contribution

It introduces a method to analyze combined effects of temperature and spatial boundaries on phase transitions in $ ext{(}\lambda ext{)} ext{phi}^4$ theory, including formulas for critical curves.

Findings

Derived temperature- and boundary-dependent mass and coupling formulas.

Established the critical curve equation in the $eta imes L$ plane.

Analyzed phase transition behavior in confined systems.

Abstract

We consider the massive vector $N$-component $(\lambda\phi^{4})_{D}$ theory in Euclidian space and, using an extended Matsubara formalism we perform a compactification on a $d$-dimensional subspace, $d\leq D$. This allows us to treat jointly the effect of temperature and spatial confinement in the effective potential of the model, setting forth grounds for an analysis of phase transitions driven by temperature and spatial boundary. For $d=2$, which corresponds to the heated system confined between two parallel planes (separation $L$), we obtain, in the large $N$ limit at one-loop order, formulas for temperature- and boundary-dependent mass and coupling constant. The equation for the critical curve in the $\beta \times L$ plane is also derived.