Critical behaviour of the compactified $\lambda \phi^4$ theory

TL;DR

This paper analyzes the critical behavior of the $N$-component Euclidean $ abla ext{lambda} ext{phi}^4$ model under various confinement geometries, deriving critical temperature equations and size limits for phase transitions.

Contribution

It provides a leading-order $1/N$ expansion analysis of the model's critical behavior in confined geometries, connecting to Ginzburg-Landau models for different sample shapes.

Findings

Derived critical temperature equations for confined geometries.

Determined size limits for phase transitions in films, wires, and grains.

Connected theoretical results to experimental sample configurations.

Abstract

We investigate the critical behaviour of the $N$-component Euclidean $\lambda \phi^4$ model at leading order in $\frac{1}{N}$-expansion. We consider it in three situations: confined between two parallel planes a distance $L$ apart from one another, confined to an infinitely long cylinder having a square cross-section of area $A$ and to a cubic box of volume $V$. Taking the mass term in the form $m_{0}^2=\alpha(T - T_{0})$, we retrieve Ginzburg-Landau models which are supposed to describe samples of a material undergoing a phase transition, respectively in the form of a film, a wire and of a grain, whose bulk transition temperature ($T_{0}$) is known. We obtain equations for the critical temperature as functions of $L$ (film), $A$ (wire), $V$ (grain) and of $T_{0}$, and determine the limiting sizes sustaining the transition.