Critical behaviour of the compactified $\lambda {\varphi }^4$ theory

Abstract

We investigate the critical behaviour of the $N$-component Euclidean $\lambda {\varphi }^4$ 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.

12 pages, no figures