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We consider the Euclidean D-dimensional N-component λ|φ|4 (λ>0) model with d (d≤D) compactified dimensions. Introducing temperature by means of the Ginzburg–Landau prescription in the mass term of the Hamiltonian, this model can be interpreted as describing a second-order phase transition for a system in a region of the D-dimensional space, limited by d pairs of parallel planes, orthogonal to the coordinates axis x1,x2,…,xd. The planes in each pair are separated by distances L1,L2,…,Ld. Making the appropriate boundary corrections to the coupling constant, we obtain in the large-N limit the transition temperature as a function of the size of the system, Tc({Li}), i=1,2,…,d. For D=3 we particularize this formula, taking L1=L2=⋯=Ld=L for the physically interesting cases d=1 (a film), d=2 (an infinitely long wire having a square cross section), and d=3 (a cubic grain).

Boundary effects on the mass and coupling constant in the compactified Ginzburg–Landau model: The boundary dependent critical temperature