Critical temperature for first-order phase transitions in confined systems

We consider the Euclidean D-dimensional -λ|ϕ |4+η|ϕ|6 (λ,η>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 first-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. We obtain an expression for 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 for d = 3 (a cube). For completeness, the corresponding formulas for second-order transitions are also presented. Comparison with experimental data for superconducting films and wires shows qualitative agreement with our theoretical expressions.