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

Abstract

We consider the Euclidean D-dimensional N-component 4 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 . © 2009 American Institute of Physics.