Infrared behavior and fixed-point structure in the compactified Ginzburg--Landau model

Abstract

We consider the Euclidean $N$-component Ginzburg--Landau model in $D$ dimensions, of which $d$ ($d\leq D$) of them are compactified. As usual, temperature is introduced through the mass term in the Hamiltonian. This model can be interpreted as describing a system in a region of the $D$-dimensional space, limited by $d$ pairs of parallel planes, orthogonal to the coordinates axis $x_1,\,x_2,\,...,\,x_d$. The planes in each pair are separated by distances $L_1,\;L_2,\; ...,\,L_d$. For $D=3$, from a physical point of view, the system can be supposed to describe, in the cases of $d=1$, $d=2$, and $d=3$, respectively, a superconducting material in the form of a film, of an infinitely long wire having a retangular cross-section and of a brick-shaped grain. We investigate in the large-$N$ limit the fixed-point structure of the model, in the absence or presence of an external magnetic field. An infrared-stable fixed point is found, whether of not an external magnetic field is applied, but for different ranges of values of the space dimension $ D$.