Some remarks concerning symmetry-breaking for the Ginzburg–Landau equation

The correlation term, introduced in [13] to describe the interaction between very far apart vortices, governs symmetry-breaking for the Ginzburg-Landau equation in R^2 or bounded domains. It is a homogeneous function of degree -2, and then for 2\pi/N-symmetric vortex configurations can be expressed in terms of the so-called correlation coefficient. Ovchinnikov and Sigal [13] have computed it in few cases and conjectured its value to be an integer multiple of \pi/4. We will disprove this conjecture by showing that the correlation coefficient always vanishes, and will discuss some of its consequences.