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 R2R2 or bounded domains. It is a homogeneous function of degree (−2)(−2), and then for 2πN-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 π4. We will disprove this conjecture by showing that the correlation coefficient always vanishes, and will discuss some of its consequences.