We will prove that if the order parameter belongs to the representation
D which is the direct sum of two complex conjugate irreducible representations, $D=\ensuremath{\Delta}\ensuremath{\bigoplus}{\ensuremath{\Delta}}^{*}$, and if $\ensuremath{\Delta}$ has at least one quartic invariant, then the symmetry SO(2) appears in the parameter space. If, in addition, $\ensuremath{\Delta}$ and ${\ensuremath{\Delta}}^{*}$ are quasiequivalent then higher symmetry than SO(2) will appear in the parameter space. Since SO(2) is a continuous group, every fixed point of the renormalization-group transformations will be either invariant under SO(2) or part of a fixed line generated by SO(2) transformations on the fixed point.
This article explores how a mathematical tool called SU(2)/SO(2) helps us understand the behavior of Cooper pairs, tiny electron couples responsible for superconductivity. Copper pairs consist of two electrons with opposite spins bound together by vibrations in the material. The article explains how SU(2)/SO(2) describes the electron spins and their spatial arrangement within the pairs. It shows how this tool helps predict their distribution, formation process, and magnetic properties. Ultimately, this allows us to understand superconductivity better and design new superconducting materials.