Unbinding of Giant Vortices in States of Competing Order

We consider a two-dimensional system with two order parameters, one with O(2) symmetry and one with $\mathrm{O}(M)$, near a point in parameter space where they couple to become a single $\mathrm{O}(2+M)$ order. While the O(2) sector supports vortex excitations, these vortices must somehow disappear as the high symmetry point is approached. We develop a variational argument which shows that the size of the vortex cores diverges as $1/\sqrt{\ensuremath{\Delta}}$ and the Berezinskii-Kosterlitz-Thouless transition temperature of the O(2) order vanishes as $1/\mathrm{ln}(1/\ensuremath{\Delta})$, where $\ensuremath{\Delta}$ denotes the distance from the high-symmetry point. Our physical picture is confirmed by a renormalization group analysis which gives further logarithmic corrections, and demonstrates full symmetry restoration within the cores.