Interaction energy of superconducting vortices

By means of a constrained variational calculation we determine the interaction energy of two-vortex configurations in the Ginzburg-Landau theory or, equivalently, in the Abelian Higgs model. The energy is evaluated as a function of the separation between vortices and of the parameter $\lambda$, which measures the relative strength of the matter self-coupling and the electromagnetic coupling. Our results provide a precise determination of the inter-vortex potential, attractive for $\lambda <1\mathrm{}$ and repulsive for $\lambda >1\mathrm{}$. They also show that for $\lambda =1\mathrm{}$ the lower bound on the energy which can be then derived is actually reached at all separations and, therefore, that in this case vortices do not interact.