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Interaction between multi components vortices at arbitrary distances using a variational method in the Ginzburg-Landau theory

We study the interaction between the vortices in multi components superconductors based on the Jacobs and Rebbi variation method using Ginzburg-Landau theory. With one condensation, we get attraction interaction between the vortices for type I and repulsion for type II superconductors. With two condensation states such as Mg B_{2} superconductors the behavior is quite different. There is attraction at large distances and repulsion when the vortices are close to each other. A stability point at distance 2.7/λ_{1} is obtained. In the case of three condensation states such as iron based superconductors,we see different behavior depending on penetration depth and correlation length. The formation energy of a vortex with three condensation states is larger than the one with one condensation state with comparable penetration and correlation length. We obtain two stability points for the superconductors with three condensation states.