Theory of superconductors with $\kappa$ close to $1/\sqrt{2}$

As was first shown by Bogomolnyi, the critical Ginzburg-Landau (GL) parameter $\kappa =1/\sqrt{2}$ at which a superconductor changes its behavior from type I to type II, is the special highly degenerate point where Abrikosov vortices do not interact and therefore all vortex states have the same energy. Developing a secular perturbation theory, we studied how this degeneracy is lifted when $\kappa$ is slightly different from $1/\sqrt{2}$ or when the GL theory is extended to the higher terms in $T-T_c.$ We constructed a simple secular functional that depends only on few experimentally measurable phenomenological parameters and therefore is quite efficient to study the vortex state of superconductor in this transitional region of $\kappa .$ On this base, we calculated such vortex state properties as critical fields, energy of the normal-superconductor interface, energy of the vortex lattice, vortex interaction energy, etc., and compared them with previous results that were based on bulky solutions of GL equations.