Extended Ginzburg-Landau formalism for two-band superconductors

Recent observation of unusual vortex patterns in MgB_2 single crystals raised speculations about a new type of superconductivity (named "type-1.5"). However, the strict application of the standard two-band Ginzburg-Landau (GL) theory results in simply proportional order parameters of the two bands - and does not support the "type-1.5" behavior. Here we derive the extended GL formalism (accounting all terms of the next order over the small t= 1- T/T_c parameter) for a two-band clean s-wave superconductor and show that the two condensates generally have different spatial scales, with difference disappearing only in the limit T -> T_c. The extended version of the two-band GL formalism improves the validity of GL theory below T_c, and suggests revisiting of the earlier calculations based on the standard model.