Grand thermodynamic potential in a two-band unconventional superconductor

In a two-band system, both conventional sign-preserving $s_{++}$ and unconventional sign-changing $s_{\pm}$ superconducting state may appear at low temperatures. Moreover, they may transform from one to another due to the impurity scattering. To study the details of such a transition here we derive the expression for the Grand thermodynamic potential $Ω$ for a two-band model with nonmagnetic impurities considered in a $\mathcal{T}$-matrix approximation. For the iron-based materials within the multiband Eliashberg theory, we show that the $s_{\pm} \to s_{++}$ transition in the vicinity of the Born limit is a first order phase transition.