Constrained weak-coupling superconductivity in multiband superconductors

We consider superconductivity in a system with $N$ Fermi surfaces, including intraband and interband effective electron-electron interactions. The effective interaction is described by an $N\ifmmode\times\else\texttimes\fi{}N$ matrix whose elements are assumed to be constant in thin momentum shells around each Fermi surface, giving rise to $s$-wave superconductivity. Starting with attractive intraband interactions in all $N$ bands, we show that too strong interband interactions are detrimental to sustaining $N$ nonzero components of the superconducting order parameter. We find similar results in systems with repulsive intraband interactions. The dimensionality reduction of the order-parameter space is given by the number of nonpositive eigenvalues of the interaction matrix. Using general models and models for superconducting transition metal dichalcogenides and iron pnictides frequently employed in the literature, we show that constraints must be imposed on the order parameter to ensure a lower bound on the free energy and that subsequent higher-order expansions around the global minimum are thermodynamically stable. We also demonstrate that similar considerations are necessary for unconventional pairing symmetries.