Identifying detrimental effects for multi-orbital superconductivity - Application to Sr$_2$RuO$_4$

We propose a general scheme to probe the compatibility of arbitrary pairing states with a given normal state Hamiltonian by the introduction of a concept called superconducting fitness. This new quantity gives a direct measure of the suppression of the superconducting critical temperature in presence of key symmetry breaking fields. A merit of the superconducting fitness is that it can be used as a tool to identify non-trivial mechanisms to suppress superconductivity under various external influences, in particular, magnetic fields or distortions, even in complex multi-orbital systems. In the light of this new concept we analyse the multi-band superconductor Sr$_2$RuO$_4$ and propose a new mechanism for the suppression of superconductivity in multi-orbital systems, which we call inter-orbital effect, as a possible explanation for the unusual limiting feature observed in the upper critical field of this material.