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We propose a general scheme to probe the compatibility of arbitrary pairingstates with a given normal state Hamiltonian by the introduction of a conceptcalled superconducting fitness. This new quantity gives a direct measure of thesuppression of the superconducting critical temperature in presence of keysymmetry breaking fields. A merit of the superconducting fitness is that it canbe used as a tool to identify non-trivial mechanisms to suppresssuperconductivity under various external influences, in particular, magneticfields or distortions, even in complex multi-orbital systems. In the light ofthis new concept we analyse the multi-band superconductor Sr$_2$RuO$_4$ andpropose a new mechanism for the suppression of superconductivity inmulti-orbital systems, which we call inter-orbital effect, as a possibleexplanation for the unusual limiting feature observed in the upper criticalfield of this material.

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