Revised symmetry rule and intrinsically time-reversal symmetry breaking pairing states in superconducting Sr₂RuO₄

We investigate the superconducting pairing symmetries in Sr₂RuO₄ by solving the frequency-dependent linearized Eliashberg equation within the DFT+DMFT framework. It is found that the ‘SPOT’ classification of the gap symmetry does not apply to inter-orbital pairing solutions in general, since the inter-orbital pairing interaction is not invariant under the orbital or frequency exchange. In the imaginary frequency domain, an inter-orbital paring interaction component transforms to its complex conjugate under the frequency exchange, so that its gap solution components with opposite frequency signs are also the complex conjugate to each other. These considerations lead to the conclusion that the gap solutions are either purely real or imaginary in the real frequency domain, each preserving the time reversal symmetry. Meanwhile, we find a pair of inter-orbital gap solutions one of which has positive imaginary frequency components only and the other has negative frequency components only in the numerical solutions of Sr₂RuO₄. We show that this unique property translates to the gap functions having real and imaginary components both non-zero in the real frequency domain, implying the inherently time-reversal symmetry broken pairing state even without the formation of a two-component pairing state in the form of Δ_a+iΔ_b. We argue that this unique property is associated with the Hundness of the material.