Revised symmetry rule and intrinsically time-reversal symmetry breaking pairing in multi-orbital superconductors

Abstract: We investigate the basic symmetry rule for the particle permutation in superconducting (SC) pairing states by examining the numerical solution of the linearized Eliashberg equation for Sr$_2$RuO$_4$. We find that the general multi-band, frequency-dependent SC gap function does not simply transform to itself up to the minus sign with either orbital ($\hat{O}$) or frequency ($\hat{T}$) exchange between two pairing electrons, contradicting the common assumption which has been used without verification. It originates from the fact that paring interactions are not invariant under the $\hat{O}$ or $\hat{T}$ operation, and is demonstrated to be essential to correctly interpret and understand multi-band SC states. One of unique properties implied by our newly found symmetry rule is the possibility of the complex eigenvalue in the linearized gap equation where the corresponding gap functions $Δ$ always have real and imaginary components both non-zero in real frequencies, inherently breaking the time-reversal symmetry. Our numerical results suggest the possibility of this unique pairing in a real material, Sr$_2$RuO$_4$, for which the Hundness of the material is found to play a key role. The revised symmetry rule not only leads us to more comprehensive understanding of the known SC states, but also opens new possibilities into exotic and unique states.