Coexistence of Even- and Odd-Frequency Superconductivity in Correlated Multi-Orbital Systems with Spin-Orbit Coupling

Abstract

The symmetry of the superconducting order parameters in multi-orbital systems involves many quantum numbers. Pairing mediated by electronic correlations being retarded, the frequency structure of superconducting order parameters bears important information. Here we generalize the frequency-dependent theory of superconductivity mediated by spin and charge fluctuations to systems with spin-orbit coupling. This formulation is applied to strontium ruthenate with a normal state obtained using density functional theory. Taking advantage of pseudospin and inversion symmetries, the inter-pseudospin sector of the normal state Eliashberg equation is mapped to a pseudospin-diagonal one. We find ubiquitous entanglement of spin and orbital quantum numbers, along with notable mixing between even-and odd-frequency correlations. We present the phase diagrams for leading and subleading symmetries in the pseudospin-orbital basis of Sr2RuO4. They are characterized using SP OT contributions, group theory, phase distributions in the complex plane and temperature dependence.

Frequency-dependent Inter-pseudospin Solutions to Superconducting Strontium Ruthenate

“…Finally, the absence of the d yz ± id zx (E + g ) [25,26], E − g [29], s * -wave (A + 1g ) [33][34][35] and g xy(x 2 −y 2 ) (A + 2g ) [42][43][44][45] solutions as dominant instabilities is explained by the requirement from SOC in 2D that all orbital components participate in pairing and transform like a unique global irrep. Instead, we find that most subleading eigenvectors have odd-ω intra-l components [63].…”

“…Linearized Eliashberg equation. -Details and derivations are given in the companion paper [63]. The pairing mechanism is contained in an effective interaction that combines with the pair susceptibility to yield a pairing function V pp that must be diagonalized to find the superconducting instabilities of the normal state.…”

“…Pseudospin basis. -The KSH conserves spins but not pseudospins [34,63]. However, it is still block diagonal and V pp can be decomposed into the intra-and inter-pseudospin (intra-ρ and inter-ρ) channels.…”

“…for α the e-p (o-p) channel, with the effective vertices V α KK given in Ref. 63. This problem does not introduce new numerical challenges compared with the spin-diagonal case.…”

“…It also has TR symmetry, which is always satisfied within a global phase. Given that even-and odd-ω contributions can mixed when pairing between different orbitals is considered [77], we label the irreps with +/− [63,78] based on the intra-l components since these are pure in even-/odd-ω.…”