Leading superconducting instabilities in three-dimensional models for Sr_{2}RuO_{4}

Abstract

The unconventional superconductor Sr_{2}RuO_{4} has been the subject of enormous interest over more than two decades, but until now the form of its order parameter has not been explicitly determined. Since groundbreaking NMR experiments revealed recently that the pairs are of dominant spin-singlet character, attention has focused on time-reversal symmetry breaking linear combinations of s-, d-, and g-wave one-dimensional (1D) irreducible representations. However, a state of the form d_{xz}+id_{yz} corresponding to the two-dimensional representation E_{g} has also been proposed based on some experiments. We present a systematic study of the stability of various superconducting candidate states, assuming that pairing is driven by the fluctuation exchange mechanism, including a realistic three-dimensional Fermi surface, full treatment of both local and nonlocal spin-orbit couplings, and a wide range of Hubbard-Kanamori interaction parameters U,J,U^{′},J^{′}. The leading superconducting instabilities are found to exhibit nodal even-parity A_{1g}(s^{′}) or B_{1g}(d_{x^{2}−y^{2}}) symmetries, similar to the findings in two-dimensional models without longer-range Coulomb interaction which tends to favor d_{xy} over d_{x^{2}−y^{2}}. Within the so-called Hund's coupling mean-field pairing scenario, the E_{g}(d_{xz}/d_{yz}) solution can be stabilized for large J and specific forms of the spin-orbit coupling, but for all cases studied here the eigenvalues of other superconducting solutions are significantly larger when the full fluctuation exchange vertex is included in the pairing kernel. Additionally, we compute the spin susceptibility in relevant superconducting candidate phases and compare to recent neutron scattering and nuclear magnetic resonance (NMR) Knight shift measurements. It is found that d_{xz}+id_{yz} order supports a neutron resonance in its superconducting phase, in contrast to a recent experiment [K. Jenni et al., Phys. Rev. B 103, 104511 (2021)2469-995010.1103/PhysRevB.103.104511], whereas s^{′}+id_{x^{2}−y^{2}} does not. Furthermore, comparison of the Knight shift reveals that s^{′}+id_{x^{2}−y^{2}} exhibits a larger low-temperature shift than d_{xz}+id_{yz}.