We study the superconducting state of multi-orbital spin-orbit coupled systems in the presence of an orbitally driven inversion asymmetry assuming that the inter-orbital attraction is the dominant pairing channel. Although the inversion symmetry is absent, we show that superconducting states that avoid mixing of spin-triplet and spin-singlet configurations are allowed, and remarkably, spintriplet states that are topologically nontrivial can be stabilized in a large portion of the phase diagram. The orbital-dependent spin-triplet pairing generally leads to topological superconductivity with point nodes that are protected by a nonvanishing winding number. We demonstrate that the disclosed topological phase can exhibit Lifshitz-type transitions upon different driving mechanisms and interactions, e.g., by tuning the strength of the atomic spin-orbit and inversion asymmetry couplings or by varying the doping and the amplitude of order parameter. Such distinctive signatures of the nodal phase manifest through an extraordinary reconstruction of the low-energy excitation spectra both in the bulk and at the edge of the superconductor.

The spin-orbital polarization of superconducting excitations in momentum space is shown to provide distinctive marks of unconventional pairing in the presence of inversion symmetry breaking. Taking the prototypical example of an electronic system with atomic spin-orbit and orbital-Rashba couplings, we provide a general description of the spin-orbital textures and their most striking changeover moving from the normal to the superconducting state. We find that the variation of the spin-texture is strongly imprinted by the combination of the misalignment of spin-triplet d-vector with the inversion asymmetry g-vector coupling and the occurrence of superconducting nodal excitations. Remarkably, the multi-orbital character of the superconducting state allows to unveil a unique type of topological transition for the spin-winding around the nodal points. This finding indicates the fundamental topological relation between chiral and spin-winding in nodal superconductors. By analogy between spin-and orbital-triplet pairing we point out how orbital polarization patterns can also be employed to assess the character of the superconducting state.

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We study the pairing symmetry of a doped Kane-Mele model on a honeycomb lattice with on-site Coulomb interaction. The pairing instability of Cooper pair is calculated based on the linearizedÉliashberg equation within the random phase approximation (RPA). When the magnitude of the spin-orbit coupling is weak, even-frequency spinsinglet even-parity (ESE) pairing is dominant. On the other hand, with the increase of the spin-orbit coupling, we show that the even-frequency spin-triplet odd-parity (ETO) f -wave pairing exceeds ESE one. ETO f -wave pairing is supported by the longitudinal spin fluctuation. Since the transverse spin fluctuation is strongly suppressed by spinorbit coupling, ETO f -wave pairing becomes dominant for large magnitude of spin-orbit coupling.

We study the Josephson effects arising in junctions made of non-centrosymmetric superconductors with spin-triplet pairing having s-wave orbital-singlet symmetry. We demonstrate that the orbital dependent character of the spin-triplet order parameter determines its non-trivial texture in the momentum space due to the inversion symmetry breaking and spin-orbit interactions. The emergence of this pattern is responsible for the occurrence of an anomalous Josephson coupling and a dominance of high-harmonics in the current phase relation. Remarkably, due to the spin-orbital couplings, variations in the electronic structure across the heterostructure can generally turn the ground state of the junction from 0- to a generic value of the Josephson phase, thus realizing the so-called φ-junction. Hallmarks of the resulting Josephson behavior, apart from non-standard current-phase relation, are provided by an unconventional temperature and magnetic field dependence of the critical current. These findings indicate the path for the design of superconducting orbitronics devices and account for several observed anomalies of the supercurrent in oxide interface superconductors.