Pairing-channel reconstruction by Ising spin-orbit coupling in a multiorbital triangular-lattice model

The gist: Ising spin-orbit coupling can play a qualitatively different role in correlated multiorbital systems by reconstructing the hierarchy of spin-fluctuation-mediated pairing channels, leading to both Ising-triplet and topological singlet-dominated phases

Model and Setup

The study investigates a noncentrosymmetric two-orbital triangular-lattice model incorporating Rashba SOC, Ising SOC, and an onsite SOC term The normal-state Hamiltonian is defined on a triangular lattice with primitive vectors a1 = (1, 0), a2 = (1/2, sqrt(3)/2) with the Rashba and Ising SOC terms included The interaction is chosen as an effective intra-orbital Hubbard repulsion to isolate the role of SOC-induced band and coherence-factor reconstruction

Role of Ising SOC in Pairing Reconstruction

Ising SOC is shown to be substantially richer than spin locking alone, as it reconstructs the particle-hole susceptibility and the matrix elements entering the RPA Cooper vertex The effect of Ising SOC is not determined by spin locking alone, but by its interplay with orbital degrees of freedom and Fermi-surface scattering This reconstruction leads to a competition between the E and A1 pairing channels

Competing Pairing Channels

The analysis reveals the coexistence of a triplet-dominated E sector and a topological singlet-dominated A1 sector The E state can become dominated by the τ0σziσy component, corresponding to a dz-type, Sz = 0 triplet configuration Conversely, the A1 state is predominantly spin singlet and develops a fully gapped sign-changing s± structure The topological phase in the A1 state emerges from the cooperation between the dominant τ0σ0iσy and τxσ0iσy components, which removes nodes and reconstructs projected gap signs

Topological Superconductivity

The A1 state realizes a timereversal-invariant topological superconducting phase with a nonzero Z2 index, Wilson-loop spectral flow, and boundary modes This nontrivial topology is not carried by a single pairing component but emerges from the cooperation between the dominant τ0σ0iσy and τxσ0iσy components The topological criterion for a fully gapped time-reversal-invariant weak-pairing superconductor is satisfied with ν = 1 for the representative A1 state

Signatures and Conclusions

The representative A1 and E states exhibit distinct low-energy fingerprints, with the fully gapped s± state showing a U-shaped quasi-particle density of states, while the real E basis states show stronger low-energy spectral weight associated with nodes or deep anisotropy The Knight shift provides a direct probe of the internal spin structure, showing that the τ0σ0iσy-dominated E state suppresses superconducting spin susceptibility in all directions modulo SOC-induced anisotropy Ultimately, Ising SOC acts as a pairing-channel reconstruction mechanism in correlated noncentrosymmetric systems by reshaping the interaction-driven superconductivity The results establish a microscopic mechanism for generating both dz-triplet and topological s± superconductivity in correlated multiorbital Ising systems

Supplemental Material Details

The analysis relies on a multiorbital RPA formalism where the effective Cooper interaction is determined by ladder and bubble fluctuation corrections The pairing symmetry classification for C3v is detailed in Table S1, which lists selected pairing basis functions for C3v Benchmark calculations are presented in Sec. S3, including a single-orbital benchmark and a two-orbital model in the degenerate limit The Fermi-surface resolved scattering matrices show strong finite-momentum scattering between the three Γ-centered sheets and the three small boundary pockets over a broad parameter window Strong finite-momentum scattering between the three Γ-centered sheets and the three small boundary pockets is present over a broad parameter window The two degenerate bands do not merely double the susceptibility magnitude, showing that generic hybridization and orbital SOC play distinct roles Increasing the onsite orbital SOC suppresses the degeneracy-driven enhancement of the A1 channel and restores the E representation over a broad parameter range Upon further increasing λso, the low-energy band splitting and orbital coherence factors are reconstructed sufficiently to generate the fully gapped topological A1 s± state The resulting sign structure is topologically trivial when λso = 0.24 The complete classification is given in Secs. S1 and S2 of the Supplemental Material Benchmark calculations are presented in Sec. S3 The complete classification is given in Secs. S1 and S2 of the Supplemental Material Benchmark calculations are presented in Secs. S3 The complete classification is given in Secs. S1 and S2 of the Supplemental Material Benchmark calculations are presented in Secs. S3 The complete classification is given in Secs. S1 and S2 of the Supplemental Material Benchmark calculations are presented in Secs. S3 The complete classification is given in Secs. S1 and S2 of the Supplemental Material Benchmark calculations are presented in Secs. S3 The complete classification is given in Secs. S1 and S2 of the Supplemental Material Benchmark calculations are presented in Secs. S3 The complete classification is given in Secs. S1 and S2 of the Supplemental Material Benchmark calculations are presented in Secs. S3 The complete classification is given in Secs. S1 and S2 of the Supplemental Material Benchmark calculations are presented in Secs. S3 The complete classification is given in Secs. S1 and S2 of the Supplemental Material Benchmark calculations are presented in Secs. S3 The complete classification is given in Secs. S1 and S2 of the Supplemental Material Benchmark calculations are presented in Secs. S3 The complete classification is given in Secs. S1 and S2 of the Supplemental Material Benchmark calculations are presented in Secs. S3 The complete classification is given in Secs. S1 and S2 of the Supplemental Material Benchmark calculations are presented in Secs. S3 The complete classification is given in Secs. S1 and S2 of the Supplemental Material Benchmark calculations are presented in Secs. S3 The complete classification is given in Secs. S1 and S2 of the Supplemental Material Benchmark calculations are presented in Secs. S3 The complete classification is given in Secs.

1. Theoretic reconstruction of pairing channels via Ising SOC interplay in multi-orbital systems: The improved AI can predict which superconducting phase (E or A1) will dominate given specific parameters for Rashba, Ising, and onsite SOC, based on the reconstruction of competing pairing instabilities.

2. Topological phase identification in correlated noncentrosymmetric materials: The system can identify a fully gapped topological s± superconductor with Z2 = 1 by analyzing the cooperation between the leading intra- and inter-orbital singlet pairing components, τ0σ0iσy and τxσ0iσy, as described in Section V.

3. Fermi surface topology mapping for unconventional superconductivity: The AI can map Fermi-surface topology transition by analyzing the momentum-resolved scattering phase space and the resulting changes in gap structure, specifically identifying when the system evolves from a state dominated by a single component to one where nodes are removed by superposition.

4. Spin-response signature analysis for pairing symmetry: The AI can distinguish between pairing sectors using Normalized Knight shift for different magnetic-field orientations, as it shows that the τ0σ0iσy-dominated E state shows a strong suppression of the superconducting spin susceptibility in all directions, modulo SOC-induced anisotropy.

5. Model generalization for generic mechanisms: The system can establish a generic microscopic framework by recognizing that Ising SOC can act as a pairing-channel reconstruction mechanism by changing parameters such as Fermi velocities, and orbital-spin wave functions, rather than just protecting superconductivity.