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We study the sodium-iridates model on the honeycomb lattice with both BCSpairing potential and Hubbard interaction term. It is shown that this model canbe exactly solved with appropriate choices of amplitude of pairing gaps, wherethe interacting terms are transformed to external field terms. The bandstructures of these exact solutions on both torus and cylinder geometry arediscussed in great details. It is found that the ground state assumes ananti-ferromagnetic configuration, which breaks the time reversal symmetryspontaneously and renders the superconductor topologically trivial. On theother hand, the nontrivial topology is preserved with ferromagneticconfiguration and can be characterized by the isospin Chern number.

Exact Solution to Sodium-Iridate-BCS-Hubbard Model along the Symmetric Line: Non-trivial topology in the ferromagnetic order