Recent studies in the realization of Majorana fermion (MF) quasiparticles have focused on engineering topological superconductivity by combining conventional superconductors and spin-textured electronic materials. We propose an effective model to create unpaired MFs at a honeycomb lattice edge by generalizing a 2-dimensional topologically nontrivial Haldane model and introducing textured pairings. The core idea is to add both the spin-singlet and textured spin-triplet pairings to a pseudospin-state dependent, time-reversal symmetry (TRS) noninvariant honeycomb lattice, and to satisfy generalized sweet spot conditions as in the Kitaev chain model. Our model has a gapped superconducting phase and a gapless phase; either phase may have zero or nonzero topological winding numbers. The discriminant that distinguishes those two phases gives a measure of TRS breaking and may have more general implications. Effective Majorana zero modes arise at edges in distinct phases with different degrees of degeneracy. Our theoretical model motivates concepts, such as textured pairings and the strength of TRS breaking, that may play important roles in future implementation of MFs with cold atoms in optical lattices.
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