  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.
