Chiral symmetries and Majorana fermions in coupled magnetic atomic chains on a superconductor

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We study the magnetic structures and their connections with topological superconductivity due to proximity effect for coupled magnetic atomic chains deposited on a superconductor. Several magnetic phases are self-consistently determined, including both the coplanar and non-coplanar ones. For a N-chain triangular atomic ladder, topologically nontrivial superconducting states can always be realized, but strongly depend on its magnetic structure and the number of atomic chains. When N is even, the topologically nontrivial states with noncoplanar structures are characterized by Z₂ invariants, while the topologically nontrivial noncoplanar states with an odd N are characterized by integer Z invariants, due to the presence of a new chiral symmetry. The new chiral symmetry for the noncoplanar states is found to be robust against the on-site disorder, as long as the crystal reflection symmetry is respected.