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

We study the magnetic structures and their connections with topological superconductivity due to the proximity effect for coupled magnetic atomic chains deposited on a superconductor. Several magnetic phases are self-consistently determined, including both the coplanar and noncoplanar ones. For an 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 $Z_2$ 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. (paper)