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Majorana fermion (MF) excitations in solid state system have non-Abelianstatistics which is essential for topological quantum computation. Previousproposals to realize MF, however, generally requires fine-tuning of parameters.Here we explore a platform which avoids the fine-tuning problem, namely aferromagnetic chain deposited on the surface of a spin-orbit coupled $s$-wavesuperconductor. We show that it generically supports zero-energy topological MFexcitations near the two ends of the chain with minimal fine-tuning. Dependingon the strength of the ferromagnetic moment in the chain, the number of MFs ateach end, $n$, can be either one or two, and should be revealed by a robustzero-bias peak (ZBP) of height $2ne^2/h$ in scanning tunneling microscopy (STM)measurements which would show strong (weak) signals at the ends (middle) of thechain. The role of an approximate chiral symmetry which gives an integertopological invariant to the system is discussed.

Majorana fermions in ferromagnetic chains on the surface of bulk spin-orbit coupled $s$-wave superconductors