Topological degeneracy (Majorana zero-mode) and 1  +  1D fermionic topological order in a magnetic chain on superconductor via spontaneous ${\text{Z}}_2^{\text{f}}$ symmetry breaking

We study a chain of ferromagnetic sites, ie nano-particles, molecules or atoms, on a substrate of fully gapped superconductors. We find that under quite realistic conditions, the fermion-number-parity symmetry $Z_2^f$ can spontaneously break. In other words, such a chain can realize a 1  +  1D fermionic topologically ordered state and the corresponding two-fold topological degeneracy on an open chain. Such a topological degeneracy becomes the so called Majorana zero mode in the non-interacting limit. (paper)