Topological degeneracy (Majorana zero-mode) and 1+1D fermionic topological order in a magnetic chain on superconductor via spontaneous Z2 symmetry breaking

We study a chain of ferromagnetic nano-particles or ferromagnetic molecule/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. More specifically, we find that $Z_2^f$ symmetry breaking or fermionic 1+1D topological order can appear if (1) the electron hopping $t_{ij}$ between nano-particles is larger than the energy splitting $\delta E_{eo}$ between the ground states of even and odd electrons on a nano-particle, (2) the Josephson coupling $J_i$ between the superconducting substrate and the nano-particle is larger than or similar to $\delta E_{eo}$, and (3) the electron hopping amplitude $t_{ij}$ is complex, or more precisely, the phase of gauge invariant combination $J_i t_{ij}^2 J_j^*$ is not zero.