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

TL;DR

This paper demonstrates that a chain of ferromagnetic nanoparticles on a superconductor can spontaneously break fermion-number-parity symmetry, leading to a 1+1D fermionic topological order with Majorana zero modes under realistic conditions.

Contribution

It reveals conditions under which a ferromagnetic chain on a superconductor exhibits fermionic topological order and Majorana zero modes through spontaneous symmetry breaking.

Findings

Fermion-number-parity symmetry can spontaneously break in the chain.

Topological degeneracy and Majorana zero modes can emerge in realistic setups.

Conditions involve electron hopping, Josephson coupling, and complex phase relationships.

Abstract

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.