Majorana Fermions in a Spin Ladder System

We consider a two-legged spin chain version of Kitaev's honeycomb model. Like its parent, this model supports $Z_2$ vortices at every plaquette. The topological phase of the system is sensitive to, {\it inter alia}, the spatial pattern of these vortices. The topological phases are gapped in the bulk but possess isolated zero energy Majorana modes at each end. The existence of such Majorana modes can be inferred from a new topological invariant. We show that there is an intimate relation between the existence of Majorana modes and the spontaneous breaking of the global $Z_2$ symmetry.