Simple Extension of Kitaev Chain for \({{\mathcal{Z}}_{3}}\) Parafermions

A simple quantum chain with a topological phase is constructed. It is formed by parafermions (quasiparticles obeying the fractional statistics) with angular parameter \(\theta = {{2\pi } \mathord{\left/ {\vphantom {{2\pi } 3}} \right. \kern-0em} 3}\). The model is derived from the one-dimensional spin model with topological order protected by the \({{\mathcal{Z}}_{3}} \times {{\mathcal{Z}}_{3}}~\) symmetry (generalized cluster model). Parafermionic zero modes arising at the chain boundaries, which remain stable because of the vacuum gap and topological order, are constructed and studied.