Topological phase, supercritical point and emergent phenomena in extended $mathbb{Z}_3$ parafermion chain

Topological orders and associated topological protected excitations satisfying non-Abelian statistics have been widely explored in various platforms. The $mathbb{Z}_3$ parafermions are regarded as the most natural generation of the Majorana fermions to realize these topological orders. Here we investigate the topological phase and emergent $mathbb{Z}_2$ spin phases in an extended parafermion chain. This model exhibits rich variety of phases, including not only topological ferromagnetic phase, which supports non-Abelian anyon excitation, but also spin-fluid, dimer and chiral phases from the emergent $mathbb{Z}_2$ spin model. We generalize the measurement tools in $mathbb{Z}_2$ spin models to fully characterize these phases in the extended parafermion model and map out the corresponding phase diagram. Surprisingly, we find that all the phase boundaries finally merge to a single supercritical point. In regarding of the rather generality of emergent phenomena in parafermion models, this approach opens a wide range of intriguing applications in investigating the exotic phases in other parafermion models.