  Parafermions are exotic quasiparticles with non-Abelian fractional statistics
that can be realized and stabilized in 1-dimensional models that are
generalizations of the Kitaev p-wave wire. We study the simplest
generalization, i.e. the $Z_3$ parafermionic chain. Using a Jordan-Wigner
transform we focus on the equivalent three-state chiral clock model, and study
its rich phase diagram using the density matrix renormalization group
technique. We perform our analyses using quantum entanglement diagnostics which
allow us to determine phase boundaries, and the nature of the phase
transitions. In particular, we study the transition between the topological and
trivial phases, as well as to an intervening incommensurate phase which appears
in a wide region of the phase diagram. The phase diagram is predicted to
contain a Lifshitz type transition which we confirm using entanglement
measures. We also attempt to locate and characterize a putative tricritical
point in the phase diagram where the three above mentioned phases meet at a
single point.
