Phase diagram of the $Z_3$ Parafermionic Chain with Chiral Interactions

TL;DR

This paper explores the phase diagram of a $Z_3$ parafermionic chain with chiral interactions, revealing complex phase transitions and incommensurate phases using advanced numerical and entanglement diagnostics.

Contribution

It provides the first detailed phase diagram of the $Z_3$ parafermionic chain with chiral interactions, identifying phase boundaries and characterizing phase transitions including Lifshitz and tricritical points.

Findings

Identification of a rich phase diagram with topological, trivial, and incommensurate phases.

Confirmation of a Lifshitz transition using entanglement measures.

Evidence suggesting a tricritical point where three phases meet.

Abstract

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.