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arXiv:1502.05049 · Physical Review B

Phase diagram of the Z3 parafermionic chain with chiral interactions

Ye Zhuang, Hitesh J. Changlani, Norm M. Tubman, Taylor L. Hughes

Published Jul 31, 2015• 73 citations
arXivHTMLPDF

Abstract

Parafermions are exotic quasiparticles with non-Abelian fractional statistics that can be realized and stabilized in one-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.

DOI
10.1103/physrevb.92.035154

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Parafermions are exotic quasiparticles with non-Abelian fractional statistics that can be realized and stabilized in one-dimensional models that are generalizations of the Kitaev ppp-wave wire. We study the simplest generalization, i.e., the Z3{Z}_{3}Z3​ 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.
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Topological orders and associated topological protected excitations with non-Abelian statistics have been widely explored in various platforms. The Z3{\mathbb{Z}}_{3}Z3​ 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 Z2{\mathbb{Z}}_{2}Z2​ spin phases in an extended parafermion chain. This model exhibits a 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 Z2{\mathbb{Z}}_{2}Z2​ spin model. These phases correspond to various metallic and insulator phases with different orders realized by parafermions. We generalize the measurement tools in Z2{\mathbb{Z}}_{2}Z2​ spin models to fully characterize these phases and map out the corresponding phase diagram. Surprisingly, all the phase boundaries finally merge to a single supercritical (SC) point with infinite degeneracy. The phases surrounding this SC point can be regarded as phases under different types of symmetry breakings. This criteria may suggest a general principle to search these SC points in the other models. Our phase diagram also reveals the asymptotic behavior of the parafermion towards canonical fermions. Regarding the rather generality of emergent phenomena in parafermion models, this approach opens a wide range of intriguing applications in investigating the various exotic phases using parafermion models.
Parafermion modes are non-Abelian anyons which were introduced as ZN{\mathbb{Z}}_{N}ZN​ generalizations of Z2{\mathbb{Z}}_{2}Z2​ Majorana states. In particular, Z3{\mathbb{Z}}_{3}Z3​ parafermions can be used to produce Fibonacci anyons, laying a path towards universal topological quantum computation. Due to their fractional nature, much of the theoretical work on Z3{\mathbb{Z}}_{3}Z3​ parafermions has relied on bosonization methods or parafermionic quasiparticles. In this paper, we introduce a representation of Z3{\mathbb{Z}}_{3}Z3​ parafermions in terms of purely fermionic models. We establish the equivalency of a family of lattice fermionic models written in the basis of the $t\ensuremath{-}J$ model with a Kitaev-like chain supporting free Z3{\mathbb{Z}}_{3}Z3​ parafermionic modes at its ends. By using density matrix renormalization group calculations, we are able to characterize the topological phase transition and study the effect of local operators (doping and magnetic fields) on the spatial localization of the parafermionic modes and their stability. Moreover, we discuss the necessary ingredients towards realizing Z3{\mathbb{Z}}_{3}Z3​ parafermions in strongly interacting electronic systems.
We study the phase diagram of an extended parafermion chain, which, in addition to terms coupling parafermions on neighbouring sites, also possesses terms involving four sites. Via a Fradkin--Kadanoff transformation the parafermion chain is shown to be equivalent to the non-chiral Z3\mathbb{Z}_3Z3​ axial next-nearest neighbour Potts model. We discuss a possible experimental realisation using hetero-nanostructures. The phase diagram contains several gapped phases, including a topological phase where the system possesses three (nearly) degenerate ground states, and a gapless Luttinger-liquid phase.
Simple Extension of Kitaev Chain for Z3{{\mathcal{Z}}_{3}}Z3​ Parafermions
A simple quantum chain with a topological phase is constructed. It is formed by parafermions (quasiparticles obeying the fractional statistics) with angular parameter θ=2π/2π33\theta = {{2\pi } \mathord{\left/ {\vphantom {{2\pi } 3}} \right. \kern-0em} 3}θ=2π/2π33 . The model is derived from the one-dimensional spin model with topological order protected by the Z3×Z3 {{\mathcal{Z}}_{3}} \times {{\mathcal{Z}}_{3}}~Z3​×Z3​  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.
We study the phase diagram of an extended parafermion chain, which, in addition to terms coupling parafermions on neighbouring sites, also possesses terms involving four sites. Via a Fradkin–Kadanoff transformation the parafermion chain is shown to be equivalent to the non-chiral \mathbf{\mathbb{Z}_3} ℤ 3 axial next-nearest neighbour Potts model. We discuss a possible experimental realisation using hetero-nanostructures. The phase diagram contains several gapped phases, including a topological phase where the system possesses three (nearly) degenerate ground states, and a gapless Luttinger-liquid phase.