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Topological phase, supercritical point and emergent fermions in the extended parafermion chain

Shunyao Zhang, Hong-Ze Xu, Yue-Xin Huang, Guang-Can Guo, Zhengwei Zhou, Ming Gong

Published Jan 10, 2018• 0 citations
OpenAlex

Abstract

Topological orders and associated topological protected excitations satisfying Abelian or non-Abelian statistics have been widely explored in various platforms. In recent years, the Majorana zero modes with these features have been realized in experiments based on semiconductor and superconducting hybrid structures, opening an exciting avenue for realization of fault-tolerant topological quantum computations. The $\mathbb{Z}_3$ parafermions are regarded as the most natural generation of the Majorana fermions for realizing of more intriguing topological orders. In this work, we investigate the topological phase and emergent fermions in an extended parafermion chain using exact numerical methods. This model exhibits rich variety of phases, including threefold degenerate topological ferromagnetic phase, trivial paramagnetic phase, XX phase, dimer phase, chiral phase and commensurate phase. We present a smooth connection between this model and the extended spin-${1\over 2}$ XX model, by which we can generalize their measurements, such as the order parameters and long-range spin-spin correlations {\it etc}, to fully characterize the properties of these phases and map out their corresponding phase diagram. Strikingly, we find that all the phase boundaries finally merge to a single supercritical point, determined by the extended three-state Potts model. This approach opens a wide range of intriguing applications in investigating the phases and emergent phenomena in other $\mathbb{Z}_k$ parafermion models, and may be also illuminating for their experimental realizations.

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Parafermion modes are non-Abelian anyons which were introduced as ${\mathbb{Z}}_{N}$ generalizations of ${\mathbb{Z}}_{2}$ Majorana states. In particular, ${\mathbb{Z}}_{3}$ 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 ${\mathbb{Z}}_{3}$ parafermions has relied on bosonization methods or parafermionic quasiparticles. In this paper, we introduce a representation of ${\mathbb{Z}}_{3}$ 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 ${\mathbb{Z}}_{3}$ 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 ${\mathbb{Z}}_{3}$ parafermions in strongly interacting electronic systems.

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Topological orders and associated topological protected excitations satisfying Abelian or non-Abelian statistics have been widely explored in various platforms. In recent years, the Majorana zero modes with these features have been realized in experiments based on semiconductor and superconducting hybrid structures, opening an exciting avenue for realization of fault-tolerant topological quantum computations. The Z3\mathbb{Z}_3Z3​ parafermions are regarded as the most natural generation of the Majorana fermions for realizing of more intriguing topological orders. In this work, we investigate the topological phase and emergent fermions in an extended parafermion chain using exact numerical methods. This model exhibits rich variety of phases, including threefold degenerate topological ferromagnetic phase, trivial paramagnetic phase, XX phase, dimer phase, chiral phase and commensurate phase. We present a smooth connection between this model and the extended spin-12{1\over 2}21​ XX model, by which we can generalize their measurements, such as the order parameters and long-range spin-spin correlations {\it etc}, to fully characterize the properties of these phases and map out their corresponding phase diagram. Strikingly, we find that all the phase boundaries finally merge to a single supercritical point, determined by the extended three-state Potts model. This approach opens a wide range of intriguing applications in investigating the phases and emergent phenomena in other Zk\mathbb{Z}_kZk​ parafermion models, and may be also illuminating for their experimental realizations.
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.
The Haldane phase represents one of the most important symmetry-protected states in modern physics. This state can be realized using spin-1 and $\mathrm{spin}\ensuremath{-}\frac{1}{2}$ Heisenberg models and bosonic particles. Here we explore the emergent Haldane phase in an alternating bond Z3{\mathbb{Z}}_{3}Z3​ parafermion chain, which is different from the previous proposals from fundamental statistics and symmetries. We show that this emergent phase can also be characterized by a modified long-range string order, as well as fourfold degeneracy in the ground-state energies and entanglement spectra. This phase is protected by both the charge conjugate and parity symmetry, and the edge modes are shown to satisfy parafermionic statistics in which braiding of the two edge modes yields a $\frac{2\ensuremath{\pi}}{3}$ phase. This model also supports rich phases, including a topological ferromagnetic parafermion (FP) phase, trivial paramagnetic parafermion phase, classical dimer phase, and gapless phase. The boundaries of the FP phase are shown to be gapless and critical with central charge c=4/5c=4/5c=4/5. Even in the topological FP phase, it is also characterized by long-range string order; thus we observe a drop of string order across the phase boundary between the FP phase and the Haldane phase. This work opens a new way for finding of exotic topological phases with parafermions.
Parafermions are emergent excitations that generalize Majorana fermions and can also realize topological order. In this Letter, we present a nontrivial and quasi-exactly-solvable model for a chain of parafermions in a topological phase. We compute and characterize the ground-state wave functions, which are matrix-product states and have a particularly elegant interpretation in terms of Fock parafermions, reflecting the factorized nature of the ground states. Using these wave functions, we demonstrate analytically several signatures of topological order. Our study provides a starting point for the nonapproximate study of topological one-dimensional parafermionic chains with spatial inversion and time-reversal symmetry in the absence of strong edge modes.
Parafermion zero modes are generalizations of Majorana modes that underlie comparatively rich non-Abelian-anyon properties. We introduce exact mappings that connect parafermion chains, which can emerge in two-dimensional fractionalized media, to strictly one-dimensional fermionic systems. In particular, we show that parafermion zero modes in the former setting translate into symmetry-enriched Majorana modes that intertwine with a bulk order parameter---yielding braiding and fusion properties that are impossible in standard Majorana platforms. Fusion characteristics of symmetry-enriched Majorana modes are directly inherited from the associated parafermion setup and can be probed via two kinds of anomalous pumping cycles that we construct. Most notably, our mappings relate Z4{\mathbb{Z}}_{4}Z4​ parafermions to conventional electrons with time-reversal symmetry. In this case, one of our pumping protocols entails fairly minimal experimental requirements: Cycling a weakly correlated wire between a trivial phase and time-reversal-invariant topological superconducting state produces an edge magnetization with quadrupled periodicity. Our work highlights new avenues for exploring beyond-Majorana physics in experimentally relevant one-dimensional electronic platforms, including proximitized ferromagnetic chains.
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.
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.