Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Poor man's parafermions in a lattice model with even multiplet pairing

Leonardo Mazza, Fernando Iemini, Marcello Dalmonte, Christophe Mora

Published Jan 25, 2018• 2 citations
OpenAlex

Abstract

We discuss a one-dimensional fermionic model with a generalized $\mathbb{Z}_{N}$ even multiplet pairing extending Kitaev $\mathbb{Z}_{2}$ chain. The system shares many features with models believed to host localized edge parafermions, the most prominent being a similar bosonized Hamiltonian and a $\mathbb{Z}_{N}$ symmetry enforcing an $N$-fold degenerate ground state robust to certain disorder. Interestingly, we show that the system supports a pair of parafermions but they are non-local instead of being boundary operators. As a result, the degeneracy of the ground state is only partly topological and coexists with spontaneous symmetry breaking by a (two-particle) pairing field. Each symmetry-breaking sector is shown to possess a pair of Majorana edge modes encoding the topological twofold degeneracy.

Related papers

Nontopological parafermions in a one-dimensional fermionic model with even multiplet pairing

Leonardo Mazza, Fernando Iemini, Marcello Dalmonte +1 more

Topological order and spontaneous symmetry breaking are typically thought of as paradigmatic, yet antipodal, phenomena in correlated quantum matter. Here, the authors show instead that they can concomitantly act to determine the ground-state properties of interacting systems, providing a novel angle for the discussion of topological phenomena and parafermions in one-dimensional fermionic systems. They show that topological Majorana modes can coexist with \ensuremath{\mathbb{Z}}${}_{n}$ spontaneous symmetry breaking by discussing a lattice model of one-dimensional spinless fermions and thoroughly characterize the presence of nontopological parafermions.

arXiv:1801.08548•Physical review. B./Physical review. B•Nov 14, 2018•30 citations

Parafermionic phases with symmetry breaking and topological order

A. Alexandradinata, Nicolas Regnault, Chen Fang +2 more

Parafermions are the simplest generalizations of Majorana fermions that realize topological order. We propose a less restrictive notion of topological order in one-dimensional open chains, which generalizes the seminal work by Fendley [J. Stat. Mech. (2012) P11020]. The first essential property is that the ground states are mutually indistinguishable by local, symmetric probes, and the second is a generalized notion of zero edge modes which cyclically permute the ground states. These two properties are shown to be topologically robust, and applicable to a wider family of topologically ordered Hamiltonians than has been previously considered. As an application of these edge modes, we formulate a notion of twisted boundary conditions on a closed chain, which guarantees that the closed-chain ground state is topological, i.e., it originates from the topological manifold of the open chain. Finally, we generalize these ideas to describe symmetry-breaking phases with a parafermionic order parameter. These exotic phases are condensates of parafermion multiplets, which generalize Cooper pairing in superconductors. The stability of these condensates is investigated on both open and closed chains.

arXiv:1506.03455•Physical review. B./Physical review. B•Sep 2, 2016•45 citations

Majorana edge states in interacting two-chain ladders of fermions

Meng Cheng, Hong-Hao Tu

In this work, we study interacting spinless fermions on a two-chain ladder with interchain pair tunneling while single-particle tunneling is suppressed at low energy. The model embodies a ${\mathbb{Z}}_{2}$ symmetry associated with the fermion parity on each chain. We find that when the system is driven to the strong-coupling phase by the pair tunneling, Majorana excitations appear on the boundary. Such Majorana edge states correspond to two-fold degeneracy of ground states distinguished by different fermion parity on each chain, thus representing a generalization of one-dimensional topological superconductors. We also characterize the stability of the ground-state degeneracy against local perturbations. Lattice fermion models realizing such an effective field theory are discussed.

arXiv:1106.2614•Physical Review B•Sep 12, 2011•103 citations

Dyonic zero-energy modes

Morten I. K. Munk, Asbjørn Rasmussen, Michele Burrello

One-dimensional systems with topological order are intimately related to the appearance of zero-energy modes localized on their boundaries. The most common example is the Kitaev chain, which displays Majorana zero-energy modes and it is characterized by a twofold ground-state degeneracy related to the global ${\mathbb{Z}}_{2}$ symmetry associated with fermionic parity. By extending the symmetry to the ${\mathbb{Z}}_{N}$ group, it is possible to engineer systems hosting topological parafermionic modes. In this work, we address one-dimensional systems with a generic discrete symmetry group $G$. We define a ladder model of gauge fluxes that generalizes the Ising and Potts models and displays a symmetry broken phase. Through a non-Abelian Jordan-Wigner transformation, we map this flux ladder into a model of dyonic operators, defined by the group elements and irreducible representations of $G$. We show that the so-obtained dyonic model has topological order, with zero-energy modes localized at its boundary. These dyonic zero-energy modes are in general weak topological modes, but strong dyonic zero modes appear when suitable position-dependent couplings are considered.

arXiv:1807.09286•Physical review. B./Physical review. B•Dec 21, 2018•11 citations

Edge Z3 parafermions in fermionic lattices

Raphael L. R. C. Teixeira, Luis G. G. V. Dias da Silva

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.

arXiv:2111.10147•Physical review. B./Physical review. B•May 13, 2022•5 citations

$\mathbb{Z}_3$ Parafermionic Chain Emerging From Yang-Baxter Equation

Li-Wei Yu, Mo‐Lin Ge

We construct the 1D $\mathbb{Z}_3$ parafermionic model based on the solution of Yang-Baxter equation and express the model by three types of fermions. It is shown that the $\mathbb{Z}_3$ parafermionic chain possesses both triple degenerate ground states and non-trivial topological winding number. Hence, the $\mathbb{Z}_3$ parafermionic model is a direct generalization of 1D $\mathbb{Z}_2$ Kitaev model. Both the $\mathbb{Z}_2$ and $\mathbb{Z}_3$ model can be obtained from Yang-Baxter equation. On the other hand, to show the algebra of parafermionic tripling intuitively, we define a new 3-body Hamiltonian $\hat{H}_{123}$ based on Yang-Baxter equation. Different from the Majorana doubling, the $\hat{H}_{123}$ holds triple degeneracy at each of energy levels. The triple degeneracy is protected by two symmetry operators of the system, $ω$-parity $P$($ω=e^{{\textrm{i}\frac{2π}{3}}}$) and emergent parafermionic operator $Γ$, which are the generalizations of parity $P_{M}$ and emergent Majorana operator in Lee-Wilczek model, respectively. Both the $\mathbb{Z}_3$ parafermionic model and $\hat{H}_{123}$ can be viewed as SU(3) models in color space. In comparison with the Majorana models for SU(2), it turns out that the SU(3) models are truly the generalization of Majorana models resultant from Yang-Baxter equation.

arXiv:1507.05269•Jul 19, 2015•1 citation
We discuss a one-dimensional fermionic model with a generalized ZN\mathbb{Z}_{N}ZN​ even multiplet pairing extending Kitaev Z2\mathbb{Z}_{2}Z2​ chain. The system shares many features with models believed to host localized edge parafermions, the most prominent being a similar bosonized Hamiltonian and a ZN\mathbb{Z}_{N}ZN​ symmetry enforcing an NNN-fold degenerate ground state robust to certain disorder. Interestingly, we show that the system supports a pair of parafermions but they are non-local instead of being boundary operators. As a result, the degeneracy of the ground state is only partly topological and coexists with spontaneous symmetry breaking by a (two-particle) pairing field. Each symmetry-breaking sector is shown to possess a pair of Majorana edge modes encoding the topological twofold degeneracy.
Topological order and spontaneous symmetry breaking are typically thought of as paradigmatic, yet antipodal, phenomena in correlated quantum matter. Here, the authors show instead that they can concomitantly act to determine the ground-state properties of interacting systems, providing a novel angle for the discussion of topological phenomena and parafermions in one-dimensional fermionic systems. They show that topological Majorana modes can coexist with \ensuremath{\mathbb{Z}}n{}_{n}n​ spontaneous symmetry breaking by discussing a lattice model of one-dimensional spinless fermions and thoroughly characterize the presence of nontopological parafermions.
Parafermions are the simplest generalizations of Majorana fermions that realize topological order. We propose a less restrictive notion of topological order in one-dimensional open chains, which generalizes the seminal work by Fendley [J. Stat. Mech. (2012) P11020]. The first essential property is that the ground states are mutually indistinguishable by local, symmetric probes, and the second is a generalized notion of zero edge modes which cyclically permute the ground states. These two properties are shown to be topologically robust, and applicable to a wider family of topologically ordered Hamiltonians than has been previously considered. As an application of these edge modes, we formulate a notion of twisted boundary conditions on a closed chain, which guarantees that the closed-chain ground state is topological, i.e., it originates from the topological manifold of the open chain. Finally, we generalize these ideas to describe symmetry-breaking phases with a parafermionic order parameter. These exotic phases are condensates of parafermion multiplets, which generalize Cooper pairing in superconductors. The stability of these condensates is investigated on both open and closed chains.
In this work, we study interacting spinless fermions on a two-chain ladder with interchain pair tunneling while single-particle tunneling is suppressed at low energy. The model embodies a Z2{\mathbb{Z}}_{2}Z2​ symmetry associated with the fermion parity on each chain. We find that when the system is driven to the strong-coupling phase by the pair tunneling, Majorana excitations appear on the boundary. Such Majorana edge states correspond to two-fold degeneracy of ground states distinguished by different fermion parity on each chain, thus representing a generalization of one-dimensional topological superconductors. We also characterize the stability of the ground-state degeneracy against local perturbations. Lattice fermion models realizing such an effective field theory are discussed.
One-dimensional systems with topological order are intimately related to the appearance of zero-energy modes localized on their boundaries. The most common example is the Kitaev chain, which displays Majorana zero-energy modes and it is characterized by a twofold ground-state degeneracy related to the global Z2{\mathbb{Z}}_{2}Z2​ symmetry associated with fermionic parity. By extending the symmetry to the ZN{\mathbb{Z}}_{N}ZN​ group, it is possible to engineer systems hosting topological parafermionic modes. In this work, we address one-dimensional systems with a generic discrete symmetry group GGG. We define a ladder model of gauge fluxes that generalizes the Ising and Potts models and displays a symmetry broken phase. Through a non-Abelian Jordan-Wigner transformation, we map this flux ladder into a model of dyonic operators, defined by the group elements and irreducible representations of GGG. We show that the so-obtained dyonic model has topological order, with zero-energy modes localized at its boundary. These dyonic zero-energy modes are in general weak topological modes, but strong dyonic zero modes appear when suitable position-dependent couplings are considered.
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.
Z3\mathbb{Z}_3Z3​ Parafermionic Chain Emerging From Yang-Baxter Equation
We construct the 1D Z3\mathbb{Z}_3Z3​ parafermionic model based on the solution of Yang-Baxter equation and express the model by three types of fermions. It is shown that the Z3\mathbb{Z}_3Z3​ parafermionic chain possesses both triple degenerate ground states and non-trivial topological winding number. Hence, the Z3\mathbb{Z}_3Z3​ parafermionic model is a direct generalization of 1D Z2\mathbb{Z}_2Z2​ Kitaev model. Both the Z2\mathbb{Z}_2Z2​ and Z3\mathbb{Z}_3Z3​ model can be obtained from Yang-Baxter equation. On the other hand, to show the algebra of parafermionic tripling intuitively, we define a new 3-body Hamiltonian H^123\hat{H}_{123}H^123​ based on Yang-Baxter equation. Different from the Majorana doubling, the H^123\hat{H}_{123}H^123​ holds triple degeneracy at each of energy levels. The triple degeneracy is protected by two symmetry operators of the system, ωωω-parity PPP(ω=ei2π3ω=e^{{\textrm{i}\frac{2π}{3}}}ω=ei32π​) and emergent parafermionic operator ΓΓΓ, which are the generalizations of parity PMP_{M}PM​ and emergent Majorana operator in Lee-Wilczek model, respectively. Both the Z3\mathbb{Z}_3Z3​ parafermionic model and H^123\hat{H}_{123}H^123​ can be viewed as SU(3) models in color space. In comparison with the Majorana models for SU(2), it turns out that the SU(3) models are truly the generalization of Majorana models resultant from Yang-Baxter equation.