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Scientific Reports

ℤ3 parafermionic chain emerging from Yang-Baxter equation

Li-Wei Yu, Mo‐Lin Ge

Published Feb 23, 2016• 10 citations
Publisher

Abstract

We construct the 1D ℤ3 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 parafermionic chain possesses both triple degenerate ground states and non-trivial topological winding number. Hence, the ℤ3 parafermionic model is a direct generalization of 1D ℤ2 Kitaev model. Both the ℤ2 and ℤ3 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 H123 based on Yang-Baxter equation. Different from the Majorana doubling, the H123 holds triple degeneracy at each of energy levels. The triple degeneracy is protected by two symmetry operators of the system, ω-parity P [formula in text] and emergent parafermionic operator Γ, which are the generalizations of parity PM and emergent Majorana operator in Lee-Wilczek model, respectively. Both the ℤ3 parafermionic model and H123 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.

DOI
10.1038/srep21497

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We construct the 1D ℤ3 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 parafermionic chain possesses both triple degenerate ground states and non-trivial topological winding number. Hence, the ℤ3 parafermionic model is a direct generalization of 1D ℤ2 Kitaev model. Both the ℤ2 and ℤ3 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 H123 based on Yang-Baxter equation. Different from the Majorana doubling, the H123 holds triple degeneracy at each of energy levels. The triple degeneracy is protected by two symmetry operators of the system, ω-parity P [formula in text] and emergent parafermionic operator Γ, which are the generalizations of parity PM and emergent Majorana operator in Lee-Wilczek model, respectively. Both the ℤ3 parafermionic model and H123 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.
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.
A new realization of doubling degeneracy based on emergent Majorana operator Γ presented by Lee-Wilczek has been made. The Hamiltonian can be obtained through the new type of solution of Yang-Baxter equation, i.e. -matrix. For 2-body interaction, R(θ) gives the "superconducting" chain that is the same as 1D Kitaev chain model. The 3-body Hamiltonian commuting with Γ is derived by 3-body R₁₂₃-matrix, we thus show that the essence of the doubling degeneracy is due to [R(θ), Γ=0]. We also show that the extended Γ'-operator is an invariant of braid group BN for odd N. Moreover, with the extended Γ'-operator, we construct the high dimensional matrix representation of solution to Yang-Baxter equation and find its application in constructing 2N-qubit Greenberger-Horne-Zeilinger state for odd N.
Starting from the Kauffman-Lomonaco braiding matrix transforming the natural basis to Bell states, the spectral parameter describing the entanglement is introduced through Yang-Baxterization. It gives rise to a new type of solutions for Yang-Baxter equation, called the type-II that differs from the familiar solution called type-I of YBE associated with the usual chain models. The Majorana fermionic version of type-II yields the Kitaev Hamiltonian. The introduced ℓ 1 -norm leads to the maximum of the entanglement by taking the extreme value and shows that it is related to the Wigner's D-function. Based on the Yang-Baxter equation the 3-body S-Matrix for type-II is explicitly given. Different from the type-I solution, the type-II solution of YBE should be considered in describing quantum information. The idea is further extended to 3 parafermion model based on SU (3) principal representation. The type-II is in difference from the familiar type-I in many respects. For example, the quantities corresponding to velocity in the chain models obey the Lorentzian additivity rather than Galilean rule ( u + v ). Most possibly, for the type-II solutions of YBE there may not exist RTT relation. Further more, for 3 parafermion model we only need the rational Yang-Baxterization, which seems like trigonometric. Similar discussions are also made in terms of generalized Yang-Baxter equation with three spin spaces .
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 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.
In the Fock representation, we propose a framework to construct the generalized matrix product states (MPS) for topological phases with Zp{\mathbb{Z}}_{p}Zp​ parafermions. Unlike the Z2{\mathbb{Z}}_{2}Z2​ Majorana fermions, the Zp{\mathbb{Z}}_{p}Zp​ parafermions form intrinsically interacting systems. Here we explicitly construct two topologically distinct classes of irreducible Z3{\mathbb{Z}}_{3}Z3​ parafermionic MPS wave functions, characterized by one or two parafermionic zero modes at each end of an open chain. Their corresponding parent Hamiltonians are found as the fixed point models of the single Z3{\mathbb{Z}}_{3}Z3​ parafermion chain and two-coupled parafermion chains with ${\mathbb{Z}}_{3}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{3}$ symmetry. Our results thus pave the road to investigate all possible topological phases with Zp{\mathbb{Z}}_{p}Zp​ parafermions within the matrix product representation in one dimension.