To study exotic excitations described by parafermions in the possible spin liquid states of $\mathrm{SU}(n)$ spin systems, we introduce a parafermion parton approach. The $\mathrm{SU}(n)$ spin operators can be represented by clock and shift matrices, which are shown to be the polynomials of parafermion operators in the parafermion representation. We find that $\mathrm{SU}(n)$ spins can be decomposed into $n$ parafermion matrices of degree one. In this decomposition, the spin has a ${{⨂\mathrm{SU}(n)}}^{n\ensuremath{-}1}$ gauge symmetry. As an application, we study the one-dimensional three-state clock model and generalized Kitaev model by a mean-field theory; both of them have been proved to be related to parafermion excitations. We find that with the symmetries of translations, sixfold rotation, and combination of parity and time reversal, there are 9 types and 102 solutions for two-dimensional ${\mathbb{Z}}_{3}$ parafermion spin liquids on the honeycomb lattice. By contrast, there are 9 types and 36 solutions if both parity and time-reversal symmetries are present. Our results provide a route for the systematic search of types of spin liquids with exotic anyon excitations.To study exotic excitations described by parafermions in the possible spin liquid states of $\mathrm{SU}(n)$ spin systems, we introduce a parafermion parton approach. The $\mathrm{SU}(n)$ spin operators can be represented by clock and shift matrices, which are shown to be the polynomials of parafermion operators in the parafermion representation. We find that $\mathrm{SU}(n)$ spins can be decomposed into $n$ parafermion matrices of degree one. In this decomposition, the spin has a ${{⨂\mathrm{SU}(n)}}^{n\ensuremath{-}1}$ gauge symmetry. As an application, we study the one-dimensional three-state clock model and generalized Kitaev model by a mean-field theory; both of them have been proved to be related to parafermion excitations. We find that with the symmetries of translations, sixfold rotation, and combination of parity and time reversal, there are 9 types and 102 solutions for two-dimensional ${\mathbb{Z}}_{3}$ parafermion spin liquids on the honeycomb lattice. By contrast, there are 9 types and 36 solutions if both parity and time-reversal symmetries are present. Our results provide a route for the systematic search of types of spin liquids with exotic anyon excitations.

arXiv:1711.07149Physical review. B./Physical review. BDec 11, 20175 citations

Spinons and parafermions in fermion cosets

D. C. Cabra

arXiv:hep-th/9704015Springer eBooksMay 15, 20070 citations

Parafermionization, bosonization, and critical parafermionic theories

Yuan Yao, Akira Furusaki

Abstract We formulate a ℤk-parafermionization/bosonization scheme for one-dimensional lattice models and field theories on a torus, starting from a generalized Jordan-Wigner transformation on a lattice, which extends the Majorana-Ising duality atk= 2. The ℤk-parafermionization enables us to investigate the critical theories of parafermionic chains whose fundamental degrees of freedom are parafermionic, and we find that their criticality cannot be described by any existing conformal field theory. The modular transformations of these parafermionic low-energy critical theories as general consistency conditions are found to be unconventional in that their partition functions on a torus transform differently from any conformal field theory whenk >2. Explicit forms of partition functions are obtained by the developed parafermionization for a large class of critical ℤk-parafermionic chains, whose operator contents are intrinsically distinct from any bosonic or fermionic model in terms of conformal spins and statistics. We also use the parafermionization to exhaust all the ℤk-parafermionic minimal models, complementing earlier works on fermionic cases.Abstract We formulate a ℤk-parafermionization/bosonization scheme for one-dimensional lattice models and field theories on a torus, starting from a generalized Jordan-Wigner transformation on a lattice, which extends the Majorana-Ising duality atk= 2. The ℤk-parafermionization enables us to investigate the critical theories of parafermionic chains whose fundamental degrees of freedom are parafermionic, and we find that their criticality cannot be described by any existing conformal field theory. The modular transformations of these parafermionic low-energy critical theories as general consistency conditions are found to be unconventional in that their partition functions on a torus transform differently from any conformal field theory whenk >2. Explicit forms of partition functions are obtained by the developed parafermionization for a large class of critical ℤk-parafermionic chains, whose operator contents are intrinsically distinct from any bosonic or fermionic model in terms of conformal spins and statistics. We also use the parafermionization to exhaust all the ℤk-parafermionic minimal models, complementing earlier works on fermionic cases.

arXiv:2012.07529Journal of High Energy PhysicsApr 1, 202125 citations

To study exotic excitations described by parafermions in the possible spin liquid states of

\mathrm{SU}(n)

spin systems, we introduce a parafermion parton approach. The

\mathrm{SU}(n)

spin operators can be represented by clock and shift matrices, which are shown to be the polynomials of parafermion operators in the parafermion representation. We find that

\mathrm{SU}(n)

spins can be decomposed into

n

parafermion matrices of degree one. In this decomposition, the spin has a ${{⨂\mathrm{SU}(n)}}^{n\ensuremath{-}1}$ gauge symmetry. As an application, we study the one-dimensional three-state clock model and generalized Kitaev model by a mean-field theory; both of them have been proved to be related to parafermion excitations. We find that with the symmetries of translations, sixfold rotation, and combination of parity and time reversal, there are 9 types and 102 solutions for two-dimensional

{\mathbb{Z}}_{3}

parafermion spin liquids on the honeycomb lattice. By contrast, there are 9 types and 36 solutions if both parity and time-reversal symmetries are present. Our results provide a route for the systematic search of types of spin liquids with exotic anyon excitations.