The second $\mathbb{Z}_{3}$ parafermionic conformal theories are associated with the coset construction $\frac{SU(2)_{k}\times SU(2)_{4}}{SU(2)_{k+4}} $. Solid-on-solid integrable lattice models obtained by fusion of the model based on level-1 representation of the affine algebra $B_1^{(1)}$ have a critical point described by these conformal theories. Explicit values for the Boltzmann weights are derived for these models, and it is shown that the Boltzmann weights can be made positive for a particular value of the spectral parameter, opening a way to eventual numerical simulations of these conformal field theories. Away from criticality, these lattice models describe an integrable, massive perturbation of the parafermionic conformal theory by the relevant field $Ψ_{-2/3}^{\dagger}D_{1,3} $.The second $\mathbb{Z}_{3}$ parafermionic conformal theories are associated with the coset construction $\frac{SU(2)_{k}\times SU(2)_{4}}{SU(2)_{k+4}} $. Solid-on-solid integrable lattice models obtained by fusion of the model based on level-1 representation of the affine algebra $B_1^{(1)}$ have a critical point described by these conformal theories. Explicit values for the Boltzmann weights are derived for these models, and it is shown that the Boltzmann weights can be made positive for a particular value of the spectral parameter, opening a way to eventual numerical simulations of these conformal field theories. Away from criticality, these lattice models describe an integrable, massive perturbation of the parafermionic conformal theory by the relevant field $Ψ_{-2/3}^{\dagger}D_{1,3} $.

arXiv:0812.3473Dec 18, 20080 citations

Critical correlation functions of the Potts model

Bernard Nienhuis

Physica A Statistical Mechanics and its ApplicationsMar 1, 19981 citation

Z3 and (×Z3)3 symmetry protected topological paramagnets

Hrant Topchyan, Vasilii Iugov, Mkhitar Mirumyan +3 more

Abstract We identify two-dimensional three-state Potts paramagnets with gapless edge modes on a triangular lattice protected by (×Z3)3 ≡ Z3 × Z3 × Z3 symmetry and smaller Z3 symmetry. We derive microscopic models for the gapless edge, uncover their symmetries and analyze the conformal properties. We study the properties of the gapless edge by employing the numerical density-matrix renormalization group (DMRG) simulation and exact diagonalization. We discuss the corresponding conformal field theory, its central charge, and the scaling dimension of the corresponding primary field. We argue, that the low energy limit of our edge modes defined by the SUk(3)/SUk(2) coset conformal field theory with the level k = 2. The discussed two-dimensional models realize a variety of symmetry-protected topological phases, opening a window for studies of the unconventional quantum criticalities between them.Abstract We identify two-dimensional three-state Potts paramagnets with gapless edge modes on a triangular lattice protected by (×Z3)3 ≡ Z3 × Z3 × Z3 symmetry and smaller Z3 symmetry. We derive microscopic models for the gapless edge, uncover their symmetries and analyze the conformal properties. We study the properties of the gapless edge by employing the numerical density-matrix renormalization group (DMRG) simulation and exact diagonalization. We discuss the corresponding conformal field theory, its central charge, and the scaling dimension of the corresponding primary field. We argue, that the low energy limit of our edge modes defined by the SUk(3)/SUk(2) coset conformal field theory with the level k = 2. The discussed two-dimensional models realize a variety of symmetry-protected topological phases, opening a window for studies of the unconventional quantum criticalities between them.

arXiv:2210.01187Journal of High Energy PhysicsDec 29, 20236 citations

Hexagon and pentagon identities for the Z3 Potts model

Shijong Ryang

Investigating the transformation properties of the conformal blocks in the ${Z}_{3}$ Potts model we derive some braid matrices. From the obtained braid matrices we explicitly show how the hexagon and pentagon identities are satisfied.Investigating the transformation properties of the conformal blocks in the ${Z}_{3}$ Potts model we derive some braid matrices. From the obtained braid matrices we explicitly show how the hexagon and pentagon identities are satisfied.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsApr 15, 19910 citations

A lattice model for the second

\mathbb{Z}_{3}

parafermionic field theory

The second

\mathbb{Z}_{3}

parafermionic conformal theories are associated with the coset construction $\frac{SU(2)_{k}\times SU(2)_{4}}{SU(2)_{k+4}} $. Solid-on-solid integrable lattice models obtained by fusion of the model based on level-1 representation of the affine algebra

B_1^{(1)}

have a critical point described by these conformal theories. Explicit values for the Boltzmann weights are derived for these models, and it is shown that the Boltzmann weights can be made positive for a particular value of the spectral parameter, opening a way to eventual numerical simulations of these conformal field theories. Away from criticality, these lattice models describe an integrable, massive perturbation of the parafermionic conformal theory by the relevant field $Ψ_{-2/3}^{\dagger}D_{1,3} $.

Investigating the transformation properties of the conformal blocks in the

{Z}_{3}

Potts model we derive some braid matrices. From the obtained braid matrices we explicitly show how the hexagon and pentagon identities are satisfied.