A lattice model for the second $\mathbb{Z}_{3}$ parafermionic field theory

TL;DR

This paper constructs a lattice model for the second $ ext{Z}_3$ parafermionic conformal field theory, providing explicit Boltzmann weights and demonstrating their positivity at criticality, enabling potential numerical simulations.

Contribution

It introduces a new lattice model based on fusion of affine algebra representations that captures the second $ ext{Z}_3$ parafermionic theory and details its critical and off-critical behavior.

Findings

Explicit Boltzmann weights derived for the lattice model

Boltzmann weights can be positive at a specific spectral parameter

Model describes an integrable massive perturbation of the conformal theory

Abstract

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 $\Psi_{-2/3}^{\dagger}D_{1,3} $.