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The second $\mathbb{Z}_{3}$ parafermionic conformal theories are associatedwith 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 basedon level-1 representation of the affine algebra $B_1^{(1)}$ have a criticalpoint described by these conformal theories. Explicit values for the Boltzmannweights are derived for these models, and it is shown that the Boltzmannweights can be made positive for a particular value of the spectral parameter,opening a way to eventual numerical simulations of these conformal fieldtheories. Away from criticality, these lattice models describe an integrable,massive perturbation of the parafermionic conformal theory by the relevantfield $\Psi_{-2/3}^{\dagger}D_{1,3} $.

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