A lattice model for the second Z3 parafermionic field theory

The second Z 3 parafermionic conformal theories are associated with the coset construction SU (2) k ×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 D 1,3. Over the past twenty years Conformal Field Theories (CFT) have proven to be a very powerful tool to describe two dimensional statistical systems at criticality. Recently a new way to study critical systems has been developed : Schramm Loewner Evolution (SLE) [1, 2]. In this approach to conformally invariant systems, clusters are described through the conformal properties of their domain walls. While the connection between minimal models (c < 1) and SLE is relatively well understood, it is still an open problem for CFTs possessing extra symmetries. Statistical systems with a discrete symmetry Z N are believed to be described by parafermionic theories. The first of such theories, constructed by Fateev and Zamolodchikov [3], have been studied in the context of SLE [5, 6], and using particular lattice realizations of the first parafermions, numerical studies of fractal interfaces have been performed [7, 8, 9].