The Yang-Baxter Equation for the Chiral Potts Model and Integrable Parafermions

Abstract

A new type of Yang-Baxter equation (YBE) for $R$-operators depending on three spectral parameters is constructed from the star-triangle relation for the chiral Potts model. As the $Z_N$ symmetric generalization to the Ising model, its Boltzmann weights are known to depend on two variables describing a curve with genus larger than one for $N>2$, except for the self-dual point corresponding to the Fateev-Zamolodchikov chain. Combined with the fact that quantum Hamiltonians of edge-type models such as the Ising model contain both nearest-neighbor interaction and onsite potential terms, this leads naturally to an additional spectral parameter in the associated $R$-operator. The construction extends the edge-vertex correspondence of solvable lattice models, and provides a bridge between the Bazhanov-Stroganov four-parameter $R$-matrix---realized as an intertwiner of cyclic representations of $U_q({\mathfrak{sl}}_2)$ at a root of unity---and Shastry's two-parameter $R$-operator obtained from the decorated YBE.

arXiv comment27 pages, 7 figures