Interaction-round-a-face and consistency-around-a-face-centered-cube

There is a correspondence between integrable lattice models of statistical mechanics and discrete integrable equations which satisfy multidimensional consistency, where the latter may be found in a quasi-classical expansion of the former. This paper extends this correspondence to interaction-round-a-face (IRF) models, resulting in a new formulation of the consistency-around-a-cube (CAC) integrability condition that is applicable to five-point equations which are defined on a vertex and its four nearest-neighbours in the square lattice. Multidimensional consistency for these equations is formulated as consistency-around-a-face-centered-cube (CAFCC), which namely involves satisfying an overdetermined system of fourteen five-point lattice equations for eight unknown variables on the face-centered cubic unit cell. From the quasi-classical limit of IRF models, which are constructed from the continuous spin solutions of the star-triangle relations associated to the Adler-Bobenko-Suris (ABS) list, fifteen sets of equations are obtained which satisfy CAFCC.