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

Journal of Mathematical Physics, Volume 62, Issue 3, March 2021. There is a correspondence between integrable lattice models of statistical mechanics and discrete integrable equations that 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 integrability condition applicable to five-point equations 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 14 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 with the Adler–Bobenko–Suris list, 15 sets of equations are obtained, which satisfy CAFCC.