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In a previous paper, the author has established an extension of theZ-invariance property for integrable edge-interaction models of statisticalmechanics, that satisfy the star-triangle relation (STR) form of theYang-Baxter equation (YBE). In the present paper, an analogous extendedZ-invariance property is shown to also hold for integrable vector models andinteraction-round-a-face (IRF) models of statistical mechanics respectively. Asfor the previous case of the STR, the Z-invariance property is shown throughthe use of local cubic-type deformations of a 2-dimensional surface associatedto the models, which allow an extension of the models onto a subset of nextnearest neighbour vertices of $\mathbb{Z}^3$, while leaving the partitionfunctions invariant. These deformations are permitted as a consequence of therespective YBE's satisfied by the models. The quasi-classical limit is alsoconsidered, and it is shown that an analogous Z-invariance property holds forthe variational formulation of classical discrete Laplace equations which arisein this limit. From this limit, new integrable 3D-consistent multi-componentquad equations are proposed, which are constructed from a degeneration of theequations of motion for IRF Boltzmann weights.

Extended Z-invariance for integrable vector and face models and multi-component integrable quad equations