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

Abstract

In a previous paper, the author has established an extension of the Z-invariance property for integrable edge-interaction models of statistical mechanics, that satisfy the star-triangle relation (STR) form of the Yang-Baxter equation (YBE). In the present paper, an analogous extended Z-invariance property is shown to also hold for integrable vector models and interaction-round-a-face (IRF) models of statistical mechanics respectively. As for the previous case of the STR, the Z-invariance property is shown through the use of local cubic-type deformations of a 2-dimensional surface associated to the models, which allow an extension of the models onto a subset of next nearest neighbour vertices of ${\mathbb{Z}}^3$, while leaving the partition functions invariant. These deformations are permitted as a consequence of the respective YBE's satisfied by the models. The quasi-classical limit is also considered, and it is shown that an analogous Z-invariance property holds for the variational formulation of classical discrete Laplace equations which arise in this limit. From this limit, new integrable 3D-consistent multi-component quad equations are proposed, which are constructed from a degeneration of the equations of motion for IRF Boltzmann weights.

arXiv comment39 pages, 26 figures; v2: typos, improvements to text; v3: additional comments

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Cited by in corpus (8)

• Interaction-round-a-face and consistency-around-a-face-centered-cube
• Non-Abelian hierarchies of compatible maps, associated integrable difference systems and Yang-Baxter maps
• Lax matrices for lattice equations which satisfy consistency-around-a-face-centered-cube
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