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

## TL;DR

This paper extends the Z-invariance property to integrable vector and face models in statistical mechanics, demonstrating invariance under local deformations and deriving new multi-component integrable quad equations from classical limits.

## Contribution

It generalizes Z-invariance to vector and IRF models and introduces new integrable multi-component quad equations from classical limits.

## Key findings

- Z-invariance holds for vector and IRF models via local cubic deformations.
- Partition functions remain invariant under certain 3D surface deformations.
- New multi-component quad equations are derived from classical limits.

## 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.