Fusion of \ade Lattice Models

TL;DR

This paper constructs fusion hierarchies of ext{ADE} face models, producing new solutions to the Yang-Baxter equations and explicit fused face weights, connecting various integrable models.

Contribution

It introduces fused ext{ADE} face models with new solutions to Yang-Baxter equations and explicit fused weights, including applications to Potts, Ising, and CSOS models.

Findings

Fused models satisfy special functional equations.

Explicit fused face weights for Potts and Ising models.

Fusion yields models with bond and spin variables.

Abstract

Fusion hierarchies of \ade face models are constructed. The fused critical $D$, $E$ and elliptic $D$ models yield new solutions of the Yang-Baxter equations with bond variables on the edges of faces in addition to the spin variables on the corners. It is shown directly that the row transfer matrices of the fused models satisfy special functional equations. Intertwiners between the fused \ade models are constructed by fusing the cells that intertwine the elementary face weights. As an example, we calculate explicitly the fused $2\times 2$ face weights of the 3-state Potts model associated with the $D_4$ diagram as well as the fused intertwiner cells for the $A_5$--$D_4$ intertwiner. Remarkably, this $2\times 2$ fusion yields the face weights of both the Ising model and 3-state CSOS models.