Quasi-classical expansion of a hyperbolic solution to the star-star relation and multicomponent 5-point difference equations

TL;DR

This paper studies the quasi-classical expansion of a multicomponent ($n$-component) hyperbolic star-star solution in edge-interaction lattice models, using hyperbolic Boltzmann weights built from the hyperbolic gamma function. The authors scale variables with $\hbar$ and derive leading-order Lagrangian densities, yielding $n-1$ component 5-point difference equations that generalize known scalar equations on face-centered cubic lattices. In the hyperbolic and rational limits, these equations provide multicomponent extensions of scalar 5-point equations (e.g., $A3_{(1)}$ for $n=2$ and $A2_{(1;1)}$ in the rational $n=2$ case), with integrability conjectured to follow from consistency-around-a-face-centered-cube (CAFCC) and IRF Yang-Baxter structures. The work highlights a deep connection between integrable lattice models and discrete integrable systems, offering a pathway to analyze Lax pairs and CAFCC for $n>2$ and to explore multicomponent reductions and related hex systems. Overall, the paper extends the bridge between star-star integrable models and multicomponent discrete equations, enriching the landscape of discrete integrability.

Abstract

The quasi-classical expansion of a multicomponent spin solution of the star-star relation with hyperbolic Boltzmann weights is investigated. The equations obtained in a quasi-classical limit provide n-1-component extensions of certain scalar 5-point equations (corresponding to n=2) that were previously investigated by the author in the context of integrability and consistency of equations on face-centered cubics.

Figures (8)

• Figure 1: Checkerboard square lattice and directed rapidity lines.
• Figure 2: Four different types of edges belonging to the sets $E^{(1)}$, $E^{(2)}$, $E^{(3)}$, and $E^{(4)}$, respectively.
• Figure 3: The two types of four-edge stars that appear in the checkerboard square lattice.
• Figure 4: Star-star relation
• Figure 5: A vertex configuration for \ref{['Adefhyp']}.

Theorems & Definitions (12)

• Theorem 3.1
• Proposition 3.2
• proof
• Proposition 3.3
• Lemma 1
• proof
• Remark 1
• Proposition 3.4
• proof
• Remark 2