Quasi-classical expansion of the star-triangle relation and integrable systems on quad-graphs

Abstract

In this paper we give an overview of exactly solved edge-interaction models, where the spins are placed on sites of a planar lattice and interact through edges connecting the sites. We only consider the case of a single spin degree of freedom at each site of the lattice. The Yang-Baxter equation for such models takes a particular simple form called the star-triangle relation. Interestigly all known solutions of this relation can be obtained as particular cases of a single "master solution", which is expressed through the elliptic gamma function and have continuous spins taking values of the circle. We show that in the low-temperature (or quasi-classical) limit these lattice models reproduce classical discrete integrable systems on planar graphs previously obtained and classified by Adler, Bobenko and Suris through the consistency-around-a-cube approach. We also discuss inversion relations, the physicical meaning of Baxter's rapidity-independent parameter in the star-triangle relations and the invariance of the action of the classical systems under the star-triangle (or cube-flip) transformation of the lattice, which is a direct consequence of Baxter's Z-invariance in the associated lattice models.

Final version

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Cited by in corpus (14)

• Integrable quad equations derived from the quantum Yang-Baxter equation
• Lens elliptic gamma function solution of the Yang-Baxter equation at roots of unity
• Interaction-round-a-face and consistency-around-a-face-centered-cube
• Extended Z-invariance for integrable vector and face models and multi-component integrable quad equations
• Lens partition function, pentagon identity and star-triangle relation
• Star-triangle type relations from $2d$ $\mathcal{N}=(0,2)$ $USp(2N)$ dualities
• Exactly solved models on planar graphs with vertices in ${\mathbb{Z}}^3$
• Pentagon identities arising in supersymmetric gauge theory computations
• Algebraic entropy for face-centered quad equations
• Two-component Yang-Baxter maps and star-triangle relations
• Lax matrices for lattice equations which satisfy consistency-around-a-face-centered-cube
• Superintegrable cellular automata and dual unitary gates from Yang-Baxter maps
• Quasi-classical expansion of a hyperbolic solution to the star-star relation and multicomponent 5-point difference equations
• Exactly Solved Models and Beyond: a special issue in honour of R J Baxter's 75th birthday