### 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 on 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.

Original language | English |
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Article number | 464001 |

Pages (from-to) | 1-44 |

Number of pages | 44 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 49 |

Issue number | 46 |

DOIs | |

Publication status | Published - 21 Oct 2016 |

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## Cite this

Bazhanov, V. V., Kels, A., & Sergeev, S. (2016). Quasi-classical expansion of the star-triangle relation and integrable systems on quad-graphs.

*Journal of Physics A: Mathematical and Theoretical*,*49*(46), 1-44. [464001]. https://doi.org/10.1088/1751-8113/49/46/464001