Bazhanov, Vladimir V.; Kels, Andrew P.; Sergeev, Sergey M. Quasi-classical expansion of the star-triangle relation and integrable systems on quad-graphs. (English) Zbl 1352.37175 J. Phys. A, Math. Theor. 49, No. 46, Article ID 464001, 44 p. (2016). Summary: 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. Cited in 19 Documents MSC: 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 37K60 Lattice dynamics; integrable lattice equations 16T25 Yang-Baxter equations 82B10 Quantum equilibrium statistical mechanics (general) 81T10 Model quantum field theories Keywords:Yang-Baxter equation; star-triangle relation; discrete evolution equations; exactly solvable lattice models; 3D consistency Software:CirclePack PDFBibTeX XMLCite \textit{V. V. Bazhanov} et al., J. Phys. A, Math. Theor. 49, No. 46, Article ID 464001, 44 p. (2016; Zbl 1352.37175) Full Text: DOI arXiv