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 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 startriangle 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.

Integrability as duality: The Gauge/YBE correspondence

“…For this reason, we are free to disregard the quiver diagram and keep only the zig-zag paths. 23 If we do this for the left figure of Figure 12 and deform the paths (while keeping the topology of the graph), we arrive at the right figure of Figure 12, which is very similar to the picture for the YBE (recall Figure 1). We can also represent the SSR and the YBE in terms of the zig-zag paths, as in Figures 14 and 15.…”

Extended Z-Invariance for Integrable Vector and Face Models and Multi-component Integrable Quad Equations

“…For the previously studied cases of the quasi-classical limit of scalar solutions of the startriangle relation [15][16][17][18][19], the equation for the critical point of the latter relation was always found to be equivalent to a 3D-consistent integrable quad equation from the ABS classification [7,28]. This connection means that the star-triangle relation itself has a natural interpretation as being a quantum counterpart (in a path integral sense) of a discrete integrable equation.…”

“…The quasi-classical limit of the IRF model is also considered in this paper. This is an important limit that connects integrable models of statistical mechanics [15][16][17][18][19], with discrete integrable systems that satisfy an integrability condition known as 3D-consistency [7,20,21]. Through this connection, the Yang-Baxter equation itself may be interpreted as a quantum counterpart of a discrete integrable equation, where the latter equation is identified as the equation of the saddle-point of the YBE.…”

“…In the quasi-classical limit, the fluctuating variables of the model are taken to approach a fixed ground state configuration, which is determined as the solution of a classical discrete integrable equation. For integrable models that satisfy a star-triangle relation, this limit has been found [15][16][17][18][19] to always lead to a classical discrete integrable equation in the Adler-Bobenko-Suris (ABS) classification [7,28], where the latter equations are equivalent to the equations of the critical point of the star-triangle relation. This correspondence has recently been extended to the entire ABS list, with the use of star-triangle relations that are based on hypergeometric integrals [18,19].…”

“…Interestingly, the hyperbolic-and elliptic-type equations in (69), and (70), don't require a point transformation of the components of the corner variables x, u, y, v, in terms of hyperbolic or elliptic functions respectively, in order to go from the four-leg form (63), to the affine-linear form (71). Particularly, such a point transformation is typically always required for the scalar cases, in order to relate a 3D-consistent quad equation to a Yang-Baxter equation [18,19] (or even simply for a three-leg equation [7]). The equations (69), and (70), are curious in this respect, and are unlikely to arise from a counterpart multi-component Yang-Baxter equation.…”