Kaleidoscope Yang-Baxter equation for Gaudin’s kaleidoscope models

Recently, researchers have proposed the Asymmetric Bethe Ansatz method, a theoretical tool that extends the scope of Bethe Ansatz-solvable models by “breaking” partial mirror symmetry via the introduction of a fully reflecting boundary. Within this framework, the integrability conditions which were originally put forward by Gaudin have been further generalized. In this work, building on Gaudin’s generalized kaleidoscope model, we present a detailed investigation of the relationship between dihedral symmetry and its Bethe-type solvability. We identify a scattering-consistency relation, which we refer to as the Kaleidoscope Yang-Baxter Equation (KYBE), and use it to organize the Bethe-type construction in the present two-dimensional setting. Furthermore, we show that the solvability of a model via the coordinate Bethe Ansatz depends not only on the consistency relations satisfied by scattering matrices, but also on the model’s boundary conditions and the symmetry of the subspace where solutions are sought. Through finite-element-based numerical studies, we further confirm that Bethe Ansatz integrability arises in a specific symmetry sector. Finally, by analyzing the algebraic structure of the Kaleidoscope Yang-Baxter Equation, we derive a family of algebraic identities within the framework of quantum torus algebra.