Bethe Ansatz and the generalized Yang-Baxter equations

R. S. Banerjee

C. Destri, Hector J. de Vega

Frank Göhmann, Er Seel, Fachbereich Physik

We solve the gl(1|2) generalized model by means of the algebraic Bethe ansatz. The resulting eigenvalue of the transfer matrix and the Bethe ansatz equations depend on three complex functions, called the parameters of the generalized model. Specifying the parameters appropriately, we obtain the Bethe ansatz equations of the supersymmetric t-J model, the Hubbard model, or of Yang’s model of electrons with delta interaction. This means that the Bethe ansatz equations of these (and many other) models can be obtained from a common algebraic source, namely from the Yang-Baxter algebra generated by the gl(1|2) invariant R-matrix.

Wen-Jie Qiu, Xi-Wen Guan, Yi-Cong Yu

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.

Chen Ning Yang

Wen-Jie Qiu, Xi-Wen Guan, Yi-Cong Yu

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 DN symmetry and its integrability. We demonstrate that the mathematical essence of integrability in this class of models is characterized by a newly proposed Kaleidoscope Yang-Baxter Equation. 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 method 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 series of novel quantum algebraic identities within the framework of quantum torus algebra.