Luc Frappat, Laboratoire d'Annecy-le-Vieux de Physique Théorique LAPTh, Univ. Grenoble Alpes, Univ. Savoie

Solving quantum integrable models with Bethe Ansatz - Application to U(1)-invariant three-state Hamiltonians

Quantum integrable systems have a long history. Originally, solving such models was done through the Coordinate Bethe Ansatz, while the underlying mathematical structure was not manifest. In the eighties, the R-matrices, solutions of the celebrated Yang-Baxter equation, has become a cornerstone of the resolution of such systems. R-matrices contain the Hamiltonian of the system and constitute the basic ingredient of the Algebraic Bethe Ansatz that provides the eigenvalues and eigenfunctions of the model. After presenting and comparing the two ansatz, we review some of the strategies that can be implemented to infer an R-matrix from the knowledge of its Hamiltonian, and apply this framework to the case of three-state Hamiltonians with rank 1 symmetry and nearest-neighbour interactions in the context of spin chains.