University of Sydney Algebra Seminar

Luc Frappat (Universite de Savoie)

Friday 31 October, 12:05-12:55pm, Place: 373

Three-state Hamiltonians solvable by the coordinate Bethe ansatz, classification and R matrices

R-matrices, solutions of the celebrated Yang-Baxter equation, are a cornerstone of the resolution of quantum integrable systems. They contain the Hamiltonian of the sytem and constitute the basic ingredient of the Algebraic Bethe Ansatz (ABA) that provides the eigenvalues and eigenfunctions of the model. Historically, solving such models was done through the Coordinate Bethe Ansatz (CBA), but the underlying mathematical structure was not manifest. After presenting and comparing the ABA and the CBA, we review some of the strategies that can be implemented to infer an R-matrix from the knowledge of its Hamiltonian. We then apply this framework to the case of three-state Hamiltonians with rank 1 symmetry and nearest-neighbour interactions in the context of spin chains. We provide the classification of such Hamiltonians and raise some open questions.