Introduction to the nested algebraic Bethe ansatz

Abstract

We give a detailed description of the nested algebraic Bethe ansatz. We consider integrable models with a ${\mathfrak{gl}}_3$-invariant $R$-matrix as the basic example, however, we also describe possible generalizations. We give recursions and explicit formulas for the Bethe vectors. We also give a representation for the Bethe vectors in the form of a trace formula.

59 pages, 6 figures, lectures at the Les Houches summer school 2018

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• Bethe ansatz inside Calogero-Sutherland models
• Superintegrable cellular automata and dual unitary gates from Yang-Baxter maps
• Fractal decompositions and tensor network representations of Bethe wavefunctions
• Symmetries of the periodic Fredkin chain
• Multiple commutation relations of the quantum affine algebra $U_q({\widehat{\mathfrak{gl}}}_N)$, nested Bethe vector and the Gelfand-Tsetlin basis