We are interested in the problem of understanding the structure of the Bethe roots for the periodic spin-1/2 Heisenberg XXZ chain. To probe this structure it is fruitful to study what happens as one varies the parameters. We start with the (ferromagnetic) Ising limit ∆ → ∞, in which ​the Bethe roots become explicit, and strings of Bethe roots become directly visible on the lattice, already examined by Gaudin. Despite the simplicity of the limiting model, there are some subtleties in the limit. Next we study what happens with the explicit Ising Bethe roots as ∆ is lowered, where bound states may form or unbind. I will present a ‘critical equation’ that determines when Bethe roots switch between real and complex for the sector with M = 2 magnons, generalising Essler–Korepin–Schoutens (1992, ∆ = 1) and Imoto–Sato–Deguchi (2019, ∆ ≥ 1), and discuss some results for M > 2. If time allows, I will outline how a similar game can be played by turning on long-range interactions, going from Heisenberg XXX, via the Inozemtsev chain, to the Haldane-Shasatry chain.

This is based on joint work J.-S. Caux and R. Koch, to appear, and with R. Klabbers (2022).