Bethe ansatz inside Calogero--Sutherland models

Abstract

The Haldane--Shastry spin chain has long-range interactions and remarkable properties including Yangian symmetry at finite length and explicit highest-weight wave functions featuring Jack polynomials. This stems from the trigonometric spin-Calogero--Sutherland model, which is intimately related to affine Hecke algebras, already enjoys these properties from affine Schur–Weyl duality and reduces to the Haldane--Shastry chain in the ‘freezing’ limit. I will present some new results for these models, including Heisenberg-like symmetries whose spectrum can be characterised by Bethe ansatz. Based on recent work with D. Serban and ongoing work with G. Ferrando, F. Levkovich-Maslyuk and D. Serban.