From fermionic spin-Calogero–Sutherland models to the Haldane–Shastry chain by freezing

Abstract

The Haldane–Shastry spin chain has a myriad of remarkable properties, including Yangian symmetry and, for spin 1/2 , explicit highest-weight eigenvectors featuring (the case α=1/2 of) Jack polynomials. This stems from the spin-Calogero–Sutherland model, which reduces to Haldane–Shastry in a special ‘freezing’ limit. In this work we clarify various points that, to the best of our knowledge, were missing in the literature. We have two main results. First, we show that freezing the fermionic spin-1/2 Calogero–Sutherland model naturally accounts for the precise form of the Haldane–Shastry wave functions, including the Vandermonde factor squared. Second, we use the fermionic framework to prove the claim of Bernard–Gaudin–Haldane–Pasquier that the Yangian highest-weight eigenvectors of the SU(r) -version of the Haldane–Shastry chain arise by freezing SU(r−1) spin-Calogero–Sutherland eigenvectors at α=1/2 .