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

TL;DR

This paper clarifies how the fermionic spin-Calogero-Sutherland model, in a freezing limit, precisely reproduces the Haldane-Shastry chain's wave functions and eigenvectors, illuminating their algebraic and symmetry properties.

Contribution

It demonstrates that freezing the fermionic spin-1/2 Calogero-Sutherland model naturally reproduces Haldane-Shastry wave functions and proves the origin of Yangian highest-weight eigenvectors in the $SU(r)$ case.

Findings

Fermionic model accounts for Vandermonde squared in wave functions

Freezing at $ ext{alpha}=1/2$ yields Haldane-Shastry eigenvectors

Yangian eigenvectors arise from frozen $SU(r-1)$ Calogero-Sutherland states

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 $\alpha = 1/2$ 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 $\mathit{fermionic}$ 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 $\alpha = 1/2$.