From fermionic spin-Calogero-Sutherland models to the Haldane-Shastry chain by freezing
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913365981396992 |
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| author | Lamers, Jules Serban, Didina |
| author_facet | Lamers, Jules Serban, Didina |
| contents | 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 $\mathit{fermionic}$ 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 $α= 1/2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2212_01373 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | From fermionic spin-Calogero-Sutherland models to the Haldane-Shastry chain by freezing Lamers, Jules Serban, Didina Mathematical Physics Strongly Correlated Electrons High Energy Physics - Theory Exactly Solvable and Integrable Systems 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 $\mathit{fermionic}$ 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 $α= 1/2$. |
| title | From fermionic spin-Calogero-Sutherland models to the Haldane-Shastry chain by freezing |
| topic | Mathematical Physics Strongly Correlated Electrons High Energy Physics - Theory Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2212.01373 |