One-Dimensional Vertex Models Associated with a Class of Yangian Invariant Haldane-Shastry Like Spin Chains

We define a class of $Y(sl_{(m|n)})$ Yangian invariant Haldane-Shastry (HS) like spin chains, by assuming that their partition functions can be written in a particular form in terms of the super Schur polynomials. Using some properties of the super Schur polynomials, we show that the partition functions of this class of spin chains are equivalent to the partition functions of a class of one-dimensional vertex models with appropriately defined energy functions. We also establish a boson-fermion duality relation for the partition functions of this class of supersymmetric HS like spin chains by using their correspondence with one-dimensional vertex models.

💡 Research Summary

The paper introduces a broad class of supersymmetric spin chains that are invariant under the Yangian algebra (Y(\mathfrak{sl}{(m|n)})). The authors start by postulating that the partition function of any model in this class can be expressed as a linear combination of super Schur polynomials (S{\lambda}(x|y)), where the two sets of variables (x) and (y) correspond to the bosonic ((m)) and fermionic ((n)) degrees of freedom, respectively. This assumption is motivated by the fact that super Schur polynomials are characters of the irreducible representations of the super‑Lie algebra (\mathfrak{sl}_{(m|n)}) and therefore naturally encode the Yangian symmetry of the system.

Using fundamental identities of super Schur functions—most notably the supersymmetric Cauchy identity and Pieri rules—the authors rewrite the spin‑chain partition function as \