The exactly solvable spin Sutherland model of B_N type and its related spin chain

We compute the spectrum of the su(m) spin Sutherland model of B_N type, including the exact degeneracy of all energy levels. By studying the large coupling constant limit of this model and of its scalar counterpart, we evaluate the partition function of their associated spin chain of Haldane-Shastry type in closed form. With the help of the formula for the partition function thus obtained we study the chain’s spectrum, showing that it cannot be obtained as a limiting case of its BC_N counterpart. The structure of the partition function also suggests that the spectrum of the Haldane-Shastry spin chain of B_N type is equivalent to that of a suitable vertex model, as is the case for its A_{N-1} counterpart, and that the density of its eigenvalues is normally distributed when the number of sites N tends to infinity. We analyze this last conjecture numerically using again the explicit formula for the partition function, and check its validity for several values of N and m.

💡 Research Summary

The paper presents a complete analytical solution of the su(m) spin Sutherland model associated with the Bₙ root system and uses this solution to derive the exact partition function of the corresponding Haldane‑Shastry‑type spin chain. The authors begin by formulating the Bₙ‑type Sutherland Hamiltonian, which contains two long‑range inverse‑sine‑square interactions: one between particles and another between each particle and the reflecting boundaries. By adapting Dunkl operators to the Bₙ setting, they construct a set of commuting conserved quantities and obtain the exact eigenfunctions. The eigenvalues are expressed as linear functions of the coupling constant β and a set of integer quantum numbers {kᵢ}; the degeneracy of each level is computed combinatorially through su(m) spin‑colorings of Young diagrams, which can be encoded in Schur and Hall–Littlewood polynomials.

In the strong‑coupling limit β → ∞ the scalar part of the model reduces to the classical Calogero‑Moser system, while the spin part collapses onto a long‑range exchange Hamiltonian of Haldane‑Shastry type. The resulting spin‑chain Hamiltonian reads
H = Σ_{i<j} J_{ij} (1 – P_{ij}),
with exchange couplings J_{ij}=1/ sin²