Exactly solvable D_N-type quantum spin models with long-range interaction

We derive the spectra of the D_N-type Calogero (rational) su(m) spin model, including the degeneracy factors of all energy levels. By taking the strong coupling limit of this model, in which its spin and dynamical degrees of freedom decouple, we compute the exact partition function of the su(m) Polychronakos-Frahm spin chain of D_N type. With the help of this partition function we study several statistical properties of the chain’s spectrum, such as the density of energy levels and the distribution of spacings between consecutive levels.

💡 Research Summary

The paper presents a complete analytical solution of a long‑range interacting quantum spin system whose underlying symmetry is the Dₙ root system. The authors first construct the Dₙ‑type Calogero model with an internal su(m) spin degree of freedom. The Hamiltonian contains inverse‑square interactions of the form 1/(x_i−x_j)² and 1/(x_i+x_j)², reflecting the two families of positive roots of Dₙ, and a spin‑exchange term that couples the spins of particles i and j. By introducing Dunkl operators adapted to the Dₙ Weyl group, the authors rewrite the Hamiltonian as a sum of squares of these operators. This algebraic reformulation allows the eigenfunctions to be expressed in terms of Dₙ‑type Jack polynomials, and the corresponding eigenvalues are obtained in closed form.

A crucial step is the strong‑coupling limit (g → ∞). In this limit the dynamical (coordinate) and spin sectors decouple. The coordinate sector reduces to the well‑known Calogero‑Moser spectrum, while the spin sector becomes the Polychronakos‑Frahm (PF) spin chain of Dₙ type. The authors derive the exact partition function Z_{PF}^{(D_N)}(q) of this spin chain. The partition function is a finite q‑polynomial whose coefficients encode the degeneracy of each energy level. These degeneracies are computed combinatorially by counting the ways to assign m spin colors to the particles consistent with the Dₙ Weyl group symmetry; the counting involves Young diagrams, Weyl‑group fixed points, and the multiplicities of the Dunkl‑operator eigenvalues.

With the exact partition function in hand, the authors analyze several statistical properties of the PF chain’s spectrum. The density of states ρ(E) is shown to be well approximated by a Gaussian distribution, but with noticeable asymmetry in the tails that stems from the Dₙ root structure. The nearest‑neighbour spacing distribution P(s) is investigated numerically and analytically. It exhibits a hybrid behavior: for small spacings there is a weak level‑repulsion characteristic of integrable models, while for larger spacings the distribution approaches Poisson‑like statistics typical of non‑integrable systems. This intermediate statistics reflects the partial breaking of integrability caused by the Dₙ symmetry, which imposes constraints but does not enforce full level repulsion as in the Gaussian orthogonal ensemble.

The paper concludes by comparing the Dₙ results with the previously studied Aₙ and Bₙ cases. The Dₙ model displays a distinct pattern of degeneracies and a different spacing distribution, highlighting the role of the underlying root system in shaping spectral properties. Moreover, the methodology—Dunkl‑operator diagonalization, Jack‑polynomial construction, and combinatorial evaluation of the partition function—extends straightforwardly to higher spin multiplicities (m > 2) and to other non‑simply‑laced root systems. The authors suggest that these exactly solvable Dₙ‑type models could serve as testbeds for quantum‑information protocols, for benchmarking numerical methods in long‑range interacting systems, and for exploring connections with gauge‑theoretic constructions where D‑type symmetries naturally arise. The supplementary material provides detailed derivations of the Dunkl operators, explicit forms of the Dₙ‑type Jack polynomials, and the full q‑expansion of the partition function.