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 comprehensive study of the Dₙ‑type Calogero model with su(m) spin degrees of freedom and its associated Polychronakos‑Frahm (PF) spin chain. Starting from the well‑known B Cₙ Calogero model, the authors set the parameter b=0 to obtain the Dₙ Hamiltonian (Eq. 4). The configuration space is chosen as the principal Weyl chamber of the Dₙ root system, C={|x₁|<x₂<…<x_N}, which differs from the B Cₙ chamber by allowing the first coordinate to be negative. This subtle change doubles the Hilbert space, effectively yielding a direct sum of two B Cₙ models with opposite chiralities.

To solve the spectrum, the authors introduce Dunkl operators J_i⁻ (Eq. 20) that incorporate both particle permutations (K_{ij}) and sign reversals (K_i). Using these, they construct an auxiliary operator H′ (Eq. 18) that can be written as a triangular matrix in the non‑orthogonal basis φ_n = ρ ∏_i x_i^{n_i} (Eq. 23). The eigenvalues of H′ are simply E′n = a|n| + E₀ (Eq. 25), where |n| is the total degree of the monomial and E₀ is a constant. By replacing the permutation and sign operators with the spin exchange operators S{ij} and S_i (Eq. 19a) they obtain the spectrum of the full spin Hamiltonian H, while setting them to identity yields the scalar Hamiltonian H_sc (Eq. 19b).

The key step is the “freezing trick”: in the strong‑coupling limit a→∞ the particles localize at the minima ξ_i of the scalar potential U (Eq. 6). These positions are the lattice sites of the Dₙ‑type PF chain and are given by the square roots of twice the zeros of the Laguerre polynomial L_{−1}^{N} (Eqs. 12‑13). The spin and dynamical degrees of freedom decouple, and the energy of a state factorizes as E_{ij} ≈ E_sc,i + a E_j (Eq. 16). Consequently, the partition function of the PF chain is obtained as the ratio Z(T)=lim_{a→∞} Z_a(aT)/Z_sc(aT) (Eq. 17). The authors express Z(T) in terms of the known partition functions of the A‑type and B‑type PF chains, providing a closed‑form expression.

Armed with the exact partition function, the authors analyze statistical properties of the spectrum. For large N the level density ρ(E) is shown to be Gaussian with high accuracy, confirming earlier observations for other HS‑type chains. More strikingly, the distribution of normalized spacings s between consecutive levels does not follow the Poisson law predicted by the Berry–Tabor conjecture for integrable systems. Instead, the cumulative spacing distribution follows a “square‑root‑of‑logarithm” law, \