An integrable BC(n) Sutherland model with two types of particles

A hyperbolic BC(n) Sutherland model involving three independent coupling constants that characterize the interactions of two types of particles moving on the half-line is derived by Hamiltonian reduction of the free geodesic motion on the group SU(n,n). The symmetry group underlying the reduction is provided by the direct product of the fixed point subgroups of two commuting involutions of SU(n,n). The derivation implies the integrability of the model and yields a simple algorithm for constructing its solutions.

💡 Research Summary

The paper presents a new integrable many‑body system of the hyperbolic BCₙ Sutherland type that describes two species of particles moving on the half‑line. The model contains three independent coupling constants κ, x and y, which respectively control the interaction strength among particles of the same species, the interaction of the particles with a fixed charge at the origin, and the interaction with their mirror images.

The construction starts from the free geodesic motion on the non‑compact Lie group G = SU(n,n). Two commuting involutions are introduced: the Cartan involution Θ and a second involution Γ defined by conjugation with the diagonal matrix Dₘ. Their fixed‑point subgroups are a maximal compact subgroup G₊ ≅ S(U(n)×U(n)) and a non‑compact subgroup G₊′ ≅ S(U(m,n−m)×U(m,n−m)), where 1 ≤ m < n determines the numbers of the two particle types. The direct product G₊×G₊′ therefore serves as the symmetry group for the Hamiltonian reduction.

A detailed decomposition of the Lie algebra su(n,n) into four orthogonal subspaces G_{±}^{±} is carried out, and a maximal Abelian subspace A⊂G_{−}^{−} is chosen. Regular elements q∈A are taken from the open Weyl chamber defined by q₁>…>q_m>0 and q_{m+1}>…>q_n>0. The authors also introduce a minimal‑dimension coadjoint orbit O_{κ,x} of G₊, parametrised by a complex vector u∈ℂⁿ with |u_j|²=2κ, and a one‑point orbit y C_r of G₊′.

The phase space is enlarged to P = TG × O_{κ,x}, equipped with the canonical symplectic form Ω = Ω_{TG}+Ω_{O}. A commuting family of Hamiltonians H_k = (1/4k) tr(J^{2k}) (k=1,…,n) is defined on P; H₁ generates the free geodesic flow. The G₊×G₊′ action on P is Hamiltonian with moment map Φ = (Φ₊,Φ₊′), where Φ₊ = π₊(J)+ζ and Φ₊′ = −π₊(g^{-1}Jg). The reduction is performed by imposing the moment‑map constraint Φ = ν = (0, −y C_r) and assuming x² ≠ y². Under this condition the reduced space P_red = Φ^{-1}(ν)/(G₊×G₊′) is a smooth symplectic manifold of dimension 2n.

A key technical result (Lemma 3.2) shows that any point satisfying the constraint can be brought, by the action of the symmetry group, to the form (e^{q}, J, x C_l+ξ(u)) with |u_j|²=2κ and q regular. Decomposing J into its four components and using the constraint equations, the authors solve for the non‑diagonal parts of J explicitly in terms of q, κ, x and y. Substituting back yields a reduced Hamiltonian that depends only on the coordinates q_i and their conjugate momenta p_i. The final expression coincides with equation (1.1) of the paper:

H = ½∑{j=1}^{n}p_j²
 −∑{1≤j≤m<k≤n}