The spin Sutherland model of type and its associated spin chain

Abstract

In this paper we study the su(m) spin Sutherland (trigonometric) model of D N type and its related spin chain of Haldane-Shastry type obtained by means of Polychronakos's freezing trick. As in the rational case recently studied by the authors, we show that these are new models, whose properties cannot be simply deduced from those of their well-known BC N counterparts by taking a suitable limit. We identify the Weyl-invariant extended configuration space of the spin dynamical model, which turns out to be the N-dimensional generalization of a rhombic dodecahedron. This is in fact one of the reasons underlying the greater complexity of the models studied in this paper in comparison with both their rational and BC N counterparts. By constructing a non-orthogonal basis of the Hilbert space of the spin dynamical model on which its Hamiltonian acts triangularly, we compute its spectrum in closed form. Using this result and applying the freezing trick, we derive an exact expression for the partition function of the associated Haldane-Shastry spin chain of D N type.

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

“…On the other hand, although the B N , C N and D N scalar Calogero-Sutherland models have been studied in the literature [45,46], their spin versions have been largely ignored. Recently, however, the D N spin Calogero [47] and Sutherland [48] models, as well as their associated spin chains, were studied by the present authors and shown to be nontrivial reductions of their BC N counterparts. …”

“…This fundamental difference explains why the spectrum of the D N -type spin Sutherland model is essentially different from that of its BC N counterpart. It also accounts for the greater complexity of the partition function of the associated chain of D N type [48] compared to its BC N version [42].…”

“…, χ N −1 are the zeros of the Jacobi polynomial P (1,1) N−2 . Using this characterization, it has been shown that in the (β, β ) → 0 limit the Hamiltonian of the HS chain of BC N type yields its D N analog, plus "impurity" terms at both ends of the latter chain [48]:…”

“…We shall employ a well-established technique [33,42,48], which is based on relating both of these models to the auxiliary differential-difference operator …”

“…[48], there is a natural isomorphism between the spaces L 2 (A (B) ) ⊗ Σ and (C (B) ) onto the space of functions symmetric under permutations and even under sign reversals, defined by the relations …”