Integrable 1/r Spin Chain with Reflecting End

A new integrable spin chain of the Haldane-Shastry type is introduced. It is interpreted as the inverse-square interacting spin chain with a reflecting end. The lattice points of this model consist of the square of the zeros of the Laguerre polynomial. Using the “exchange operator formalism”, the integrals of motion for the model are explicitly constructed. Typeset using REVTEX ∗E-mail address : [email protected] †E-mail address : [email protected] 1 Studies of the Calogero-Sutherland model [1], the Haldane-Shastry spin chain [2] and their variants [3] have provided many new links with other areas of physics and mathematics. In particular, these models provide exactly solvable models in which the ideas of the fractional exclusion statistics can be tested [4,5]. In ref. [6], with a view to proving the quantum integrability of the Calogero-Sutherland model and, its rational version, the Calogero-Moser model confined in a harmonic potential (we call the Calogero model), Polychronakos had proposed the so-called exchange operator formalism. His clever formalism could be applicable not only to the continuum models but to the lattice models and has become a standard technique to study the integrability and the spectrum of the inverse-square interacting systems [7–12]. Within the exchange operator formalism, all of the inverse-square interacting lattice models can be related to the appropriate continuum inverse-square interacting models with the internal degrees of freedom (spin). More precisely, the lattice models are obtained by freezing out the kinematic degrees of freedom in the corresponding continuum models, and lattice sites lie at the classical staticequilibrium positions of the continuum models [13–15]. For example [8,18], the HaldaneShastry model is related to the spin Calogero-Sutherland model [16,7,17] whose classical equilibrium positions form a regular lattice on the circle. Polychronakos [18] has applied his formalism to construct the new lattice model related to the spin Calogero model [7,10,11,19,20]. We call this model the Polychronakos-Frahm (PF) model [21,22]. The lattice sites of the PF model are positioned at the zeros of the Hermite polynomial, i.e., the spins are no more equidistant. Against this unusual property, the spectra of the PF model are equally spaced and therefore simpler than those of the HaldaneShastry model. Thus the fractional exclusion statistics for the elementary excitations of the PF model is more tractable than one of the Haldane-Shastry model [22]. On the other hands, in ref. [23,24], an another generalization of the spin chain model, the Haldane-Shastry model with open boundary conditions (BCN -type Haldane-Shastry model), has been introduced. This model is related to the BCN -type spin Calogero-Sutherland model [25,26]. It is now well known that such BCN -type models can be applicable to analyze the 2 physics with boundaries [27–30]. In particular, one of the authors and his collaborators have shown that the above models possess the properties of the chiral Tomonage-Luttinger liquids [30]. The aim of this letter is twofold. The first is to prove the integrability of the BN type spin Calogero model [31] within the exchange operator formalism. The second is to construct the new integrable lattice model related to the BN -type spin Calogero model. This lattice model is thought of as the “intersection” of the PF model and the BCN -type Haldane-Shastry model. Before turning to the explicit calculation, we shall briefly mention this new integrable spin chain. The Hamiltonian is given by,