Lattice integrable systems of Haldane-Shastry type.

We present a new lattice integrable system in one dimension of the Haldane-Shastry type. It consists of spins positioned at the static equilibrium positions of particles in a corresponding classical Calogero system and interacting through an exchange term with strength inversely proportional to the square of their distance. We achieve this by viewing the Haldane-Shastry system as a high-interaction limit of the Sutherland system of particles with internal degrees of freedom and identifying the same limit in a corresponding Calogero system. The commuting integrals of motion of this system are found using the exchange operator formalism. ∗ Address after December 1, 1992. Recently, the interest in spin systems of the Haldane-Shastry type as well as in integrable systems of particles with internal degrees of freedom has been revived [1-8]. The Haldane-Shastry model for spin chains and its SU(n) generalization consists of spins or, in general, SU(n) color degrees of freedom equally spaced around the unit circle with the hamiltonian [1] H = ∑ i<j 1 sin( xi−xj 2 ) Pij (1) where xi are the positions of the spins and Pij is the operator which exchanges the spins or colors of sites i and j. Haldane and Shastry found the antiferromagnetic ground state wavefunction of the system, which is similar in form to the ground state wavefunction of the Sutherland system of particles on the circle [9], as well as all energy levels for the system. Although the above system was suspected to be integrable, and particular commuting integrals of the motion were sporadically found [2,10], a complete proof was lacking. Recently, however, Fowler and Minahan [11] showed the integrability of the system and derived the conserved quantities using a recently developed exchange operator formalism [12]. Their approach consists of working initially with a system of N bosons with internal degrees of freedom and no kinematics which sit on the N lattice sites and only allowing states with exactly one particle per site. Then every operator which is invariant under particle permutation must involve degrees of freedom on all lattice sites and can thus be substituted with a corresponding lattice operator. Integrability of this particle system, then, translates into integrability for the lattice system. These authors, then, consider the operators πi = ∑ j 6=i zj zij Mij (2) where the indices i, j now refer to particles, zi = exp(2πixi) and zij = zi − zj . Mij are the operators which exchange the positions of particles, satisfying Mijxi = xjMij , Mijxk = xkMij (for i 6= k 6= j) (3) 2 as well as the standard permutation group commutation relations among themselves. The hamiltonian of the system is taken to be H = ∑ i<j 1 sin( xi−xj 2 ) Mij (4) Using the commutation properties of Mij and zi one can show that the quantities In = ∑ i π i (5) commute among themselves and, if the lattice sites are equidistant, they also commute with the hamiltonian. Therefore this system is integrable. Each operator Mij, now, acting on a bosonic state translates into a spin exchange operator σij for the particles [8]. Since the In are symmetric under particle permutation, every particle spin exchange operator they contain will translate into a site spin exchange operator Pij and will reduce to the commuting conserved quantities of the corresponding Haldane-Shastry lattice system. The above operators πi considered by Fowler and Minahan are, in fact, identical in form to the corresponding operators considered by this author in the exchange operator formalism of the Sutherland problem [12], only lacking an explicit kinetic term. These operators are π̄i = pi + il ∑ j 6=i cot (xi − xj 2 )