Invariants of the Haldane-Shastry SU(N) chain.

Using a formalism developed by Polychronakos, we explicitly construct a set of invariants of the motion for the Haldane-Shastry SU(N) chain. Submitted to Physical Review Letters, PACS numbers: 75.10.Jm, 05.30.-d, 03.65.Ca † [email protected] ‡ [email protected] There have been several recent papers on the Haldane-Shastry model for spin chains and its SU(N) generalization [1-7]. This model is described by the hamiltonian H = ∑ i<j 1 sin π L (xi − xj) Pij , (1) where xi are the positions of the spins, equally spaced around a ring, and Pij is the operator that exchanges the spins at sites i and j. Haldane and Shastry found the wave functions for the antiferromagnetic ground state [1], showing it to be identical in form to the Sutherland ground state wave function for particles on a line with the inverse square potential [9]. These authors also found all possible energy levels for the system. It would thus seem that the Haldane-Shastry model and its generalizations are integrable systems. If the model is integrable, there must exist a set of operators that commute among themselves and with the Hamiltonian. Inozemtsev found the first such nontrivial operator, one involving the exchange of three spins [2]. Haldane later found two others, a four-spin-exchange operator that commutes with both the Hamiltonian and with Inozemtsev’s operator, and a more basic two-spin-exchange vector operator he refers to as the rapidity [8]. In this paper we explicitly show that the Haldane-Shastry model is integrable by constructing a complete set of operators that commute among themselves and with the Hamiltonian. These operators are very similar in structure to those used by Polychronakos [10] in his exchange operator approach to the Sutherland and Calogero models [9,11,12], and by Polychronakos and one of the authors in generalizations of these models [13]. The key to our approach is that we consider the system as a set of N bosons with internal degrees of freedom which sit on the N sites of the lattice, only allowing states with one particle per site, as in the infinite-U Hubbard model. The exchange terms making up the Hamiltonian provide both the kinetic energy, from hopping exchange of particles with different internal quantum states, and the po-