Valence bond solids for SU(n) spin chains:: Exact models, spinon confinement, and the Haldane gap

To begin with, we introduce several exact models for SU(3) spin chains: First is a translationally invariant parent Hamiltonian involving four-site interactions for the trimer chain, with a threefold degenerate ground state. We provide numerical evidence that the elementary excitations of this model transform under representation 3 of SU(3) if the original spins of the model transform under representation 3. Second is a family of parent Hamiltonians for valence bond solids of SU(3) chains with spin representations 6, 10, and 8 on each lattice site. We argue that of these three models, only the latter two exhibit spinon confinement and, hence, a Haldane gap in the excitation spectrum. We generalize some of our models to SU(n). Finally, we use the emerging rules for the construction of valence bond solid states to argue that models of antiferromagnetic chains of SU(n) spins, in general, possess a Haldane gap if the spins transform under a representation corresponding to a Young tableau consisting of a number of boxes lambda which is divisible by n. If lambda and n have no common divisor, the spin chain will support deconfined spinons and not exhibit a Haldane gap. If lambda and n have a common divisor different from n, it will depend on the specifics of the model including the range of the interaction.