Simplex solid states of SU(N) quantum antiferromagnets

I define a set of wave functions for SU(N) lattice antiferromagnets, analogous to the valence bond solid states of Affleck et al. [ Phys. Rev. Lett. 59, 799 (1987); Commun. Math. Phys. 115, 477 ( 1988 ) ], in which the singlets are extended over N-site simplixes. As with the valence bond solids, the new simplex solid (SS) states are extinguished by certain local projection operators, allowing one to construct Hamiltonians with local interactions which render the SS states exact ground states. Using a coherent state representation, I show that the quantum correlations in each SS state are calculable as the finite temperature correlations of an associated classical model with N-spin interactions on the same lattice. In three and higher dimensions, the SS states can spontaneously break SU(N) and exhibit N-sublattice long-ranged order as a function of a discrete parameter which fixes the local representation of SU ( N ). I analyze this transition using a classical mean field approach. For N > 2, the ordered state is selected via an "order by disorder" mechanism. As in the Affleck-Kennedy-Lieb-Tasaki case, the bulk representations fractionalize at an edge, and the ground state entropy is proportional to the volume of the boundary.