Resonating singlet valence plaquettes

We consider the simplest generalizations of the valence bond physics of SU(2) singlets to SU(N) singlets that comprise objects with N sites---these are SU(N) singlet plaquettes with $N=3$ and $N=4$ in three spatial dimensions. Specifically, we search for a quantum mechanical liquid of such objects---a resonating singlet valence plaquette phase that generalizes the celebrated resonating valence bond phase for SU(2) spins. We extend the Rokhsar-Kivelson construction [Phys. Rev. Lett. 61, 2376 (1988)] of the quantum dimer model to the simplest SU(4) model for valence plaquette dynamics on a cubic lattice. The phase diagram of the resulting quantum plaquette model is analyzed both analytically and numerically. We find that the ground state is solid everywhere, including at the Rokhsar-Kivelson point where the ground state is an equal amplitude sum. By contrast, the equal amplitude sum of SU(3) singlet triangular plaquettes on the face-centered cubic lattice is liquid and is thus a candidate for describing a resonating single valence plaquette phase, given a suitably defined local Hamiltonian.