Order and disorder in SU(N) simplex solid antiferromagnets

We study the structure of quantum ground states of simplex solid models, which are generalizations of the valence bond construction for quantum antiferromagnets originally proposed by Affleck, Kennedy, Lieb, and Tasaki (AKLT) (1987 Phys. Rev. Lett. 59 799). Whereas the AKLT states are created by application of bond singlet operators for $\text{SU}\left(2\right)$ spins, the simplex solid construction is based on N-simplex singlet operators for $\text{SU}\left(N\right)$ spins. In both cases, a discrete one-parameter family of translationally-invariant models with exactly solvable ground states is defined on any regular lattice, and the equal time ground state correlations are given by the finite temperature correlations of an associated classical model on the same lattice, owing to the product form of the wave functions when expressed in a $\text{C}{\text{P}}^{N-1}$ coherent state representation. We study these classical companion models via a mix of Monte Carlo simulations, mean-field arguments, and low-temperature effective field theories. Our analysis reveals that the ground states of $\text{SU}\left(4\right)$ edge- and $\text{SU}\left(8\right)$ face-sharing cubic lattice simplex solid models are long range ordered for sufficiently large values of the discrete parameter, whereas the ground states of the $\text{SU}\left(3\right)$ models on the kagome (2D) and hyperkagome (3D) lattices are always quantum disordered. The kagome simplex solid exhibits strong local order absent in its three-dimensional hyperkagome counterpart, a contrast that we rationalize with arguments similar to those leading to 'order by disorder'. (paper: quantum statistical physics, condensed matter, integrable systems)