Topological liquids and valence cluster states in two-dimensional SU(N) magnets

We study the zero-temperature phase diagram of a class of two-dimensional SU(N) antiferromagnets. These models are characterized by having the same type of SU(N) spin placed at each site of the lattice, and share the property that, in general, more than two spins must be combined to form a singlet. An important motivation to study these systems is that they may be realized naturally in Mott insulators of alkaline-earth atoms placed on optical lattices; indeed, such Mott insulators have already been obtained experimentally, although the temperatures are still high compared to the magnetic exchange energy. We study these antiferromagnets in a large-N limit, finding a variety of ground states. Some of the models studied here have a valence-bond solid ground state, as was found in prior studies, yet we find that many others have a richer variety of ground states. Focusing on the two-dimensional square lattice, in addition to valence cluster states (which are analogous to valence-bond states), we find both Abelian and non-Abelian chiral spin liquid ground states, which are magnetic counterparts of the fractional quantum Hall effect. We also find a doubled chiral spin liquid ground state that preserves time-reversal symmetry. These results are based on a combination of rigorous lower bounds on the large-N ground-state energy and a systematic numerical ground-state search. We conclude by discussing whether experimentally relevant SU(N) antiferromagnets, away from the large-N limit, may be chiral spin liquids.