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

Abstract

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.