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We study a class of SU(N) Heisenberg models, describing Mott insulators offermionic ultra-cold alkaline earth atoms on the three-dimensional simple cubiclattice. Based on an earlier semiclassical analysis, magnetic order isunlikely, and we focus instead on a solvable large-N limit designed to addressthe competition among non-magnetic ground states. We find a rich phase diagramas a function of the filling parameter k, composed of a variety of groundstates spontaneously breaking lattice symmetries, and in some cases also timereversal symmetry. One particularly striking example is a state spontaneouslybreaking lattice rotation symmetry, where the cubic lattice breaks up intobilayers, each of which forms a two-dimensional chiral spin liquid state.

Mott insulators of ultracold fermionic alkaline earth atoms in three dimensions