Topological order in ultra-cold atoms

Optical lattices are remarkable for their capacity to host rich physics. Examples include lattices having moat-like band structures, i.e., a band with infinitely degenerate energy minima attained along a closed line in the Brillouin zone. It is entirely the effect of competing correlations which lifts this degeneracy and leads to an amazing variety of completely new quantum many-body states. If such a lattice is populated with bosons, the degeneracy prevents their condensation. Such degeneracy of the kinetic energy favors fermionic quasiparticles, leading to statistical transmutation and topological order. At hard-core repulsion, the system is equivalent to the spin-1/2 XY model, while the absence of condensation translates into the absence of magnetic order in the XY plane. In this talk I will show that the frustration in such lattices stabilizes a variety of novel quantum spin liquid phases including a composite fermion state and a chiral spin liquid. These are topologically ordered states, which may be viewed as states of fermions subject to Chern-Simons gauge fields. They have a bulk gap and chiral gapless edge excitations. The talk includes a suggestion for the chiral spin liquid realizations in experiments with cold atoms. The velocity distribution of the released bosons is a sensitive probe of the statistical transmutation and a chiral spin-liquid state.