Mott Insulators of Ultracold Fermionic Alkaline Earth Atoms: Underconstrained Magnetism and Chiral Spin Liquid

We study Mott insulators of fermionic alkaline earth atoms, described by Heisenberg spin models with enhanced $\mathrm{SU}(N)$ symmetry. In dramatic contrast to SU(2) magnetism, more than two spins are required to form a singlet. On the square lattice, the classical ground state is highly degenerate and magnetic order is thus unlikely. In a large-$N$ limit, we find a chiral spin liquid ground state with topological order and Abelian fractional statistics. We discuss its experimental detection. Chiral spin liquids with non-Abelian anyons may also be realizable with alkaline earth atoms.

Figure 1

Large-$N$ dimer and plaquette ground states for $k=2$ (a), $k=3$ (b) and $k=4$ (c). ${\chi }_{\mathit{r}{\mathit{r}}^{\prime }}$ has constant magnitude on the dark bonds and is zero on the others. For $k=3$ ($k=4$), the phase of ${\chi }_{\mathit{r}{\mathit{r}}^{\prime }}$ is chosen so that the flux through each plaquette is $\pi$ (zero). The patterns shown are not necessarily those selected by $1/N$ corrections.

Figure 2

Lowest-energy competing state for $k=6$ (the $k=5$ state has a similar pattern). The lattice is covered by stripes, of which one is shown. The shading of bonds represents $|{\chi }_{\mathit{r}{\mathit{r}}^{\prime }}|$, interpolating between the maximum $|{\chi }_{\mathit{r}{\mathit{r}}^{\prime }}|$ (black), and $|{\chi }_{\mathit{r}{\mathit{r}}^{\prime }}|=0$ (white). Some regions enclose $\pi$ flux, as indicated.