Topological liquids and valence cluster states in two-dimensional SU$\left(N\right)$ magnets

We study the zero-temperature phase diagram of a class of two-dimensional SU$\left(N\right)$ antiferromagnets. These models are characterized by having the same type of SU$\left(N\right)$ 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$\left(N\right)$ antiferromagnets, away from the large-$N$ limit, may be chiral spin liquids.

Figure 1

The Young tableau corresponding to the $m\times n_c$ irreducible representation of $\mathrm{SU}(N)$. We consider spin models where the spin at each lattice site transforms in this representation.

Figure 2

Illustration of three-cluster and three-simplex VCS states on the triangular lattice (for $n_c=1$). ${\chi }_{\mathit{r}{\mathit{r}}^{\text{'}}}$ is nonzero on the highlighted bonds and zero elsewhere. Both states (a) and (b) are three-cluster states. State (b) is a three-simplex state and is a $N=\infty$ ground state of the $k=3$ triangular lattice model. State (a) is not a three-simplex state and therefore has higher energy than (b) following the discussion in the text.

Figure 3

An example $k$-simplex state with $n_c=2$ and $k=3$ on the kagome lattice. Simplices of one color are the triangles marked with solid lines (red online), and those of the other color are triangles marked with dashed lines (blue online). This state was discussed (for $N=3$) in Ref.51.

Figure 4

Cluster states ($n_c=1$) with energies saturating the lower bound (94) on the square lattice for (a) $k=2$, (b) $k=3$, and (c) $k=4$. ${\chi }_{\mathit{r}{\mathit{r}}^{\text{'}}}$ has constant magnitude on the dark bonds and is zero on the others. In the $k=3$ state, the flux through each six-site plaquette is $\pi$, while it is zero for each four-site plaquette in the $k=4$ state. For each value of $k$, in the $N=\infty$ limit, every tiling of the square lattice by the type of clusters shown is a ground state. This large degeneracy is expected to be lifted upon computing perturbative $1/N$ corrections to the ground-state energy (Ref.34).

Figure 5

Illustration of two $N=\infty$ cluster ground states on the square lattice for $n_c=2$ and $k=4$, which saturate the lower bound (94). Square clusters of one color are marked with solid lines (red online), while those of the other color are marked with dashed lines (blue online). Any configuration where clusters of the two colors separately tile the lattice is a $N=\infty$ ground state; as in the $n_c=1$ case, the degeneracy among these states is expected to be lifted upon computing perturbative $1/N$ corrections to the ground-state energy.

Figure 6

Phase diagram of the SU$(N)$ Heisenberg model in two dimensions on the square lattice with $n_c=1$ and with $N=\mathit{mk}$. In terms of an underlying Hubbard model, $m$ is the number of fermions per site, while $k$ is the inverse filling. Regions where there is substantial evidence for a given ground state, or where the ground state is known, are shaded. The Abelian chiral spin liquid (ACSL) and valence cluster state (VCS) regions on the right are established by our large-$N$ analysis; the boundary between these regions in large $N$ is shown by a dashed line. For $k=2$, $m=1$, the Neel state is the well-known ground state. There is also evidence for magnetic order at $k=3$, $m=1$ (Ref.37) and $k=4$, $m=1$ (Ref.40). Valence-bond solid (VBS) order (which is a type of VCS) was found for $k=2$ and $m=3,4$ (Ref.104). The dashed-dotted line separates the range of parameters beyond the reach of current experiments (above and to the right of the line) and the range within the reach of the experiments (below and to the left of the line). The experimentally relevant part of the phase diagram with the greatest potential for novel ground states, in particular, the Abelian chiral spin liquid, is indicated with a question mark.

Figure 7

Illustration of Young tableaux occurring in the decomposition of the $n_{c}m\times 1$ representation of $\mathrm{SU}(n_{c}N)$ into irreducible representations of the $\mathrm{SU}(N)\times \mathrm{SU}(n_c)$ subgroup for the case $n_c=m=2$. If, as described in the text, we project out the $\mathrm{SU}(n_c)$ irreducible representation corresponding to the tableau on the left, then the corresponding $\mathrm{SU}(N)$ tableau is as shown on the right. Note that the rows (columns) of the $\mathrm{SU}(n_c)$ tableau become the columns (rows) of the $\mathrm{SU}(N)$ tableau.