Antiferromagnetic to valence-bond-solid transitions in two-dimensional $\text{SU}\left(N\right)$ Heisenberg models with multispin interactions

We study two-dimensional Heisenberg antiferromagnets with additional multispin interactions which can drive the system into a valence-bond-solid state. For standard SU(2) spins, we consider both four- and six-spin interactions. We find continuous quantum phase transitions with the same critical exponents. Extending the symmetry to $\text{SU}\left(N\right)$, we also find continuous transitions for $N=3$ and 4. In addition, we also study quantitatively the crossover of the order-parameter symmetry from $Z_4$ deep inside the valence-bond-solid phase to $U\left(1\right)$ as the phase transition is approached.

Figure 1

(Color online) Interactions involving $p$ singlet projectors (illustrated by ovals enclosing two sites) on the square lattice. The two-spin $\left(p=1\right)$ interaction $J$ is the Heisenberg exchange. Higher-order $Q_p$ terms with $p=2$ and 3 are considered here. All translations and $90°$ rotations of the site groupings shown here are included in the hamiltonian.

Figure 2

(Color online) Finite-size scaling of the squared AF and VBS order parameters of the $J\text{−}{Q}_2$ and $J\text{−}{Q}_3$ models.

Figure 3

(Color online) Scaling of the spin and dimer order parameters of the SU(3) and SU(4) $J\text{−}{Q}_2$ models.

Figure 4

(Color online) Dimer order distribution $P\left(D_x,D_y\right)$ for $L=32$ systems. The left panels are for the $J\text{−}{Q}_3$ model at (a) $q=0.635$ and (b) $q=0.85$, and the right panels are for the SU(3) $J\text{−}{Q}_2$ model at (c) $q=0.45$ and (d) $q=0.65$.

Figure 5

(Color online) Finite-size scaling of the square of the anisotropic order parameter $D_4$ in the $J\text{−}{Q}_3$ model (upper panel) and SU(3) $J\text{−}{Q}_2$ model (lower panel).