Possibility of deconfined criticality in SU($N$) Heisenberg models at small $N$

To examine the validity of the scenario of the deconfined critical phenomena, we carry out a quantum Monte Carlo simulation for the SU($N$) generalization of the Heisenberg model with four-body and six-body interactions. The quantum phase transition between the SU($N$) Néel and valence-bond solid phases is characterized for $N=2$, 3, and 4 on the square and honeycomb lattices. While finite-size scaling analysis works well up to the maximum lattice size ($L=256$) and indicates the continuous nature of the phase transition, a clear systematic change towards the first-order transition is observed in the estimates of the critical exponent $y\equiv 1/\nu$ as the system size increases. We also confirm the relevance of a squared valence-bond solid field ${\Psi }^2$ for the SU(3) model.

Figure 1

The dimer coverings of (a) unit square and (b) unit hexagon in $Q$ terms and the classification of the bonds in the definition of the complex order parameter ${\Psi }_r$, characterizing the VBS phase on (c) the square and (d) the honeycomb lattice.

Figure 2

The FSS plot for the magnetization of the SU(3) $J$-$Q$ model on the square and the honeycomb lattices with the system size being restricted to $L\le 96$. The scaling factors $A=2.218$ and $B=1.752$ are multiplied to the horizontal axis and the vertical axis, respectively, for the results of the honeycomb lattice ($A=B=1$ for the square lattice.) Inset: The two-point correlation function of magnetic order for the SU(3) models on a square lattice of $L=256$ as a function of the distance at various $q$ near the critical point. The straight line corresponds to the estimate of $2x_m$ from the squared magnetization.

Figure 3

The FSS plot for the squared amplitude of total VBS order for the SU(3) $J$-$Q$ model on the square and the honeycomb lattices with the system size being restricted to $L\le 96$. We set the scaling factors $A=2.422$ and $B=0.6388$ for the honeycomb lattice. Inset: The two-point correlation function of VBS order for the SU(3) models on a square lattice of $L=256$ as a function of the distance at various $q$ near the critical point. The straight line corresponds to the estimate for $2x_{\Psi }$ obtained from the squared amplitude of the total VBS order.

Figure 4

The estimate of the scaling dimensions $y\equiv 1/\nu$ and $2x$ for the SU($N$) model on the square and honeycomb lattices as a function of the system size used in the FSS. At $L_{\max }=256,q_c=0.9568\left(2\right),0.3343\left(1\right)$, and 0.0814(3) for $\mathrm{SU}(N=2,3,4)$ models on a square lattice, respectively.

Figure 5

The two-point correlation function of the square of block-averaged order parameter ${\left(\overline{\Psi }\right)}^2$ for the SU(3) models as a function of the distance at $q=0.3343$, which is the critical point estimated from the data of $L\le 256$. The block size $b$ is 8.