SU($N$) Quantum Spin Model with Weak and Strong First-Order Néel to Valence-Bond Solid Transitions

Abstract

We introduce an SU($N$) symmetric two-dimensional quantum spin model, the $X$-$Q$ model, which hosts a ground state transition between Néel antiferromagnetic and spontaneously dimerized states. The $Q$ terms are products of two adjacent singlet projectors on nearest-neighbor sites, as in the often studied $J$-$Q$ model (where $J$ is the Heisenberg exchange), while the $X$ terms are products of two permutation operators on second-neighbor sites. Quantum Monte Carlo simulations reveal close proximity to a deconfined quantum critical point for $N=2$, as in the $J$-$Q$ model. However, for $N>2$ the transition becomes strongly first-order, contrary to conventional expectations that increasing $N$ should weaken discontinuities. We attribute this behavior to the inability of the $X$ term, which dominates at the transition for large $N$, to induce U(1) fluctuations of the dimer pattern. These results provide insights into the microscopic interactions that support deconfined criticality.

Figures (4)

• Figure 1: Binder cumulant for (a) SU(2) systems with $L=16\text{–}256$ and (b) SU(3) systems with $L=8\text{–}20$. In (a) the fitted curves cross at the near-critical point $(X/Q)_c\approx 0.115$ while in (b) the negative peak flows to a clearly first-order transition at $(X/Q)_c\approx 5.2$. (c) Extrapolations of the order parameters $M_s^2$ and $|\psi|^2$ at the transition point [defined by cumulant crossings for SU(2) and cumulant minima for SU(3)], with the non-zero $L \to \infty$ values demonstrating phase coexistence in both models. Polynomial fits were used for all order parameters, except $|\psi|^2$ for SU($3$), where the manifestly discrete symmetry breaking implies exponential convergence. The inset in (c) zooms in on the large-$L$ SU($2$) data.
• Figure 2: $Q$-energy histograms relative to the mean for the SU($N$) models. For $N=2$ in (a), the distribution for $L=64$ has a single peak, while for $N = 3,4,5$ in (b), (c), and (d), respectively, the large peak splitting for the smaller sizes (which are the largest systems studied) signals two-phase coexistence. The indicated peak splitting is almost converged versus $L$.
• Figure 3: (a) Distribution of the VBS order parameter $\psi=D_x+iD_y$ at the transition point for the SU(3) $L=20$ system. Peaks at $(D_x,D_y) = (\pm D,0),(0,\pm D)$ correspond to the VBS order and the peak at $|D|=0$ arises from coexistence with the AFM phase. (b) The anisotropy parameter increases with $L$ for SU(3) and SU(4) but decreases for SU($2$), with the line indicating the expected $L^{-\mu}$ form with $\mu = 0.72$Takahashi_2024 for a system sufficiently close to the DQC point. (c) Anisotropy parameter (red, left axis) for $L=8$ at the estimated $L \to \infty$ transition points (blue, right axis) for SU(2)-SU(5). Both increase with $N$.
• Figure 4: Action of the $X$ permutation and $Q$ singlet-projector on a single VBS plaquette. The $X$ term swaps singlets but does not mix VBS orientations. The $Q$ term rotates VBS orientations, leading to U(1) resonances. Red/Blue color is only for visual clarity, all singlets are equivalent.