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

We introduce an SU($N$) symmetric two-dimensional quantum spin model, the X-Q model, which hosts a ground state transition between N\'eel antiferromagnetic and spontaneously dimerized states. The Q terms are products of two adjacent singlet projectors on first-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 transformation 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 significant U(1) fluctuations of the dimer pattern. These results provide insights into the microscopic interactions that support deconfined criticality.