Tunable deconfined quantum criticality and interplay of different valence-bond solid phases

We use quantum Monte Carlo simulations to study a quantum $S=1/2$ spin model with competing multi-spin interactions. We find a quantum phase transition between a columnar valence-bond solid (cVBS) and a N\'eel antiferromagnet (AFM), as in the scenario of deconfined quantum-critical points, as well as a transition between the AFM and a staggered valence-bond solid (sVBS). By continuously varying a parameter, the sVBS--AFM and AFM--cVBS boundaries merge into a direct sVBS--cVBS transition. Unlike previous models with putative deconfined AFM--cVBS transitions, e.g., the standard $J$-$Q$ model, in our extended $J$-$Q$ model with competing cVBS and sVBS inducing terms the transition can be tuned from continuous to first-order. We find the expected emergent U(1) symmetry of the microscopically $Z_4$ symmetric cVBS order parameter when the transition is continuous. In contrast, when the transition changes to first-order the clock-like $Z_4$ fluctuations are absent and there is no emergent higher symmetry. We argue that the confined spinons in the sVBS phase are fracton-like. We also present results for an SU(3) symmetric model with a similar phase diagram. The new family of models can serve as a useful tool for further investigating open questions related to deconfined quantum criticality and its associated emergent symmetries.