Deconfined Boundary Phase Transition of a Quantum Critical Heisenberg Model

We investigate the boundary phases of a (2+1)-dimensional quantum critical Heisenberg model with a dangling spin chain. By introducing a multispin $Q$-term along the boundary, we drive a continuous boundary transition from an antiferromagnetic (AF) order to a valence-bond solid (VBS) order. Using large-scale quantum Monte Carlo simulations, we locate the critical point at $Q_{c}=0.310(11)$, and obtain the critical exponents at $Q_{c}$, including $y_{s}=0.81(4)$ and the scaling dimensions of AF and VBS order parameters $\Delta_{s}=0.660(15)$ and $\Delta_{v}=0.204(14)$. The weak long-range AF order for $Q<Q_{c}$ is stabilized by quasi-long-range effective interactions mediated by the critical bulk state, while the VBS phase restores the ordinary critical behavior. Our findings highlight the synergy between topological terms and quasi-long-range interactions in low-dimensional quantum many-body systems.