This study investigates the scaling behavior of the Rényi entanglement entropy (EE) with smooth boundaries at the phase transition point of the two-dimensional $ J - Q_3 $ model. Using a recently developed scaling formula, the researchers find a subleading logarithmic term with a coefficient indicating the presence of four Goldstone modes. This suggests the spontaneous breaking of an emergent $ SO(5) $ symmetry to $ O(4) $ in the thermodynamic limit, but restoration in finite systems. The results indicate that the believed deconfined quantum critical point (DQCP) of the $ J - Q_3 $ model is actually a weak first-order transition point. The study provides a new method to distinguish states with spontaneously broken continuous symmetry from critical states, particularly useful for identifying weak first-order phase transitions. The $ J - Q_3 $ model, a variant of the $ J - Q_2 $ model, exhibits a similar VBS-Néel transition but with enhanced VBS order. Previous studies suggest the transition is continuous, but this work shows it is first-order. The Néel-VBS DQC can be described by a nonlinear sigma model with a Wess-Zumino-Witten term for the five-component superspin. The leading anisotropy acts as a mass term in the field theory, driving the transition between Néel and VBS phases. The $ SO(5) $ symmetry is conjectured to emerge at the DQCP, but the transition is shown to be strongly first-order. The study uses entanglement entropy to resolve the puzzle of the phase transition. At the critical point of a (2+1)D system, the scaling of EE has a logarithmic term with a negative coefficient for sharp boundaries but no logarithmic term for smooth boundaries. The scaling behavior of the Rényi EE of the $ J - Q_3 $ model at the transition point was studied, revealing a positive corner logarithmic term, contradicting the prediction of unitary CFT. However, a recent paper shows that the positive logarithmic term due to corners becomes negative when tilted bipartitioning is applied. The study finds that the number of Goldstone modes, related to the coefficient of the logarithmic term, is four. This indicates the presence of an emergent $ SO(5) $ symmetry at the transition point, which is spontaneously broken to $ O(4) $ in the thermodynamic limit but restored in finite systems. The results show that the transition is first-order, similar to the checker-board $ J - Q $ model. The study uses quantum Monte Carlo simulations to calculate the Rényi EE and determine the spin stiffness and inertia moment density. The results support the presence of an emergent $ SO(5) $ symmetry and confirm the first-order nature of the transition. The study also shows that the transition point of the $ J - Q_2 $ model has similarThis study investigates the scaling behavior of the Rényi entanglement entropy (EE) with smooth boundaries at the phase transition point of the two-dimensional $ J - Q_3 $ model. Using a recently developed scaling formula, the researchers find a subleading logarithmic term with a coefficient indicating the presence of four Goldstone modes. This suggests the spontaneous breaking of an emergent $ SO(5) $ symmetry to $ O(4) $ in the thermodynamic limit, but restoration in finite systems. The results indicate that the believed deconfined quantum critical point (DQCP) of the $ J - Q_3 $ model is actually a weak first-order transition point. The study provides a new method to distinguish states with spontaneously broken continuous symmetry from critical states, particularly useful for identifying weak first-order phase transitions. The $ J - Q_3 $ model, a variant of the $ J - Q_2 $ model, exhibits a similar VBS-Néel transition but with enhanced VBS order. Previous studies suggest the transition is continuous, but this work shows it is first-order. The Néel-VBS DQC can be described by a nonlinear sigma model with a Wess-Zumino-Witten term for the five-component superspin. The leading anisotropy acts as a mass term in the field theory, driving the transition between Néel and VBS phases. The $ SO(5) $ symmetry is conjectured to emerge at the DQCP, but the transition is shown to be strongly first-order. The study uses entanglement entropy to resolve the puzzle of the phase transition. At the critical point of a (2+1)D system, the scaling of EE has a logarithmic term with a negative coefficient for sharp boundaries but no logarithmic term for smooth boundaries. The scaling behavior of the Rényi EE of the $ J - Q_3 $ model at the transition point was studied, revealing a positive corner logarithmic term, contradicting the prediction of unitary CFT. However, a recent paper shows that the positive logarithmic term due to corners becomes negative when tilted bipartitioning is applied. The study finds that the number of Goldstone modes, related to the coefficient of the logarithmic term, is four. This indicates the presence of an emergent $ SO(5) $ symmetry at the transition point, which is spontaneously broken to $ O(4) $ in the thermodynamic limit but restored in finite systems. The results show that the transition is first-order, similar to the checker-board $ J - Q $ model. The study uses quantum Monte Carlo simulations to calculate the Rényi EE and determine the spin stiffness and inertia moment density. The results support the presence of an emergent $ SO(5) $ symmetry and confirm the first-order nature of the transition. The study also shows that the transition point of the $ J - Q_2 $ model has similar