Title: Deconfined quantum criticality lost

Abstract: Over the past two decades, the enigma of the deconfined quantum critical point (DQCP) attracted broad attention across condensed matter and quantum materials to quantum field theory and high-energy physics communities, as it is expected to offer a new paradigm in theory, experiment, and numerical simulations that goes beyond the Landau-Ginzburg-Wilson framework of symmetry breaking and phase transitions. However, the lattice realizations of DQCP have been controversial. For instance, in the square-lattice spin-1/2 $J$-$Q$ model, believed to realize the DQCP between N\'eel and valence bond solid states, conflicting results, such as first-order versus continuous transition, and drifting critical exponents incompatible with conformal bootstrap bounds, have been reported. Here, we solve this two-decades-long mystery by taking a new viewpoint, in that we systematically study the entanglement entropy of square-lattice SU($N$) DQCP spin models, from $N=2,3,4$ within the $J$-$Q$ model to $N=5,6,\dots,12,15,20$ within the $J_1$-$J_2$ model. We unambiguously show that for $N \le 6$, the previously determined DQCPs do not belong to unitary conformal fixed points. In contrast, when $N\ge N_c$ with a finite $N_c \ge 7$, the DQCPs correspond to unitary conformal fixed points that can be understood within the Abelian Higgs field theory with $N$ complex components. From the viewpoint of quantum entanglement, our results suggest the realization of a genuine DQCP between N\'eel and valence bond solid phases at finite $N$, and yet explain why the SU(2) case is ultimately weakly-first-order, as a consequence of a collision and annihilation of the stable critical fixed point of the $N$-component Abelian Higgs field theory with another, bicritical, fixed point, in agreement with four-loop renormalization group calculations. The experimental relevance of our findings is discussed.