Deconfined quantum criticality lost

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.