Recent numerical and theoretical studies on the two-dimensional
J-
Q3 model suggests that the deconfined quantum critical point is actually a
SO(5)-symmetry-enhanced first-order phase transition that is spontaneously broken to
O(4). However, this conclusion has mainly relied on finite-size scaling of the entanglement entropy, lacking direct evidence from physical observables.} Here, we investigate the dynamical spectra of spin and bond operators at the deconfined critical point of the
J-
Q3 model using large-scale quantum Monte Carlo simulations, and contrasting them with the well-established
O(3) Wilson-Fisher criticality in the
J1-
J2 Heisenberg model. Although both models exhibit two gapless magnon modes in the Néel phase, their critical behaviors diverge strikingly. At the
J1-
J2 critical point, the Higgs mode becomes gapless, yielding three gapless modes that reflect the full restoration of the
O(3) symmetry. {In the
J-
Q3 model, we instead observe four gapless transverse modes at the either side of the transition. This spectral feature, together with the entanglement entropy results, provides direct evidence for the weakly first-order scenario that the deconfined quantum critical point exhibits an emergent
SO(5) symmetry that spontaneously breaks to
O(4).
Deconfined quantum criticality with emergent SO(5) symmetry in correlated systems remains elusive. Here, by performing numerically-exact state-of-the-art quantum Monte Carlo (QMC) simulations, we show convincing evidences of deconfined quantum critical points (DQCP) between antiferromagnetic and valence-bond-solid phases in the extended Hubbard model of fermions on the honeycomb lattice with large system sizes. We further demonstrate evidences of the SO(5) symmetry at the DQCP. It is important to note that the critical exponents obtained by finite-size scaling at the DQCP here are consistent with the rigourous conformal bounds. Consequently, we established a promising arena of DQCP with emergent SO(5) symmetry in interacting systems of fermions. Its possible experimental relevances in correlated systems of Dirac fermions will be discussed briefly.