By using a variational Monte Carlo technique based upon Gutzwiller-projected fermionic states, we investigate the dynamical structure factor of the antiferromagnetic S = 1/2 Heisenberg model on the honeycomb lattice, in presence of first-neighbor (J 1) and second-neighbor (J 2) couplings, for J 2 < 0.5J 1. The ground state of the system shows long-range antiferromagnetic order for J 2/J 1 ≲ 0.23 (Néel phase), plaquette valence-bond order for 0.23 ≲ J 2/J 1 ≲ 0.36, and columnar dimer order for J 2/J 1 ⪆ 0.36. Within the Néel phase, a well-defined magnon mode is observed, whose dispersion is in relatively good agreement with linear spin-wave approximation for J 2 = 0. When a nonzero second-neighbor super-exchange is included, a roton-like mode develops around the K point (i.e., the corner of the Brillouin zone). This mode softens when J 2/J 1 is increased and becomes gapless at the transition point, J 2/J 1 ≈ 0.23. Here, a broad continuum of states is clearly visible in the dynamical spectrum, suggesting that nearly-deconfined spinon excitations could exist, at least at relatively high energies. For larger values of J 2/J 1, valence-bond order is detected and the spectrum of the system becomes clearly gapped, with a triplon mode at low energies. This is particularly evident for the spectrum of the dimer valence-bond phase, in which the triplon mode is rather well separated from the continuum of excitations that appears at higher energies.
