  We present a comprehensive computational study of the phase diagram of the
frustrated S=1/2 Heisenberg antiferromagnet on the honeycomb lattice, with
second-nearest (J2) and third-neighbor (J3) couplings. Using a combination of
exact diagonalizations of the original spin model, of the Hamiltonian projected
into the nearest neighbor short range valence bond basis, and of an effective
quantum dimer model, as well as a self-consistent cluster mean-field theory, we
determine the boundaries of several magnetically ordered phases in the region
J2,J3\in [0,1], and find a sizable magnetically disordered region in between.
We characterize part of this magnetically disordered phase as a plaquette
valence bond crystal phase. At larger J2, we locate a sizable region in which
staggered valence bond crystal correlations are found to be important, either
due to genuine valence bond crystal ordering or as a consequence of
magnetically ordered phases which break lattice rotational symmetry.
Furthermore we find that a particular parameter-free Gutzwiller projected
tight-binding wave function has remarkably accurate energies compared to
finite-size extrapolated ED energies along the transition line from
conventional N\'eel to plaquette VBC phases, a fact that points to possibly
interesting critical behavior - such as a deconfined critical point - across
this transition. We also comment on the relevance of this spin model to model
the spin liquid region found in the half-filled Hubbard model on the honeycomb
lattice.
