  Using the density matrix renormalization group, we investigate the $S=1/2$
Heisenberg model on the honeycomb lattice with first- ($J_1$) and
second-neighbor ($J_2$) interactions. We are able to study long open cylinders
with widths up to 12 lattice spacings. For $J_2/J_1$ near 0.3, we find an
apparently paramagnetic phase, bordered by an antiferromagnetic phase for
$J_2\lesssim 0.26$ and by a valence bond crystal for $J_2\gtrsim 0.36$. The
longest correlation length that we find in this intermediate phase is for
plaquette valence bond (PVB) order. This correlation length grows strongly with
cylinder circumference, indicating either quantum criticality or weak PVB
order.
