  We study the spin-1/2 Heisenberg model on the square lattice with first- and
second-neighbor antiferromagnetic interactions J1 and J2, which possesses a
nonmagnetic region that has been debated for many years and might realize the
interesting Z2 spin liquid. We use the density matrix renormalization group
approach with explicit implementation of SU(2) spin rotation symmetry and study
the model accurately on open cylinders with different boundary conditions. With
increasing J2, we find a Neel phase, a plaquette valence-bond (PVB) phase with
a finite spin gap, and a possible spin liquid in a small region of J2 between
these two phases. From the finite-size scaling of the magnetic order parameter,
we estimate that the Neel order vanishes at J2/J1~0.44. For 0.5<J2/J1<0.61, we
find dimer correlations and PVB textures whose decay lengths grow strongly with
increasing system width, consistent with a long-range PVB order in the
two-dimensional limit. The dimer-dimer correlations reveal the s-wave character
of the PVB order. For 0.44<J2/J1<0.5, spin order, dimer order, and spin gap are
small on finite-size systems and appear to scale to zero with increasing system
width, which is consistent with a possible gapless SL or a near-critical
behavior. We compare and contrast our results with earlier numerical studies.
