  The nature of quantum spin liquids is studied for the spin-$1/2$
antiferromagnetic Heisenberg model on a square lattice containing exchange
interactions between nearest-neighbor sites, $J_1$, and those between
next-nearest-neighbor sites, $J_2$. We perform variational Monte Carlo
simulations together with the quantum-number-projection technique and clarify
the phase diagram in the ground state together with its excitation spectra. We
obtain the nonmagnetic phase in the region $0.4< J_2/J_1\le 0.6$ sandwiched by
the staggered and stripe antiferromagnetic (AF) phases. Our direct calculations
of the spin gap support the notion that the triplet excitation from the singlet
ground state is gapless in the range of $0.4 < J_2/J_1 \le 0.5$, while the
gapped valence-bond-crystal (VBC) phase is stabilized for $0.5 < J_2/J_1 \le
0.6$. The VBC order is likely to have the columnar symmetry with a spontaneous
symmetry breaking of the $C_{4v}$ symmetry. The power-law behaviors of the
spin-spin and dimer-dimer correlation functions in the gapless region are
consistent with the emergence of the algebraic quantum-spin-liquid phase
(critical phase). The exponent of the spin correlation $\langle S(0)S(r)\rangle
\propto 1/r^{z+\eta}$ at a long distance $r$ appears to increase from
$z+\eta\sim 1$ at $J_2/J_1\sim0.4$ toward the continuous transition to the VBC
phase at $J_1/J_1\sim0.5$. Our results, however, do not fully exclude the
possibility of a direct quantum transition between the staggered AF and VBC
phases with a wide critical region and deconfined criticality.
