  The spin-1/2 $J_1$-$J_2$ Heisenberg model on square lattices are investigated
via the finite projected entangled pair states (PEPS) method. Using the
recently developed gradient optimization method combining with Monte Carlo
sampling techniques, we are able to obtain the ground states energies that are
competitive to the best results. The calculations show that there is no N\'eel
order, dimer order and plaquette order in the region of 0.42 $\lesssim
J_2/J_1\lesssim$ 0.6, suggesting a single spin liquid phase in the intermediate
region. Furthermore, the calculated staggered spin, dimer and plaquette
correlation functions all have power law decay behaviours, which provide strong
evidences that the intermediate nonmagnetic phase is a single gapless spin
liquid state.
