  We construct a class of projected entangled pair states (PEPS) which is
exactly the resonating valence bond (RVB) wavefunctions endowed with both short
range and long range valence bonds. With an energetically preferred RVB
pattern, the wavefunction is simplified to live in a one parameter variational
space. We tune this variational parameter to minimize the energy for the
frustrated spin 1/2 J1-J2 antiferromagnetic Heisenberg model on the square
lattice. Taking a cylindrical geometry, we are able to construct four
topological sectors with even or odd number of fluxes penetrating the cylinder
and even or odd number of spinons on the boundary. The energy splitting in
different topological sectors is exponentially small with the cylinder
perimeter. We find a power law decay of the dimer correlation function on a
torus, and a lnL correction to the entanglement entropy, indicating a gapless
spin liquid phase at the optimum parameter.
