  The frustrated Heisenberg $J_{1}-J_{2}$ model on a square lattice is
numerically investigated by variational Monte Carlo simulations. We propose a
antiferromagnetic fermion resonating-valence-bond (AF-fRVB) state that has
ability to examine the entire phase diagram in the $J_{1}-J_{2}$ model. Two
phase transition points, the second order around $J_{2}/J_{1}=0.45$ and the
first order around $J_{2}/J_{1}=0.6$, can be extracted more clearly than the
conventional bosonic RVB state. At the maximally frustrated point
($J_{2}/J_{1}=0.5$), the AF-fRVB state shows the variational ground-state
energy in the thermodynamic limit very close to the one estimated by the
projected entangled pair state at the largest bond dimension available. On the
other hand, in the frustrated regime $0.4\lesssim J_{2}/J_{1}\leq0.5$, AF-fRVB
states with $s_{+-}$ (using the terminology in the field of iron-based
superconductors) and $d_{xy}$ pairing symmetries are degenerate in the
thermodynamic limit, implying the existence of gapless Dirac excitations in the
spinon spectrum.
