,width=3in]{fig4}
\caption{(a) The percentage of the energy difference between
$d_{x^{2}-y^{2}}+d_{xy}$ ($d+d$) and $d_{x^{2}-y^{2}}+s_{+-}$
($d+s$) AF-fRVB states as a function of $J_{2}$ for different
lattice size. (b) $d_{xy}$ and $s_{+-}$ pairing structures
plotted in the first Brillouin zone. The black diamond is the spinon
Fermi surface. White (Blue) regions present the positive (negative)
pairing gap. Red lines mean the nodal line.}\label{fig4}
\end{figure}

In order to investigate possible pairing symmetries in the
spin-liquid phase of the frustrated model, we also consider
different pairing structures of the AF-fRVB state. Due to the
pattern of the local Hamiltonian, we examine both
$s_{++}$/$d_{x^{2}-y^{2}}$ symmetry for the nearest-neighbor pairing
$\Delta_{1}$ and $s_{+-}$/$d_{xy}$ for the next-nearest-neighbor
$\Delta_{2}$. It is worth pointing out that the variational energy
cannot be optimized if we consider $s_{++}$ symmetry in $\Delta_{1}$
(not shown). The reason is that the $d_{x^{2}-y^{2}}$ singlet pair
would avoid the on-site Coulomb repulsion so that the
$d_{x^{2}-y^{2}}$ symmetry is more favorable than $s_{++}$ in the
Heisenberg model \cite{ScalapinoPR95}. As for $\Delta_{2}$,
interestingly, the $s_{+-}$ symmetry always competes with the
$d_{xy}$ symmetry for $J_{2}\leq0.4$ as shown in Fig.\ref{fig4}(a).

In the intermediate regime ($0.4<J_{2}\leq0.5$), nevertheless, the
AF-fRVB wave function with $s_{+-}$ symmetry shows much lower energy
than the one with $d_{xy}$ symmetry in finite-size calculations.
Notably, a further finite-size analysis illustrates that the energy
difference becomes smaller as increasing the size of lattice. Thus
it is reasonable to infer that the AF-fRVB wave functions with
$s_{+-}$ and $d_{xy}$ symmetry are always degenerate in the
thermodynamic limit. Since Fig.\ref{fig2}(a) shows that there is no
long-range magnetic order ($m_{i}=0$) in the intermediate regime
($0.4<J_{2}\leq0.5$), Eq.(\ref{e:eq03}) is just the mean-field BCS
Hamiltonian that can be easily diagonalized,
\begin{eqnarray}
E_{k}=\sqrt{\varepsilon_{k}^{2}+\Delta_{k}^{2}},\label{e:mf}
\end{eqnarray}
where $\varepsilon_{k}=-2\left(\cos(k_{x})+\cos(k_{y})\right)$ and
$\Delta_{k}$ is the gap function consisting of both the
nearest-neighbor pairing $\Delta_{1}$ and the next-nearest-neighbor
pairing $\Delta_{2}$. According to the spinon spectrum $E_{k}$, the
degeneracy can be understood by their equivalent nodal structures if
we simply shift the spinon Fermi surface (black diamond,
$\varepsilon_{k}=0$) by $(\pi,\pi)$ shown in Fig.\ref{fig4}(b). Note
that plus the $d_{x^{2}-y^{2}}$-wave form factor, the energy
dispersion $E_{k}$ with both $d_{x^{2}-y^{2}}$ and $s_{+-}$ pairing
symmetries ($d+s$) clearly displays four nodes at
$(\pm\pi/2,\pm\pi/2)$. Therefore, the spin-liquid state should be
gapless because of the mean-field spinon spectrum with four Dirac
points at $(\pm\pi/2,\pm\pi/2)$ for the $d+s$ pairing structure.

\textit{Conclusions.} We have numerically studied the ground-state
phase diagram of the $J_{1}-J_{2}$ Heisenberg model on a square
lattice based on the AF-fRVB wave function. The fermionic ansatz
with long-range AF orders has successfully reproduced the
ground-state phase diagram from ED \cite{RichterEPJB10} and DMRG
\cite{JiangPRB12} calculations. The AF-fRVB wave function also
captures a second-order transition at $J_{2}=0.45$ and a first-order
transition at $J_{2}=0.6$ that is consistent with the conclusion
made by the bosonic RVB states \cite{LiPRB12}. The AF-fRVB state
naturally solves the problem that the purely fermionic RVB state
cannot describe the magnetic ordered state for $J_{2}<0.45$ and
$J_{2}>0.6$. We have shown that the AF-fRVB wave function with few
variational parameters can reach almost the same energy as the PEPS
with very large bond dimension. In addition, although the mean-field
spinon spectrum is gapful for $d_{x^{2}-y^{2}}+d_{xy}$ symmetry, the
degeneracy from $s_{+-}$ and $d_{xy}$ pairing symmetries in the
frustrated regime suggests that the spin-liquid phase can also
exhibit Dirac spinon spectrum as a result of $d+s$ pairing symmetry.

We would like to thank F. Yang and T. Ma for useful discussions.
C.P.C. is supported by Chinese Academy of Engineering Physics and
Ministry of Science and Technology. The calculations are performed
in the National Center for High-performance Computing. H.Y.C. is
supported by National Science Council of Taiwan under Grant No.
NSC-101-2112-M-003-005-MY3 and National Center for Theoretical
Science of Taiwan.

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