superconducting pairing.

Now, we can write Heisenberg interaction term into its momentum-space version as[@Coleman] $$\begin{aligned}
H_{H}=\frac{1}{2}\sum_{ q}J_{q}\vec{S}_{q}^{f}\cdot \vec{S}_{-q}^{f}\nonumber\end{aligned}$$ where $J_{q}=2J_{H}(\cos q_{x}+\cos q_{y})$ for square lattice. Then, using the auxiliary fermionic representation of spin operator, the Heisenberg term is rewritten as $$\begin{aligned}
H_{H}&&=\sum_{k,k',q}J_{q}(\frac{-1}{2}f_{k+q\uparrow}^{\dag}f_{k'-q\downarrow}^{\dag}f_{k'\downarrow}f_{k\uparrow}\nonumber\\
&&+\frac{\sigma\sigma'}{8}f_{k+q\sigma}^{\dag}f_{k'-q\sigma'}^{\dag}f_{k'\sigma'}f_{k\sigma}).\nonumber\end{aligned}$$

Usually, it is well-known that in heavy fermion system the dominated magnetic correlation has antiferromagnetic feature, thus the spin-triplet pairing channel is not favored and we will only consider the spin-singlet channel in our remaining parts. Therefore, the effective interaction in spin-singlet pairing channel is $$\begin{aligned}
H_{H}=\sum_{k,k',q}\frac{-3}{4}J_{q}f_{k+q\uparrow}^{\dag}f_{k'-q\downarrow}^{\dag}f_{k'\downarrow}f_{k\uparrow}.\label{eq4}\end{aligned}$$ On physical ground, such term describes that a pair of fermions with opposite spins feels a pairing interaction as $V_{q}=\frac{-3}{4}J_{q}$.

Then, let $q\rightarrow k-k'$ and symmetrize with momentum, we have the following pairing interaction[@Coleman] $$\begin{aligned}
V_{k,k'}=\frac{1}{2}(V_{k-k'}+V_{k+k'})=-\frac{3}{2}J(\cos k_{x}\cos k_{x}'+\cos k_{y}\cos k_{y}'),\nonumber\end{aligned}$$ which leads to an effective BCS pairing Hamiltonian as $$\begin{aligned}
H_{pairing}=\sum_{k,k'}V_{k,k'}f_{k\uparrow}^{\dag}f_{-k\downarrow}^{\dag}f_{-k'\downarrow}f_{k'\uparrow}.\nonumber\end{aligned}$$ Considering the C$_{4}$ rotation symmetry of square lattice,[@Kotliar1988] the present pairing interaction can be split into extended $s$-wave ($\gamma_{k}^{s}=\cos k_{x}+\cos k_{y}$) and $d_{x^{2}-y^{2}}$-wave ($\gamma_{k}^{d}=\cos k_{x}-\cos k_{y}$) pairing channel as $$\begin{aligned}
&&V_{k,k'}=V_{k,k'}^{s}+V_{k,k'}^{d},\nonumber \\
&&V_{k,k'}^{s}=-\frac{3}{4}J_{H}\gamma_{k}^{s}\gamma_{k'}^{s},\nonumber \\
&&V_{k,k'}^{d}=-\frac{3}{4}J_{H}\gamma_{k}^{d}\gamma_{k'}^{d}. \nonumber\end{aligned}$$ In other words, Heisenberg-like local antiferromagnetic exchange interaction may mediate extended-s and $d_{x^{2}-y^{2}}$-wave pairing on square lattice.

For each pairing symmetry, we obtain 