gap functions $\Delta_{k}^{A}=\Delta\gamma_{k}\beta_{k}^{2}$ and $\Delta_{k}^{B}=\Delta\gamma_{k}\alpha_{k}^{2}$ for quasi-particle $A_{\sigma}$ and $B_{\sigma}$, respectively. Note that such gap function is modified by the normal-state coherent factor, whose momentum-dependence deviates from the pure pairing function $\gamma_{k}$ but the qualitative physics is unchanged by such modulation. Now, we see that the advantage of such simplification is to decouple each heavy fermion quasi-particle band, which leads to two independent BCS-like mean-field Hamiltonian. Therefore, after introducing Bogoliubov transformation for each quasi-particle band as $$\begin{aligned}
&&A_{k\sigma}=\mu_{k}^{A}\tilde{A}_{k\sigma}-\nu_{k}^{A}\tilde{A}_{-k-\sigma}^{\dag}\nonumber\\
&&B_{k\sigma}=\mu_{k}^{A}\tilde{B}_{k\sigma}-\nu_{k}^{A}\tilde{B}_{-k-\sigma}^{\dag}\label{eq7}\end{aligned}$$ with $$\begin{aligned}
&&\mu_{k}^{A}=1-\nu_{k}^{A}=\frac{1}{2}(1+\frac{E_{k}^{+}}{E_{k}^{A}})\nonumber\\
&&\mu_{k}^{B}=1-\nu_{k}^{B}=\frac{1}{2}(1+\frac{E_{k}^{-}}{E_{k}^{B}}),\nonumber\end{aligned}$$ the resultant Hamiltonian is $$\begin{aligned}
H&&=\sum_{k}[E_{k}^{A}\tilde{A}_{k\sigma}^{\dag}\tilde{A}_{k\sigma}+E_{k}^{B}\tilde{B}_{k\sigma}^{\dag}\tilde{B}_{k\sigma}]\nonumber\\
&&+\sum_{k}(E_{k}^{+}+E_{k}^{-}-E_{k}^{A}-E_{k}^{B})+N_{s}E_{0}',\label{eq8}\end{aligned}$$ where the energy spectrum of superconducting quasi-particle has the following form $$\begin{aligned}
&&E_{k}^{A}=\sqrt{(E_{k}^{+})^{2}+(J_{H}\Delta_{k}^{A})^{2}}\nonumber\\
&&E_{k}^{B}=\sqrt{(E_{k}^{-})^{2}+(J_{H}\Delta_{k}^{B})^{2}}.\nonumber\end{aligned}$$ The corresponding free energy and mean-field self-consistent equations are given in Appendix B.

![\[fig:SC\] Above $T_{c}$, the itinerate heavy fermion quasi-particle forms heavy fermion liquid (HFL), which undergos a pairing instability into heavy fermion superconductor below $T_{c}$. When $J_{H}/J_{K}$ increases, the magnetic fluctuation is strengthened and magnetic or spin liquid states may ultimately dominate beyond critical ratio of $J_{H}/J_{K}$.](SC.eps){width="0.6\columnwidth"}

Now, the mean-field superconducting Hamiltonian for Kondo-Heisenberg model is obtained. When the temperature is above critical temperature $T_{c}$, there still exists the normal state described by Eq. \[eq3\], where heavy fermion quasi-particle moves freely in the renormalized 