heavy Fermi liquid. If we cools the system below $T_{c}$, the normal heavy Fermi liquid transits into the heavy fermion superconducting state with pairing of heavy quasi-particle. To avoid the instability to quantum spin liquid or fractionalized Fermi liquid states,[@Senthil2004] the ratio of $J_{H}/J_{K}$ should not be larger than certain critical value. These are summarized in Fig. \[fig:SC\]

Superfluid response formula {#sec4}
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In this section, we will study the superfluid response function in heavy fermion superconductors. The main point of this section is for superfluid response in heavy fermion superconductors, which one is practically coupled to external electromagnetic field $\vec{A}$, conduction electron, auxiliary fermion or heavy fermion quasi-particle? As shown in the detailed discussion in this section, it is found that only the heavy fermion quasi-particle is most likely to couple with $\vec{A}$ strongly, which is also depicted in Fig. \[fig:ansatz\].

Superfluid response without auxiliary fermion
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For Kondo-Heisenberg model, by definition, the external electromagnetic potential $\vec{A}$ is exerted on conduction electron by substitution $t_{ij}\rightarrow t_{ij}e^{ieA_{ij}}$, where $t_{ij}$ is the hopping energy in real-space. Thus, the Hamiltonian is invariant under local U(1) charge gauge transformation via $$\begin{aligned}
&&c_{\sigma}\rightarrow c_{\sigma}e^{i\theta_{i}},\nonumber\\
&&A_{ij}\rightarrow A_{ij}+(\theta_{i}-\theta_{j})/e\nonumber\end{aligned}$$ while the auxiliary fermion $f_{\sigma}$ does not directly 