for superfluid response can be applied to understand other heavy fermion superconductors like the prototypical compound CeCu$_{2}$Si$_{2}$ and PrOs$_{4}$Sb$_{12}$ with possible time-reversal-symmetry breaking,[@Pfleiderer2009; @Shu2009] if proper modification and extensions are included. More generally, it is expected our theory may also be useful for other multi-band superconductors, i.e. iron pnictide and LaOs$_{4}$Sb$_{12}$.[@Prozorov; @Tee2012]

The remainder of this paper is organized as follows. In Sec. \[sec2\], Kondo-Heisenberg model is introduced and its mean-field normal-state Hamiltonian is derived. In Sec. \[sec3\], the pairing interaction is discussed and the corresponding pairing Hamiltonian is derived. In Sec. \[sec4\], three different superfluid density formulas are derived and analyzed. We focus on the application of our theory to heavy fermion superconductor CeCoIn$_{5}$ and its doped compound Ce$_{1-x}$Yb$_{x}$CoIn$_{5}$ in Sec.\[sec5\]. In Sec. \[sec6\], both the impurity effect and non-local effect are discussed. Finally, Sec. \[sec7\] is devoted to a brief conclusion.

Kondo-Heisenberg model and normal state Hamiltonian {#sec2}
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The definition of Kondo-Heisenberg model is standard, which reads,[@Coleman1989; @Senthil2004] $$\begin{aligned}
H=\sum_{k\sigma}\varepsilon_{k}c_{k\sigma}^{\dag}c_{k\sigma}+J_{K}\sum_{i}S_{i}^{c}\cdot S_{i}^{f}+J_{H}\sum_{<i,j>}S_{i}^{f}\cdot S_{j}^{f} \nonumber\end{aligned}$$ For simplicity, the system is located on a regular square lattice and the resulting conduction electron band is $\varepsilon_{k}=-2t(\cos k_{x}+\cos k_{y})+4t'\cos k_{x}\cos k_{y}-\mu$ with chemical potential $\mu$. (Extension to other lattices like triangular or 