heavy fermion superconducting models. As an example, we consider the effective model proposed in Ref., which is acquired from fitting to quasi-particle interference experiments in CeCoIn$_{5}$.[@Allan2013; @Zhou2013] The details of their model is not given in this paper and we refer the reader to their original article, here only the calculation result is shown in Fig. \[fig:CeCoIn5\].

![\[fig:CeCoIn5\] Normalized superfluid density $\rho_{s}(T)/\rho_{s}(0)$ versus temperature T for heavy fermion superconductor CeCoIn$_{5}$ by using Eq. \[eq11\].](cci55.eps){width="0.5\columnwidth"}

When comparing to the experimental data of undoped CeCoIn$_{5}$ in Ref. , (See also Fig. \[fig:CeCoIn5\].) we see that the calculated $\rho_{s}(T)$ does not agree with the global behavior in the experimental measurement. This may be due to the fact that the model parameters in Ref.  are not temperature-dependent, thus certain physical quantities like superfluid density cannot be faithfully calculated. Therefore, it is important to include the thermal effect and we will revisit this issue in next section.

Superfluid response in heavy fermion superconductor CeCoIn$_{5}$ and Ce$_{1-x}$Yb$_{x}$CoIn$_{5}$ {#sec5}
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Superfluid response in heavy fermion superconductor CeCoIn$_{5}$
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![\[fig:10\] (a) Calculated normalized superfluid density $\rho_{s}(T)/\rho_{s}(0)$ in the superconducting state versus $T$. (b) The normalized superfluid density $\rho_{s}(T)/\rho_{s}(0)$ of CeCoIn$_{5}$ in Ref. .](111.eps){width="1.0\columnwidth"}

In this section, we proceed 