$x_{c}=0.2$. In addition, it is noted that the pairing symmetry transition here is likely a first-order transition since the corresponding derivative of ground-state energy is discontinuous at the putative transition point $x=0.2$. Thus, no radical change of observable like $T_{c}$ is expected around this transition, which is also consistent with the doping evolution of Yb in Ref. .

Before ending this section, it is interesting to note that for the heavy fermion superconductor CeRhIn$_{5}$,[@Hegger2000] where coexistence of antiferromagnetism and superconductivity is firmly established,[@Park2006] the present superfluid response formula may also be applicable if we use a four-quasi-particle energy-band model resulted from magnetic and pairing orders. Thus, we should sum the contributions from all of four bands with corresponding quasi-particle effective mass and velocity. It will be a good test for our theory to fit to the future London penetration depth measurement in this compound.

Nonmagnetic impurity effect and non-local effect on superfluid response  {#sec6}
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Nonmagnetic impurity effect
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For realistic materials, the nonmagnetic impurity scattering is unavoidable and is ubiquitous. Particularly, it plays a key role in the low temperature regime where linear-$T$ dependence of nodal d-wave is replaced by the $T^{2}$ law.[@Goldenfeld1993] Here, we give the corresponding formalism, 