with $\eta_k = \cos k_{x}+\cos k_{y}$ due to the formation of nearest-neighbor valence-bond order $\chi=\sum_{\sigma}\langle f_{i\sigma}^{\dag}f_{j\sigma}\rangle$. Intuitively, such valence-bond order reflects the quantum dynamics of disordered local spins, which competes with magnetic long-ranged order. In addition, Lagrangian multiplier $\lambda$ is introduced to impose the local constraint on average and a constant energy shift $E_{0}=N_{s}[J_{K}V^{2}/2+J_{H}\chi^{2}-\lambda]$ with the number of lattice sites $N_{s}$ is added.

Using the following quasi-particle transformation relation $$\begin{aligned}
&&c_{k\sigma}=\alpha_{k}A_{k\sigma}-\beta_{k}B_{k\sigma}\nonumber \\
&&f_{k\sigma}=\beta_{k}A_{k\sigma}+\alpha_{k}B_{k\sigma}\label{eq2}\end{aligned}$$ with $\alpha_{k}^{2}=\frac{1}{2}(1+\frac{\varepsilon_{k}-\chi_{k}}{E_{0k}})$, $\beta_{k}^{2}=\frac{1}{2}(1-\frac{\varepsilon_{k}-\chi_{k}}{E_{0k}})$, $\alpha_{k}\beta_{k}=\frac{J_{K}V}{2E_{0k}}$ and $E_{0k}=\sqrt{(\varepsilon_{k}-\chi_{k})^{2}+(J_{K}V)^{2}}$, the original Hamiltonian Eq. (\[eq1\]) is transformed into a diagonalized Hamiltonian $$\begin{aligned}
H=\sum_{k\sigma}[E_{k}^{+}A_{k\sigma}^{\dag}A_{k\sigma}+E_{k}^{-}B_{k\sigma}^{\dag}B_{k\sigma}]+E_{0}\label{eq3}\end{aligned}$$ with the quasi-particle energy $E_{k}^{\pm}=\frac{1}{2}(\varepsilon_{k}+\chi_{k}\pm E_{0k})$ for quasi-particle $A_{k\sigma}$ and $B_{k\sigma}$, respectively.

Pairing interaction in Kondo-Heisenberg model {#sec3}
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Because we are mainly interested in the superfluid response in corresponding superconducting pairing phase, a proper superconducting model is required to perform any realistic and sensible calculation.

For the present Kondo-Heisenberg model, it has long been proposed that the introduced Heisenberg interaction is able to mediate spin-singlet pairing between conduction electrons via the preformed pairing of auxiliary fermions.[@Coleman1989; @Liu2012; @Asadzadeh2014; @Morr2014] Microscopically, the underlying mechanism of preformed pairing of auxiliary fermions can be attributed to the celebrated RVB state or alternatively the spin-fluctuation-exchange theory.[@Scalapino2012; @Anderson2004; @Lee2006; @Coleman2010] However, to 