response to $\vec{A}$ because it originates from the fermionic representation of spin and has no electric charge.[@Coleman]

Therefore, we assume that only conduction electron is involved in the calculation of superfluid response.

To discuss the superfluid response function, we consider conduction electron is coupled to electromagnetic field $\vec{A}_{q}$ via the following Hamiltonian[@Coleman] $$\begin{aligned}
H'&&=\sum_{kq\sigma}e\vec{v}_{k}\cdot\vec{A}_{q}c_{k+q\sigma}^{\dag}c_{k\sigma}\nonumber\\
&&=\sum_{kq}e\vec{v}_{k}\cdot\vec{A}_{q}(c_{k+q\uparrow}^{\dag}c_{k\uparrow}+c_{k+q\downarrow}^{\dag}c_{k\downarrow})\nonumber\\
&&=\sum_{kq}e\vec{v}_{k}\cdot\vec{A}_{q}(c_{k+q\uparrow}^{\dag}c_{k\uparrow}+c_{-k\downarrow}c_{-k-q\downarrow}^{\dag})\nonumber\\
&&=\sum_{kq}e\vec{v}_{k}\cdot\vec{A}_{q}\psi_{k+q}^{\dag}\psi_{k}.\nonumber\end{aligned}$$ Here, we have defined conduction electron velocity $\vec{v}_{k}=\frac{\partial \varepsilon_{k}}{\partial k_{i}}$ and the Nambu spinor $\psi_{k}^{\dag}=(c_{k\uparrow}^{\dag},c_{-k\downarrow})$. Note that the bare vertex is simply $ie\vec{v}_{k}$, which leads to paramagnetic current-current correlation function as $$\begin{aligned}
\Pi(q)_{ij}^{p}&&=+e^{2}\sum_{k}v_{k}^{i}v_{k+q}^{j}Tr[G(k)G(k+q)]\nonumber\\
&&=+e^{2}\sum_{k}v_{k}^{i}v_{k+q}^{j}[G_{11}(k)G_{11}(k+q)+G_{12}(k)G_{21}(k+q)\nonumber\\
&&+(1\leftrightarrow2)],\nonumber\end{aligned}$$ where $G$ is $2\times2$-matrix Nambu Green’s function for conduction electron and its detail formalism is given in Appendix A.

Inserting the explicit expression of Green’s function and setting zero-energy limit $i\Omega_{n}=0$ and zero-momentum limit $\vec{q}=0$, we get the contribution of paramagnetic current is $$\begin{aligned}
Re\Pi(q=0)_{ij}^{p}=+2e^{2}\sum_{k}v_{k}^{i}v_{k}^{j}\left[\alpha_{k}^{4}\frac{\partial f_ {F}(E_{k}^{A})}{\partial E_{k}^{A}}+\beta_{k}^{4}\frac{\partial f_ {F}(E_{k}^{B})}{\partial E_{k}^{B}}\right].\nonumber\end{aligned}$$

On the other hand, the diamagnetic current is coupled to $\vec{A}$ through $$\begin{aligned}
H'&&=\sum_{kq\sigma}\frac{e^{2}}{2m}\vec{A}_{k'-q}^{\dag}\vec{A}_{k'}c_{k+q\sigma}^{\dag}c_{k\sigma}\nonumber\\
&&=\sum_{kq}\frac{e^{2}}{2m}\vec{A}_{k'-q}^{\dag}\cdot\vec{A}_{k'}(c_{k+q\uparrow}^{\dag}c_{k\uparrow}-c_{-k-q\downarrow}c_{-k\downarrow}^{\dag})\nonumber\\
&&=\sum_{kq}\frac{e^{2}}{2m}\vec{A}_{k'-q}^{\dag}\cdot\vec{A}_{k'}\psi_{k+q}^{\dag}\hat{\tau}_{z}\psi_{k}.\nonumber\end{aligned}$$ The effective mass tensor for conduction electron is defined as $(\frac{1}{m})_{ij}=\frac{\partial^{2} \varepsilon_{k}}{\partial k_{i}\partial k_{j}}$. Here, the interacting vertex is $-(\frac{e^{2}}{2m})_{ij}\hat{\tau}_{z}$ and its corresponding zero-frequency diamagnetic current-current correlation is $$\begin{aligned}
\Pi(q=0)_{ij}^{d}&&=+e^{2}\sum_{k}(\frac{1}{m})_{ij}Tr[\hat{\tau}_{z}G(k)]\nonumber\\
&&=e^{2}\sum_{k}(\frac{1}{m})_{ij}(G_{11}-G_{22}) \nonumber \\
&&=e^{2}\sum_{k}(\frac{-1}{m})_{ij}(\alpha_{k}^{2}\frac{E_{k}^{+}}{E_{k}^{A}}\tanh\frac{E_{k}^{A}}{2T}+\beta_{k}^{2}\frac{E_{k}^{-}}{E_{k}^{B}}\tanh\frac{E_{k}^{B}}{2T}). \nonumber\end{aligned}$$

Combining the contribution from paramagnetic and 