it successfully includes certain kinds of diagrams to all orders of $K$ thanks to a self-consistent approach. We thus expect that it works to figure out existence of Aoki phase. We note that this approximation especially works well for a small hopping parameter $K\ll1$. In Fig. \[FR-H\], we depict Feynman rules in the HPE for this fermion. The fundamental Feynman rules contain contributions from 0-hopping (mass term), 1-hopping (kinetic term) and 2-hopping (flavored-mass term) terms.

First, by using these Feynman rules, we derive meson condensates from an one-point function of the meson operator $\mathcal{M}_{x}=\bar{\chi}_{x}\chi_{x}$ in the mean-field approximation. The one-point function is defined as $$\begin{aligned}
\langle \chi_x^a \bar{\chi}_x^b \rangle
\equiv& - \delta_{ab} \Sigma_x
= \frac{\int \mathcal{D}[\chi,\bar{\chi},U] \chi_x^a \bar{\chi}_x^b\ e^{S}
}{
\int \mathcal{D}[\chi,\bar{\chi},U] e^{S}
}\ .\end{aligned}$$ Note that we use the partition function $Z=\int \mathcal{D}[\chi,\bar{\chi},U] e^{S}$, not $Z=\int \mathcal{D}[\chi,\bar{\chi},U] e^{-S}$, following the convention for the partition function in the strong-coupling analysis [@KaS]. The leading term in the hopping parameter expansion is given by $$\begin{aligned}
\langle \chi_x^a \bar{\chi}_x^b \rangle_0
= \frac{\int \mathcal{D}[\chi,\bar{\chi},U]\, \chi_x^a \bar{\chi}_x^b\ e^{S_0}}
       {\int \mathcal{D}[\chi,\bar{\chi},U]\, e^{S_0}}
= - \delta^{ab}
\ ,\end{aligned}$$ where $S_0=\sum_x \bar{\chi}_x \chi_x$. By using the Feynman rules, we can evaluate the diagrams in Fig. \[One-H\]. $$\begin{aligned}
\langle 