{1}{2^3 \sqrt{3}} \right)^2 
   (-1) \left( K \eta_{\rho,x} \right)^2 
   \langle (\chi^a \bar{\chi})_x \rangle_0  
   \mathcal{W}_{\mu\nu,x}^{(2)} 
{\nonumber}\\  
& \times
   \langle (\chi \bar{\chi})_{x+\hat{\mu}+\hat{\nu}} \rangle_0 
   U_{\rho,x+\hat{\mu}+\hat{\nu}}
   \langle (\chi \bar{\chi})_{x+\hat{\mu}+\hat{\nu}+\hat{\rho}} \rangle_0 
   U_{\rho,x+\hat{\mu}+\hat{\nu}}^\dagger
   \langle (\chi \bar{\chi})_{x+\hat{\mu}+\hat{\nu}} \rangle_0 
   \mathcal{W}_{\mu\nu,x}^{(2)\dagger} 
   \langle (\chi \bar{\chi}^b)_x \rangle_0 
{\nonumber}\\  
& + \mathcal{O}(K^5)
\ ,
\label{Eq:HPE-Hoelbling1-detail}\end{aligned}$$ where $(\chi \bar{\chi})_x$ stands for $\chi_x \bar{\chi}_x$ and $\mathcal{W}_{\mu\nu,x}^{(2)}=\mathcal{W}_{+\mu+\nu,x}^{(2)}$ in Table. \[Table:U2\]. Note that we consider only connected diagrams in Fig. \[One-H\]. By summing higher hopping terms, the one-point function is obtained as shown in Fig. \[One-H\], which is given by $$- \Sigma_x  \equiv - \langle \mathcal{M}_x \rangle =
 - \langle \mathcal{M}_x \rangle_0 
 + 2K^2 \sum_\mu \Sigma_x \Sigma_{x+\hat{\mu}} 
 - 2 \cdot \displaystyle \frac {1}{24} (Kr)^2 \sum_{\mu \neq \nu} \Sigma_x 
 \Sigma_{x+\hat{\mu}+\hat{\nu}}
\ .
\label{Eq:HPE-Hoelbling1}$$ The equation contains terms to $\mathcal{O}(K^{2})$, and $\mathcal{O}(K^{3})$ diagrams are found to vanish due to cancellation between the diagrams. Here we solve it in a self-consistent way for condensate $\Sigma$ within mean-field approximation. We here assume $\Sigma_x=\sigma_x +i \epsilon_x \pi_x$ as the condensate. $\sigma$ and $\pi$ correspond to chiral and pion condensates, respectively. We substitute this form of $\Sigma_{x}$ in Eq. (\[Eq:HPE-Hoelbling1\]) and obtain a self-consistent equation $$- \left( \sigma +i \epsilon_x \pi \right)=-1 + 2K^2 \cdot 4 \left( \sigma^2 
+ \pi^2 \right) -2 \cdot 