\displaystyle \frac {1}{24} (Kr)^2 \cdot 4 \cdot 3 
\left( \sigma +i \epsilon_x \pi \right)^2
\ ,
\label{HPE-HoelSelf1}$$ which yields $- \sigma = -1 + 16 K^2 \pi^2$ and $- i \pi = - 8K^2 \cdot 2 i \sigma \pi$. For simplicity, we have set $r=2\sqrt{2}$ to make the equation Eq. (\[HPE-HoelSelf1\]) simpler. Of course we can also discuss for other values of $r$ in general.

Now we have two solutions depending on $\pi=0$ or $\pi\not=0$: For $\pi=0$, we have a trivial solution $\sigma=1$. For $\pi \neq 0$, we have a solution as $$\sigma = \displaystyle \frac{1}{16K^2},\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\pi = \pm \sqrt{ \displaystyle \frac{1}{16K^2} \left( 1- \displaystyle 
\frac{1}{16K^2} \right) }
\ .
\label{cond}$$ Nonzero pion condensate implies spontaneous parity breaking for the range $\mid{K}\mid > 1/4$. The sign of the pion condensate in Eq. (\[cond\]) reflects the $Z_2$ parity symmetry of the theory. Thus the parity-broken phase, if it exists, appears in a parameter range $-4\sqrt{2}-2<M<-4\sqrt{2}+2$ in the strong-coupling limit. We note that the critical hopping parameter $|K_c|=1/4$ is small, and we speculate that the $\mathcal{O}(K^{3})$ self-consistent equation is valid around the value.

Next, we discuss the meson mass from a two-point function of the meson operator $\mathcal{S}(0,x)\equiv\langle\mathcal{M}_{0}\mathcal{M}_{x}\rangle$. From Fig. \[Two-H\], we 