the Hoelbling fermion. Black circles stand for the leading one-point function $\langle \chi_x \bar{\chi}_x \rangle_0$ while white circles stand for $\langle \chi_x \bar{\chi}_x \rangle$ which include next-leading and higher hopping terms. By summing up higher contributions, we obtain the second equality.[]{data-label="One-H"}](HPEwHoelbling.eps){height="6cm"}

We begin with the Hoelbling-type fermion, which contains two-hopping terms in the action. One reason that we start with Hoelbling type regardless of the lower rotation symmetry is that 2-hopping calculation can be a good exercise for 4-hopping case in Adams type. To perform the HPE for the Hoelbling-type fermion, we rewrite the action Eq. (\[HoelS\]) by redefining $\chi \rightarrow \sqrt{2K} \chi$ with the hopping parameter $K=1/[2(M+2r)]$, $$S =
\sum_{x} {{\bar{\chi}}}_x \chi_x + 2K \sum_{x, y} {{\bar{\chi}}}_{x}(\eta_{\mu}  
D_\mu )_{xy}\chi_{y}+ 2 K r \sum_{x,y} {{\bar{\chi}}}_x (M_{H})_{xy} \chi_y
\ ,
\label{HPE-Hoel}$$ where $M_{H}$ is given by Eq. (\[HoelM\]). The plaquette action is $1/g^2$ term and we can omit it in the strong-coupling limit. In this section, we derive one- and two-point functions by using a $\mathcal{O}(K^{3})$ self-consistent equation: Solving this equation leads to truncation of diagrams as the ladder approximation having all diagrams to $\mathcal{O}(K^3)$ are taken into account. More precisely, this approximation does not take account of all diagrams, but 