section we again begin with the Hoelbling case as exercise, and go on to Adams fermion with better discrete symmetry.

Hoelbling type {#hoelbling-type}
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In the strong-coupling limit we can drop plaquette action. Then the partition function for meson fields $\mathcal{M}_x=({{\bar{\chi}}}_x \chi_x)/N$ with the source $J_{x}$ is given by $$\begin{aligned}
Z(J) &= \displaystyle \int {{\cal D}}\left[ \chi, {{\bar{\chi}}}, U \right]
 \exp \left[ N \sum_x J_x \mathcal{M}_x + S_F \right]
\ .
\label{ZJ}\end{aligned}$$ where $S_{F}$ stands for the fermion action. Here we have defined $\mathcal{M}_x$ with a $1/N$ factor to ensure it to have a certain large $N$ limit. In this case, $S_F$ is the Hoelbling-type staggered-Wilson action Eq. (\[HoelS\]). $N$ stands for the number of color. In the large $N$ limit, we can perform the link integral. We here consider the effective action up to $\mathcal{O}(\mathcal{M}^{3})$ for meson field $\mathcal{M}$. This order corresponds to the $\mathcal{O}(K^{3})$ self-consistent equation in the hopping parameter expansion.

We develop a method to perform the link-variable integral with multi-hopping fermion action terms. In our method, we perform the link integral by introducing two kinds of link-variable measures. Now we formally rewrite the partition function as, $$\begin{aligned}
Z(J) &= \displaystyle \int {{\cal D}}\left[ \chi, {{\bar{\chi}}}\right]
 \exp 