symmetric form case (\[EQ:breaking\_pattern\_sym\]). The first breaking pattern $\mathbb{Z}_N^4\xrightarrow{\beta_c^L}\mathbb{Z}_N^0$ is, however, the same as that in the symmetric twist. (We note that for $N\le 81$ the first breaking pattern is $\mathbb{Z}_N^4\xrightarrow{\beta_c^L}\mathbb{Z}_N^2$, which resembles the pattern $\mathbb{Z}_N^4\xrightarrow{\beta_c^L}\mathbb{Z}_N^3$ at $N=100$ for the symmetric form [@Teper:2006sp].)

Table \[TAB:minimal skew-diagonal\] shows the first breaking points for the cold start $\beta_c^L$ and for the hot start $\beta_c^H$. These data are plotted in figure \[FIG:m=1B\_low\] for $\beta_c^L$ and figure \[FIG:m=1B\_high\] for $\beta_c^H$. Again, we find clear linear dependence on $N$ for $\beta_c^L$, as we found for the symmetric form. Additionally, we also find clear dependence on $1/N$ for $\beta_c^H$. The fitted results are $$\begin{aligned}
\beta_c^L &\sim& 0.0034N+0.25,\label{EQ:bc_L_minimal_skew}\\
\beta_c^H &\sim& \frac{2.9}{N}+0.18,\label{EQ:bc_H_minimal_skew}\end{aligned}$$ where we used only $N\geq64$ data for $\beta_c^L$, whereas all data are used for $\beta_c^H$. As $N$ is increased the $\beta_c^H$ approaches a point $0.190$, where the phase transition $\mathbb{Z}_N^4\xrightarrow{\beta_c^H}\mathbb{Z}_N^3$ takes place in the original EK model. These results suggest that the quantum fluctuation is so large that the $\mathbb{Z}_N^4$ symmetry is broken in exactly the same region as that in the original EK model. The lines for transitions $\beta_c^L$ and $\beta_c^H$ seem to intersect around the bulk transition point $\beta_c^B\sim0.35$, which corresponds to $N\sim 20$ 