&
\multicolumn{3}{c}{\longrightarrow}      & \mathbb{Z}_N^0 \;\; &
\mbox{($N=144$, standard, hot start)},\\
\mathbb{Z}_N^4 & \longleftarrow          & \mathbb{Z}_N^3 & \longleftarrow  &
\mathbb{Z}_N^2 & \longleftarrow          & \mathbb{Z}_N^1 & \longleftarrow  &
\mathbb{Z}_N^0 & \xleftarrow{\beta_c^L}  & \mathbb{Z}_N^4 \;\; &
\mbox{($N=144$, standard, cold start)},
\end{array}
\label{EQ:breaking_pattern_sym}\end{aligned}$$ where $\beta_c^H$ and $\beta_c^L$ are the first breaking point for the hot start and that for cold start, respectively. Note that although there is recovery of the symmetry for the cold start, the symmetry remains broken for the hot start.

In this section we show the results of the numerical simulation for this symmetry breaking phenomena. In order to argue about the possibility of the continuum and large-$N$ limiting procedure for this model, we mainly focus on the first breaking point for the cold start $\beta_c^L$, which depends on $N$.[^2]

Simulation method
-----------------

In our simulation we use the pseudo-heatbath algorithm. The algorithm is based on [@Fabricius:1984wp], and in each sweep over-relaxation is performed five times after multiplying $SU(2)$ matrices. The number of sweeps is $O(1000)$ for each $\beta$. We scan the symmetry breaking on the resolution of $\Delta\beta=0.005$, and then we always quote the value $\pm 0.0025$ as the error due to the resolution. Note that the breaking points are 