0.0025$  $225$   $15$    $0.4575\pm 0.0025$
  $121$   $11$    $0.3625\pm 0.0025$  $256$   $16$    $0.4875\pm 0.0025$
  $144$   $12$    $0.3775\pm 0.0025$  $289$   $17$    $0.5275\pm 0.0025$
  $169$   $13$    $0.3975\pm 0.0025$  $324$   $18$    $0.5675\pm 0.0025$
  $196$   $14$    $0.4225\pm 0.0025$                 

  : Critical lattice coupling from the weak coupling side $\beta_c^L$ for the minimal symmetric twist.[]{data-label="TAB:minimal symmetric twist"}

### Minimal skew-diagonal twist {#minimal-skew-diagonal-twist .unnumbered}

Twists can be always transformed into the skew-diagonal form by $SL(4,\mathbb{Z})$ transformation as we mentioned in section \[SEC:twist prescription\]. As it were, the minimal symmetric twist (\[EQ:minimal\_sym\_form\]) is equivalent to the minimal skew-diagonal twist (\[EQ:minimal\_skew\_form\]) in the weak coupling limit. However, both forms can represent different features by taking into account the quantum fluctuation. Actually, the $\mathbb{Z}_N^4$ symmetry is already broken at $N=25$. This fact enables us to observe the $N$-dependence of the critical points easily. Not only is the symmetry breaking point different from the symmetric form, so is the breaking and restoration pattern. Figure \[FIF:pattern\_N100\_cold-start\] shows the expectation value of the plaquette (top) and the Polyakov lines (besides the top) versus $\beta$ for the cold start at $N=100$. For $N\geq100$ we find the $\mathbb{Z}_N^4$ symmetry breaking and restoration pattern: $$\mathbb{Z}_N^4\leftarrow\mathbb{Z}_N^3\leftarrow\mathbb{Z}_N^2\leftarrow
\mathbb{Z}_N^0\xleftarrow{\beta_c^L}\mathbb{Z}_N^4\quad
\mbox{($N=100$, minimal skew-diagonal, cold start)},$$ which represents a difference from the 