for the twist considered here. For smaller values than $N\sim 20$, we did not observe a signal of breakdown of the $\mathbb{Z}_N^4$ symmetry.

  $N$    $L$       $\beta_c^H$          $\beta_c^L$      $N$     $L$        $\beta_c^H$          $\beta_c^L$
  ------ ----- -------------------- -------------------- ------- ------ -------------------- --------------------
  $9$    $3$            -                    -           $64$    $8$     $0.2225\pm 0.0025$   $0.4625\pm 0.0025$
  $16$   $4$            -                    -           $81$    $9$     $0.2125\pm 0.0025$   $0.5175\pm 0.0025$
  $25$   $5$    $0.2925\pm 0.0025$   $0.3625\pm 0.0025$  $100$   $10$    $0.2075\pm 0.0025$   $0.5875\pm 0.0025$
  $36$   $6$    $0.2575\pm 0.0025$   $0.3925\pm 0.0025$  $121$   $11$    $0.2025\pm 0.0025$   $0.6525\pm 0.0025$
  $49$   $7$    $0.2375\pm 0.0025$   $0.4225\pm 0.0025$  $144$   $12$    $0.1975\pm 0.0025$   $0.7325\pm 0.0025$

  : Critical lattice coupling from the weak coupling side $\beta_c^L$ and from strong coupling side $\beta_c^H$ for the minimal skew-diagonal twist $(m=1)$.[]{data-label="TAB:minimal skew-diagonal"}

  ------------------- --
  \[FIG:m=1B\_low\]   
  ------------------- --

### Generic skew-diagonal twist {#generic-skew-diagonal-twist .unnumbered}

Here, we show the numerical result of the generic twist (\[EQ:generic\_twist\]). For this twist we use the skew-diagonal form because the $\mathbb{Z}_N^4$ symmetry breaking occurs at smaller $N$ than that in the symmetric form, which makes our investigation much easier.

We measure $\beta_c^L$ for this twist up to $m=4$. Table \[TAB:m=234\] shows the $\beta_c^L$ for $m=2, 3, 4$ and that for $m=1$ is presented in table \[TAB:minimal skew-diagonal\]. These data are 