-------------------

  : $\beta_c^L$ for the generic skew-diagonal twist ($m=2, 3, 4$). See also Tab. \[TAB:minimal skew-diagonal\] for $m=1$.[]{data-label="TAB:m=234"}

[lr]{}

\[FIG:bc\_L\_mall\_skew1\]

&

Discussions {#SEC:discussions}
===========

In this section we discuss the numerical results obtained in the previous section and the validity of taking the large-$N$ and continuum limit for this model.

Theoretical estimation of the $\mathbb{Z}_N^4$ symmetry breaking point {#SEC:theoretical_estimation}
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In the previous section we showed our numerical results. In particular, we elaborately investigated $\beta_c^L$, the first $\mathbb{Z}_N^4$ breaking point from the cold start. From our investigation, we found the clear linear behavior like (\[EQ:bc\_L\_minimal\_sym\]), (\[EQ:bc\_L\_minimal\_skew\]) and (\[EQ:bc\_L\_generic\_skew\]). These behaviors can be obtained through the following consideration.

### Energy difference between twist-eater $\Gamma_{\mu}$ and identity $\mathbbm{1}_N$ configurations {#energy-difference-between-twist-eater-gamma_mu-and-identity-mathbbm1_n-configurations .unnumbered}

We simply assume that the $\mathbb{Z}_N^4$ breaking is a transition from twist-eater phase $U_{\mu}=\Gamma_{\mu}$ to identity configuration phase $U_{\mu}=\mathbbm{1}_N$. For plainness, we consider $\mathbb{Z}_N^4\xrightarrow{\beta_c^L}\mathbb{Z}_N^0$ type breaking here. Of course we can treat $\mathbb{Z}_N^4\xrightarrow{\beta_c^L}\mathbb{Z}_N^3\xrightarrow{\beta_c^L}
 \mathbb{Z}_N^2\xrightarrow{\beta_c^L}\mathbb{Z}_N^1\xrightarrow{\beta_c^L}
 \mathbb{Z}_N^0$ (cascade) type breaking at a $\beta_c^L$, but the obtained behavior is not different from the former type. Firstly, we focus on the classical energy difference between these configurations. The energy can be easily calculated from the action (\[EQ:TEK\_action\]) as $$\begin{aligned}
\Delta S
&=&S_{TEK}[U_{\mu}=\mathbbm{1}_N]-S_{TEK}[U_{\mu}=\Gamma_{\mu}]\nonumber\\
&=&\beta N^2\sum_{\mu\neq\nu}
   \left\{ 