by $N/p$. In order to keep the physical message of this section clear and concise, we do not delve here in these number theoretic considerations, and assume $N$ to be prime.

To analyze the model we first expose the hidden order. To this end we resolve the constraint using $$\begin{aligned}
U_{ij} &= \mu_i \mu^{-1}_j\end{aligned}$$ yielding $$\begin{aligned}
  \mathscr{H}_{p} 
  &= 
  \sum_{i}\sum_{j\sim i}
  \sigma^p_i \mu_i \mu^{-1}_j \sigma^{-p}_j\, ,\end{aligned}$$ and $$\begin{aligned}
  Z &= \frac{1}{N}\sum_{\sigma,\mu}e^{-\beta\mathscr{H}}\, ,\end{aligned}$$ where the factor of $\frac{1}{N}$ comes from the $1$ to $N$ mapping between $U_{ij}$ which respect the constraint and $\mu_i$.

Next we wish to go to the composite variables $$\begin{aligned}
\tilde{\sigma}_{i;p} = \mu_i \sigma^p_i\, ,\quad
\tilde{\mu}_{i;p'} = \mu_i \sigma_i^{p'} \, .\end{aligned}$$ The assumption of $N$ prime guarantees that they are in ${\mathbb{Z}}_N$, and the assumption of $p\neq p'$ and a non-zero $p$ guarantees the mapping to be invertible. The indices $p,p'$ make explicit the dependence on $p$ and $p'$ in the definition of $\tilde{\sigma}$ and $\tilde{\mu}$.

We thus find two decoupled $Z_N$ clock models, $$\begin{aligned}
  -\beta\mathscr{H}_{p,p'} &= 
  \sum_i\sum_{j\sim i}
  \left(K_1 \tilde{\sigma}_{i;p}  \tilde{\sigma}_{j;p}^{-1} 
  + 
  K_2 \tilde{\mu}_{i;p'}\tilde{\mu}_{j;p'}^{-1}\right)\, ,\end{aligned}$$ one in the composite variable $\tilde{\sigma}_p$ and the other in the composite variable $\tilde{\mu}_{p'}$. Now we suppose that the couplings are such that 