two decoupled Ising models, and has thus ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ symmetry: $$\begin{aligned}
  \label{eq:Zsmu}
  Z
  &=
  \sum_{\sigma,\mu}
  \exp
  \sum_{\langle i,j\rangle}
  \left\{
    K_1 \rho_i\rho_j + 
    K_2 \mu_i\mu_j\right\}\, ,
  &
  \rho_i &= \sigma_i \mu_i
  \, .\end{aligned}$$ Denoting $K_c = -\frac{1}{2}\log\tanh K_c$ the critical coupling of the Ising model on a square lattice, there are two regimes which are of interest to us: The trivial phase ($K_2>K_c>K_1\ge 0$) where $\langle \mu_i\rangle\neq 0$, and the non-trivial phase ($K_1>K_c>K_2\ge 0$) where $\langle \rho_i\rangle\neq 0$. The other variables, $\rho$ and $\sigma$ for the trivial case and $\mu$ and $\sigma$ for the non-trivial, are disordered. Notably, in both cases $U_{ij}$’s are uncorrelated, namely $\left\langle (U_{ij}-\langle U_{ij} \rangle) (U_{kl}-\langle U_{kl}
  \rangle) \right\rangle$ is exponentially decaying. [^1] We remark that the partition function of Eq. (\[eq:model\]) with constraint violations at two plaquettes equals that of Eq.  where the sign of both couplings $K_1,K_2$ is reversed along a path connecting the two plaquettes [@Savit]. Thus for both regimes, the presence of order parameters with long range order implies linear confinement of the defects.

In terms of $\rho$ and $\mu$, the model is simply two decouple ferromagnets that exhibit symmetry broken phases. However, in the original degrees of freedom, $U,\sigma$, the physical properties 