**What are the examples of "confined" $Z_N$ gauge theory?**

From these two questions:

Phase Structure of (Quantum) Gauge Theory

http://physics.stackexchange.com/questions/102541

http://physics.stackexchange.com/questions/29359/

We learn that there are deconfined $Z_N$ gauge theory such as the $Z_N$-toric code (Kitaev) or $Z_N$-topological order (Wen). There are fractionalized anyons as excitations separated from the ground state by an order O(J) gap where $J$ is the coupling of lattice Hamiltonian. These "deconfined" $Z_N$ gauge theories are beyond Landau-Ginzburg theory, and the "deconfined" $Z_N$ gauge theories cannot be classified by global symmetry-breaking pattern.

However, there are discussions in the posts above concerning **"confined" $Z_N$ gauge theory. What are the examples of "confined" $Z_N$ gauge theory? Are "confined" $Z_N$ gauge theory within Landau-Ginzburg theory, and the "confined" $Z_N$ gauge theories can be classified by global symmetry-breaking pattern? **

**For example, is there an example of "confined" $Z_N$ gauge theory in 1+1d, 2+1d, 3+1d, etc? **

**Is the 1+1d example accessible through the transverse magnetic field on Ising Hamiltonian:**

$$H(\sigma) = - \sum_{\langle i~j\rangle} J_{ij} \sigma_i \sigma_j -\mu \sum_{j} h_j\sigma_j$$