Question

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

How to determine if an emergent gauge theory is deconfined or not?

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:

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

Answer 1

"Confined" $Z_N$ gauge theory may not be a well defined concept.  To make this question well defined, one need to first using measurable quantities to define the meaning of "confined" $Z_N$ gauge theory.

According one definition, a "confined" $Z_N$ gauge theory is a theory whose ground state has no topological order. In this case, any product state is a "confined" $Z_N$ gauge theory.