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

2+1D lattice gauge theory can emerge in a spin system through fractionalization. Usually if the gauge structure is broken down to $\mathbb{Z}_N$, it is believed that the fractionalized spinons are deconfined. However in general, $\mathbb{Z}_N$ gauge theory also have a confined phase. The question is how to determine if the discrete emergent gauge theory is really deconfined or not?

For example, I am considering a $\mathbb{Z}_3$ gauge-Higgs model defined on the Kagome lattice with the Hamiltonian $H=J\sum_{\langle i j\rangle}\cos(\theta_i-\theta_j-A_{ij})$, where $\theta_i=0,\pm2\pi/3$ is the matter field and $A_{ij}=0,\pm2\pi/3$ is the gauge field. If the matter field is in a ferromagnetic phase, then I can understand that the gauge field will be Higgs out. But the matter field here is a Kagome antiferromagnet, which is strongly frustrated and may not order at low temperature. So in this case, I would suspect that the effective $\mathbb{Z}_3$ gauge theory will be driven into a confined phase. Is my conjecture right? How to prove or disprove that?

Hope I'm not raising the dead here: but naively thinking, couldn't you try and compute the $\beta$-function and find out its sign? Like you do in QFTs normally? –  A friendly helper Apr 18 at 12:25
The only way I know to "prove" or "disprove" confinement is simulating the system on a computer. Some other techniques do exist, but every time I attend some confinement-related conference, there's some people fighting each other about the validity of these methods. BTW, computing the $\beta$-function won't work, as (if I'm not mistaken) a Higgs-phase gauge theory may still have negative $\beta$-function while being completely and utterly deconfined. –  David Vercauteren Aug 14 at 15:00