# 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?

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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 '13 at 12:25
@Afriendlyhelper Thanks, but I am not sure what is the RG scheme for a lattice gauge theory. The lattice geometry is very important. Like the Kagome lattice I considered here is highly frustrated. Shouldn't that make a difference with the usual QFT RG? – Everett You Apr 19 '13 at 4:03
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 '13 at 15:00
@Afriendlyhelper: Figuring out the sign of the beta function is an indication of asymptotic freedom, but confinement requires non-peturbative methods. – JamalS Apr 7 '14 at 15:12

I have a question on the Wilson loop approach, do we have to choose the loop to have the shape like a rectangle whose one edge is time $T$ and the other edge is a distance $R$ and require $T\gg R$? In a theory with Lorentz invariance, this choice may not be too important because time and space are not really different. But in a discrete lattice gauge theory we may not have Lorentz invariance, so it seems this choice of the loop is relevant? Another question: can we say confinement comes if the gauge fluctuation is so strong that electric flux can penetrate the system and charges can interact? – Mr. Gentleman Aug 24 '14 at 3:02