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

+ 6 like - 0 dislike
967 views

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?

This post imported from StackExchange Physics at 2014-04-05 03:24 (UCT), posted by SE-user Everett You
asked Jun 2, 2012
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?

This post imported from StackExchange Physics at 2014-04-05 03:24 (UCT), posted by SE-user A friendly helper
@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?

This post imported from StackExchange Physics at 2014-04-05 03:24 (UCT), posted by SE-user Everett You
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.

This post imported from StackExchange Physics at 2014-04-05 03:24 (UCT), posted by SE-user David Vercauteren

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar\varnothing$sicsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.