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Please provide some examples of confinement-deconfinement phases of U(1) gauge theory in 2 spacetime dimensions (Low dimwnsional physics).

U(1) gauge theory can be:

pure U(1) gauge theory, or

U(1) gauge theory with matter fields (bosonic/fermionic).

It is the best to provide examples for both cases.

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  • $\begingroup$ ...what is your notion of "confinement" in a pure $\mathrm{U}(1)$ gauge theory? There are no charged particles in it to begin with. $\endgroup$
    – ACuriousMind ♦
    Commented Nov 10, 2016 at 16:46
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    $\begingroup$ Break the closed Wilson loop to an open Wilson line, you get two ends of line as the gauge charge. For example, the Z2 gauge theory breaks the Wilson line to Z2 charge of $e$ particle, such as Z2 toric code. Namely, I dont think we need to have the matter field to discuss confinement or not. $\endgroup$
    – user32229
    Commented Nov 10, 2016 at 17:09

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I don't think there can exist such example; deconfinement implies the existence of fractionalized excitation, which again implies the intrinsic topological order. However, in (1+1) dimension, it has been proved that there cannot exist any intrinsic topological order.

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  • $\begingroup$ Fractionalization does not imply topological order in 1+1D: the Ising chain has fractionalization (a single spin flip = two deconfined domain walls). $\endgroup$
    – Ruben Verresen
    Commented Jul 26, 2021 at 1:46