condensed matter - Low dim physics: Examples of confinement-deconfinement phases of U(1) gauge theory in 2 dimensions

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.