Phase structure of U(1) lattice gauge theory by variational study with independent plaquette effective action

In order to compare independent plaquette effective action with link effective action in variational cumulant expansion, as a simple example, we calculate the plaquette internal energy E(p) of U(1) LGT in three dimensions with independent plaquette effective action in variational cumulant expansion approach. The E(p) - beta curve is continous, which is consistent with MC data. It shows U(1) theory has only one confine phase. It also shows the independent plaquete effective action is better than the link effective action.