a compact U(1) gauge theory in the presence of a charge 2 Higgs field. The presence of the background electric charges implies that the monopoles of the U(1) field carry Berry phases, and transform non-trivially under the space group of the lattice. As the monopoles condense in the confining phase, an immediate consequence is that the confining phase must break the space group by the development of valence bond solid (VBS) order. Furthermore, the Berry phases lead to suppression of monopoles at the critical point, so that, on the square lattice, the critical theory has a deconfined U(1) gauge field coupled to a critical charged scalar. Note that the critical theory is not expressed in terms of the VBS order as would be required by the LGW paradigm (which ignores the Z2 topological order in the deconfined phase). Instead, a dual version of the U(1) gauge theory is written in terms of a "square root" of the VBS order.
