2d Cardy-Rabinovici model with the modified Villain lattice formulation

One of the most famous scenarios of the quark confinement problem is the dual superconductor picture. In this picture, the quark confinement is induced by monopole condensation, but in the theory with a θ term, we expect that not only monopole but also dyon condensation is induced, as suggested by Cardy and Rabinovici through their intuitive arguments. In this study, the Witten effect of the theory of two-dimensional compact bosons with the θ term is examined using a modified Villain-type lattice theory that can treat the θ term and dion in a rigorous manner. In addition, we construct the 2d Cardy-Rabinovici model and analyze the phase diagram through the scaling dimension argument and the anomaly matching constraint.