Dual simulation of the 2d U(1) gauge Higgs model at topological angle $\theta = \pi\,$: Critical endpoint behavior

Understand

We simulate the 2d U(1) gauge Higgs model on the lattice with a topological angle $\theta$.

• The corresponding complex action problem is overcome by using a dual representation based on the Villain action appropriately endowed with a $\theta$-term.
• The Villain action is interpreted as a non-compact gauge theory whose center symmetry is gauged and has the advantage that the topological term is correctly quantized so that $2\pi$ periodicity in $\theta$ is intact.
• Because of this the $\theta = \pi$ theory has an exact $Z_2$ charge-conjugation symmetry $C$, which is spontaneously broken when the mass-squared of the scalars is large and positive.