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

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. Lowering the mass squared the symmetry becomes restored in a second order phase transition. Simulating the system at $\theta = \pi$ in its dual form we determine the corresponding critical endpoint as a function of the mass parameter. Using a finite size scaling analysis we determine the critical exponents and show that the transition is in the 2d Ising universality class, as expected.