  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.
