  We study 2d U(1) gauge Higgs systems with a $\theta$-term. For properly
discretizing the topological charge as an integer we introduce a mixed group-
and algebra-valued discretization (MGA scheme) for the gauge fields, such that
the charge conjugation symmetry at $\theta = \pi$ is implemented exactly. The
complex action problem from the $\theta$-term is overcome by exactly mapping
the partition sum to a worldline/worldsheet representation. Using Monte Carlo
simulation of the worldline/worldsheet representation we study the system at
$\theta = \pi$ and show that as a function of the mass parameter the system
undergoes a phase transition. Determining the critical exponents from a finite
size scaling analysis we show that the transition is in the 2d Ising
universality class. We furthermore study the U(1) gauge Higgs systems at
$\theta = \pi$ also with charge 2 matter fields, where an additional $Z_2$
symmetry is expected to alter the phase structure. Our results indicate that
for charge 2 a true phase transition is absent and only a rapid crossover
separates the large and small mass regions.
