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We study 2d U(1) gauge Higgs systems with a $\theta$-term. For properlydiscretizing the topological charge as an integer we introduce a mixed group-and algebra-valued discretization (MGA scheme) for the gauge fields, such thatthe charge conjugation symmetry at $\theta = \pi$ is implemented exactly. Thecomplex action problem from the $\theta$-term is overcome by exactly mappingthe partition sum to a worldline/worldsheet representation. Using Monte Carlosimulation of the worldline/worldsheet representation we study the system at$\theta = \pi$ and show that as a function of the mass parameter the systemundergoes a phase transition. Determining the critical exponents from a finitesize scaling analysis we show that the transition is in the 2d Isinguniversality 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 thatfor charge 2 a true phase transition is absent and only a rapid crossoverseparates the large and small mass regions.

The critical endpoint in the 2d U(1) gauge-Higgs model at topological angle $\theta=\pi$