\theta-angle monodromy in two dimensions

"\theta-angle monodromy" occurs when a theory possesses a landscape of metastable vacua which reshuffle as one shifts a periodic coupling \theta by a single period. "Axion monodromy" models arise when this parameter is promoted to a dynamical pseudoscalar field. This paper studies the phenomenon in two-dimensional gauge theories which possess a U(1) factor at low energies: the massive Schwinger and gauged massive Thirring models, the U(N) 't Hooft model, and the {\mathbb CP}^N model. In all of these models, the energy dependence of a given metastable false vacuum deviates significantly from quadratic dependence on \theta just as the branch becomes completely unstable (distinct from some four-dimensional axion monodromy models). In the Schwinger, Thirring, and 't Hooft models, the meson masses decrease as a function of \theta. In the U(N) models, the landscape is enriched by sectors with nonabelian \theta terms. In the {\mathbb CP}^N model, we compute the effective action and the size of the mass gap is computed along a metastable branch.