Instanton-sphaleron transition in the d=2 Abelian-Higgs model on a circle

The transition from the instanton-dominated quantum regime to the sphaleron -dominated classical regime is studied in the d = 2 Abelian-Higgs model whe n the spatial coordinate is compactified to S-1. Contrary to the noncompact ified case, this model allows both sharp first-order and smooth second-orde r transitions depending on the size of the circle. This finding may make th e model a useful toy model for the analysis of baryon number violating proc esses. Since the model can to a large extent be treated analytically, it ca n also serve as a transparent prototype for the application of our method t o more complicated cases, such as those in higher dimensions.