Lattice techniques to investigate the strong $CP$ problem: lessons from a toy model

Recent studies have claimed that the strong $CP$ problem does not occur in QCD, proposing a new order of limits in volume and topological sectors when studying observables on the lattice. We study the effect of the topological term on a simple quantum mechanical rotor that allows a lattice description. We particularly focus on recent proposals to face the challenging problems that this study poses in lattice QCD and that are also present in the quantum rotor, such as topology freezing and the sign problem.