Perfect lattice topology: the quantum rotor as a test case

Lattice actions and topological charges that are classically and quantum mechanically perfect (i.e. free of lattice artifacts) are constructed analytically for the quantum rotor. It is demonstrated that the Manton action is classically perfect while the Villain action is quantum perfect. The geometric construction for the topological charge is only perfect at the classical level. The quantum perfect lattice topology associates a topological charge distribution, not just a single charge, with each lattice field configuration. For the quantum rotor with the classically perfect action and topological charge, the remaining cut-off effects are exponentially suppressed.