Planar rotor: Theθ-vacuum structure, and some approximate methods in quantum mechanics

Abstract

The quantum planar rotor is chosen as a model to test different approximate techniques, emphasizing the analogies between this simple system and the non-Abelian quantum gauge field theories (instantons, $\theta$ vacuum, etc.). In particular, variational and perturbative methods, path-integral techniques, and Monte Carlo simulations are applied. It is pointed out that the maximal destructive interference between instantons takes place in the $\theta$-vacuum realization of the interacting rotor system with $\theta =\frac{1}{2}$, where the tunneling effects are shown to be severely diminished.