Vacuum Nodes and Anomalies in Quantum Theories

Abstract:

We show that nodal points of ground states of some quantum systems with magnetic interactions can be identified in simple geometric terms. We analyse in detail two different archetypical systems: i) the planar rotor with a non-trivial magnetic flux Φ and ii) the Hall effect on a torus. In the case of the planar rotor we show that the level repulsion generated by any reflection invariant potential V is encoded in the nodal structure of the unique vacuum for θ=π. In the second case we prove that the nodes of the first Landau level for unit magnetic charge appear at the crossing of the two non-contractible circles α₋, β₋ with holonomies h _α-(A)=h _β-(A)=−1 for any reflection invariant potential V. This property illustrates the geometric origin of the quantum translation anomaly.