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) a planar rotor with a non-trivial magnetic flux $Φ$, ii) 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.

14 pages, 2 ps-figures, to appear in Commun. Math. Phys