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 Ξ¦, 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 ΞΈ=Ο. 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 Ξ±ββ, Ξ²ββ with holonomies
hΞ±βββ(A)=hΞ²βββ(A)=β1 for any reflection invariant potential
V. This property illustrates the geometric origin of the quantum translation
anomaly.Comment: 14 pages, 2 ps-figures, to appear in Commun. Math. Phy