  The local current flow through three small aromatic carbon molecules, namely
benzene, naphthalene and anthracene, is studied. Applying density functional
theory and the non-equilibrium Green's function method for transport, we
demonstrate that pronounced current vortices exist at certain electron energies
for these molecules. The intensity of these circular currents, which appear not
only at the anti-resonances of the transmission but also in vicinity of its
maxima, can exceed the total current flowing through the molecular junction and
generate considerable magnetic fields. The $\pi$ electron system of the
molecular junctions is emulated experimentally by a network of macroscopic
microwave resonators. The local current flows in these experiments confirm the
existence of current vortices as a robust property of ring structures. The
circular currents can be understood in terms of a simple nearest-neighbor
tight-binding H\"uckel model. Current vortices are caused by the interplay of
the complex eigenstates of the open system which have energies close-by the
considered electron energy. Degeneracies, as observed in benzene and
anthracene, can thus generate strong circular currents, but also non-degenerate
systems like naphthalene exhibit current vortices. Small imperfections and
perturbations can couple otherwise uncoupled states and induce circular
currents.
