  The scale (conformal) anomaly can generate an electric current near the
boundary of a system in the presence of a static magnetic field. The magnitude
of this magnetization current, produced at zero temperature and in the absence
of matter, is proportional to a beta function associated with the
renormalization of the electric charge. Using first-principle lattice
simulations, we investigate how the breaking of the scale symmetry affects this
``scale magnetic effect'' near a Dirichlet boundary in scalar QED (Abelian
Higgs model). We demonstrate the interplay of the generated current with vortex
excitations both in symmetric (normal) and broken (superconducting) phases and
compare the results with the anomalous current produced in the conformal,
scale-invariant regime. Possible experimental signatures of the effect in Dirac
semimetals are discussed.
