However, this orbital motion constraint alone is not sufficient (for example, there is no dimensional reduction for charged scalar particles, carrying spin 0, although their orbital motion is constrained in the same way.) It is also important that the fermions have spin 1/2 and, as follows from the Atiyah–Singer index theorem, their lowest Landau level states have an energy independent of the magnetic field. (The corresponding energy vanishes in the case of massless particles.) This is in contrast to the energies in the higher Landau levels, which are proportional to the square root of the magnetic field. Therefore, if the field is sufficiently strong, only the lowest Landau level states are dynamically accessible at low energies. The states in the higher Landau levels decouple and become almost irrelevant. The phenomenon of magnetic catalysis has applications in particle physics, nuclear physics and condensed matter physics.

Positively or negatively charged? Positively charged gold nanoparticles can penetrate deep into cell membranes while negatively charged particles do not enter the cell wall at all, but instead prevent it breaking down under certain conditions.