The Quantum Hall Effect (QHE) is one of the most remarkable, fascinating and, yes, complex phenomena in physics. Its essence, nevertheless, is quite simple: given a steady electric current, whenever we apply a perpendicular uniform magnetic field, a spontaneous electric voltage difference can be measured in the direction perpendicular to the current-magnetic-field plane. Requiring a simple setup, the classical version of the effect was observed for the first time by Edwin Hall in 1879. The ratio between the transverse voltage and the current yields the “Hall resistance,” which increases linearly with the applied magnetic field. On general grounds, the effect is a natural consequence of the Lorentz force acting on moving charges forming the current and it should not come to be a surprise.

One century later, in 1980, von Klitzing [226] repeated the Hall experiment under specific conditions. The electric current was injected in a metal slice 3nm wide, squeezed between an insulator and a semiconductor, in a device calledMOSFET, at a temperature of the order of 1 K and under an applied magnetic field of the order of 10 T. The result was stunning. The simple straight line, which represented the magnetic field dependence of the Hall resistance, was replaced by a complex pattern, in which one could observe a sequence of plateaus corresponding to integer multiples of a basic resistance unit.

Two years later, in 1982, Tsui and Störmer [227] repeated the experiment, this time injecting the current in a gas of electrons trapped in the interfaces of a multiple junction alternating GaAs and GaAs1−x Alx, called heterostructure, at a temperature of the order of 0.1 K, under a magnetic field of up to 30 T. The curve representing the magnetic field dependence of the Hall resistance became even more complex, now exhibiting plateaus at rational multiples, mostly with odd denominators, of the same resistance unit.

Both the discoveries described above were laureated with the Nobel Prize and became known, respectively, as the integer and fractional QHE. The theoretical understanding of the two new phenomena required the use of ideas and methods that mobilized the interplay of deep properties of quantum mechanics, the quantum theory of disordered systems, topologically driven physical mechanisms and deep theorems of mathematics.