Editing Quantum Hall effect (section)

== Topological classification ==
[[Image:Hofstadter's butterfly.png|thumb|[[Hofstadter's butterfly]]]]
The integers that appear in the Hall effect are examples of [[topological quantum number]]s. They are known in mathematics as the first [[Chern class#Chern numbers|Chern numbers]] and are closely related to [[Geometric phase|Berry's phase]]. A striking model of much interest in this context is the Azbel–Harper–Hofstadter model whose quantum phase diagram is the [[Hofstadter butterfly]] shown in the figure. The vertical axis is the strength of the [[magnetic field]] and the horizontal axis is the [[chemical potential]], which fixes the electron density. The colors represent the integer Hall conductances. Warm colors represent positive integers and cold colors negative integers. Note, however, that the density of states in these regions of quantized Hall conductance is zero; hence, they cannot produce the plateaus observed in the experiments. The phase diagram is fractal and has structure on all scales. In the figure there is an obvious [[self-similarity]]. In the presence of disorder, which is the source of the plateaus seen in the experiments, this diagram is very different and the fractal structure is mostly washed away. Also, the experiments control the filling factor and not the Fermi energy. If this diagram is plotted as a function of filling factor, all the features are completely washed away, hence, it has very little to do with the actual Hall physics.

Concerning physical mechanisms, impurities and/or particular states (e.g., edge currents) are important for both the 'integer' and 'fractional' effects. In addition, Coulomb interaction is also essential in the [[fractional quantum Hall effect]]. The observed strong similarity between integer and fractional quantum Hall effects is explained by the tendency of electrons to form bound states with an even number of magnetic flux quanta, called ''[[composite fermions]]''.