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Quantum Hall transitions
Quantum Hall transitions are the quantum phase transitions that occur between different robustly quantized electronic phases of the quantum Hall effect. The robust quantization of these electronic phases is due to strong localization of electrons in their disordered, two-dimensional potential (see Anderson localization). But, at the quantum Hall transition, the electron gas delocalizes as can be observed in the laboratory. This phenomenon is understood in the language of topological field theory. Here, a vacuum angle (or 'theta angle') distinguishes between topologically different sectors in the vacuum. These topological sectors correspond to the robustly quantized phases. The quantum Hall transitions can then be understood by looking at the topological excitations (instantons) that occur between those phases.
== Historical perspective ==
Just after the first measurements on the quantum Hall effect in 1980,〔K. von Klitzing, G. Dorda, and M. Pepper. New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Physical Review Letters, 45(6):494-497, 1980.〕 physicists wondered how the strongly localized electrons in the disordered potential were able to delocalize at their phase transitions. At that time, the field theory of Anderson localization didn't yet include a topological angle and hence it predicted that: "for any given amount of disorder, all states in two dimensions are localized". A result that was irreconcilable with the observations on delocalization.〔H.P. Wei, D.C. Tsui, and A.M.M. Pruisken. Localization and scaling in the quantum Hall regime. Physical Review B, 33(2):1488-1491, 1985.〕 Without knowing the solution to this problem, physicists resorted to a semi-classical picture of localized electrons that, given a certain energy, were able to percolate through the disorder.〔R.F. Kazarinov and S. Luryi. Quantum percolation and quantization of Hall resistance in two-dimensional electron gas. Physical Review B, 25(12):7626,1982.〕 This percolation mechanism was what assumed to delocalize the electrons
As a result of this semi-classical idea, many numerical computations were done based on the percolation picture.〔J.T. Chalker and P.D. Coddington. Percolation, quantum tunnelling and the integer Hall effect. Journal of Physics C: Solid State Physics, 21:2665,1988.〕 On top of the classical percolation phase transition, quantum tunneling was included in computer simulations to calculate the critical exponent of the `semi-classical percolation phase transition'. To compare this result with the measured critical exponent, the Fermi-liquid approximation was used, where the Coulomb interactions between electrons are assumed to be ''finite''. Under this assumption, the ground state of the free electron gas can be adiabatically transformed into the ground state of the interacting system and this gives rise to an inelastic scattering length so that the canonical correlation length exponent can be compared to the measured critical exponent.
But, at the quantum phase transition, the localization lengths of the electrons becomes infinite (i.e. they delocalize) and this compromises the Fermi-liquid assumption of an inherently free electron gas (where individual electrons must be well-distinguished). The quantum Hall transition will therefore not be in the Fermi-liquid universality class, but in the F''-invariant' universality class that has a different value for the critical exponent.〔A.M.M. Pruisken and I.S. Burmistrov. Theta renormalization, electron-electron interactions and super universality in the quantum Hall regime. Annals of Physics, 322(6):1265-1334, 2007.〕 The semi-classical percolation picture of the quantum Hall transition is therefore outdated (although still widely used) and we need to understand the delocalization mechanism as an instanton effect.
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