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The fractional quantum Hall effect is more complicated and still considered an open research problem. [2] Its existence relies fundamentally on electron–electron interactions. In 1988, it was proposed that there was a quantum Hall effect without Landau levels. [3] This quantum Hall effect is referred to as the quantum anomalous Hall (QAH) effect.
Scaling of the longitudinal and Hall conductivities in a renormalization group flow-diagram of the quantum Hall effect. On the basis of the Renormalization Group Theory of the instanton vacuum one can form a general flow diagram where the topological sectors are represented by attractive fixed points. When scaling the effective system to larger ...
The fractional quantum Hall effect (FQHE) is a collective behavior in a 2D system of electrons. In particular magnetic fields, the electron gas condenses into a remarkable liquid state, which is very delicate, requiring high quality material with a low carrier concentration, and extremely low temperatures.
In ferromagnetic materials (and paramagnetic materials in a magnetic field), the Hall resistivity includes an additional contribution, known as the anomalous Hall effect (or the extraordinary Hall effect), which depends directly on the magnetization of the material, and is often much larger than the ordinary Hall effect.
In quantum mechanics, the energies of cyclotron orbits of charged particles in a uniform magnetic field are quantized to discrete values, thus known as Landau levels.These levels are degenerate, with the number of electrons per level directly proportional to the strength of the applied magnetic field.
The fractional quantum Hall effect of electrons is thus explained as the integer quantum Hall effect of composite fermions. [5] It results in fractionally quantized Hall plateaus at =, with given by above quantized values. These sequences terminate at the composite fermion Fermi sea.
The Hofstadter butterfly plays an important role in the theory of the integer quantum Hall effect and the theory of topological quantum numbers. History [ edit ]
The quantum Hall effect is a quantum mechanical version of the Hall effect, The Hall effect occurs when a magnetic field causes a perpendicular (transverse) current in a material. In the quantum Hall effect , the transverse conductivity σ x y {\displaystyle \sigma _{xy}} is quantized in integer multiples of a basic quantity: