What is "quantum hall effect"

n. (context physics English) An effect marked by the quantization of the Hall resistance, observed in two-dimensional electron systems subjected to low temperatures and strong magnetic field.

The quantum Hall effect (or integer quantum Hall effect) is a quantum-mechanical version of the Hall effect, observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall conductance σ undergoes certain quantum Hall transitions to take on the quantized values

$$\sigma = \frac{I_\text{channel}}{V_\text{Hall}} = \nu \; \frac{e^2}{h},$$

where I is the channel current, V is the Hall voltage, e is the elementary charge and h is Planck's constant. The prefactor ν is known as the "filling factor", and can take on either integer (ν = 1, 2, 3, ...) or fractional (ν = 1/3, 2/5, 3/7, 2/3, 3/5, 1/5, 2/9, 3/13, 5/2, 12/5, ...) values. The quantum Hall effect is referred to as the integer or fractional quantum Hall effect depending on whether ν is an integer or fraction, respectively.

The striking feature of the integer quantum Hall effect is the persistence of the quantization (i.e. the Hall plateau) as the electron density is varied. Since the latter remains constant when the Fermi level is in a clean spectral gap, it must be that the Fermi level is immersed in essential spectrum (i.e. of finite density of states), though Anderson localized. The quantization and stability of the Hall conductance in such conditions was achieved in 1994 by Bellissard, van Elst and Schulz-Baldes using the methods of Non-Commutative Geometry.

The fractional quantum Hall effect is more complicated, as its existence relies fundamentally on electron–electron interactions. Although the microscopic origins of the fractional quantum Hall effect are unknown, there are several phenomenological approaches that provide accurate approximations. For example the effect can be thought of as an integer quantum Hall effect, not of electrons but of charge-flux composites known as composite fermions. In 1988, it was proposed that there was quantum Hall effect without Landau levels. This quantum Hall effect is referred to as the quantum anomalous Hall (QAH) effect. There is also a new concept of the quantum spin Hall effect which is an analogue of the quantum Hall effect, where spin currents flow instead of charge currents.