The quantum Hall effect describes a process by which electric resistance in layers of materials of just a few atoms thick can be measured accurately. The quantum Hall effect is a remarkable quantization of resistance observed in a two-dimensional electronic system such as graphene, in the presence of a large applied magnetic field. Thin films of topological magnetic insulators can show the almost perfect quantum Hall effect without requiring the application of magnetic fields. Two independent studies, one led by a group led by David Goldhaber-Gordon at Stanford University, California, the other led by Jagadeesh S. Moodera and collaborators at MIT, Cambridge, now show that thin films of topological insulators can display a near-ideal quantum anomaly Hall effect, i.e., quantum Hall effect with zero external fields [3-5]. [Sources: 3, 5]

Early theoretical studies [3,4] suggested that the quantum anomalous Hall effect might be possible in materials naturally having a topologically nontrivial band structure, analogous to the band structure caused by a magnetic field (see Figure ). In 1980, Klaus von Klitzing, working in a high-magnitude lab in Grenoble on samples of a silicon-based MOSFET developed by Michael Pepper and Gerhard Dorda, made the surprising discovery that the Hall conductance was precisely quantized. The original Hall effect, discovered in 1879 by physicist Edwin Hall, describes how a magnetic field applied perpendicularly to a metal strip causes electrons to collect at either end of the strip, creating a voltage3. Working at lower temperatures with atomically thinner layers of crystal materials–known as two-dimensional electron systems–physicist Claus von Klitzing discovered that this voltage is quantized. [Sources: 3, 5, 6]

In the quantum fractional Hall effect, complicated interactions between electrons result in a quantized Hall resistance at a value that is only a fraction of the charge on an electron. The fractional quantum Hall effect is also understood as the integer quantum Hall effect, though it is not about electrons, but about charge-flux combinations known as composite fermions. The observed strong similarities between the integer quantum Hall and the fractional quantum Hall effects are explained by the tendency of electrons to form bound states with an equal number of quantum quantities of magnetic flux, called composite fermions. [Sources: 3, 6]

While in a circular orbit the centrifugal force is balanced by the Lorentz force, which is responsible for transverse inducer voltages and quantum Hall effects, it is possible to view Coulomb potential differences in a Bohr atom as the Hall voltages caused by the individual atoms, and periodic electron motion on the roundabout as the Hall current. The fine-structure constant values can already be obtained on the level of a single atom in a Bohr model while looking at it as a single-electrode Hall effect. [Sources: 6]

In the quantum Hall system, transverse resistance (measured along the width of a sample) takes a quantized value h/n E 2, with h/e 2 being Planck’s constant, E an elementary charge, and N either an integer or a fraction. Whereas the normal resistance artifact depends on the material and the geometrical dimensions, changes over time, temperature, and atmospheric pressure, and is susceptible to mechanical shocks, the Von-Klitzing constant depends on fundamental constants alone, is invariant, and is repeatable to remarkably high precision, with at least 10-10. [Sources: 2, 5]

Based on new material, graphene, discovered in 2004, in the future, a precisely measured quantum Hall resistance may be realized with a much lower temperature (>=4 k) and a much lower magnetic field (=4 T), simplifying measurement setup. Since quantum Hall effects occur only at lower temperatures and under stronger magnetic fields, the appropriate cryogenics and a superconducting solenoid are required. The resistance drops only to zero at a few teslas of applied magnetic fields, which is not as weak as required by the normal quantum Hall effect. [Sources: 2, 5]

Ultimately, the most rigorous mathematical proof may very well have failed if resistance had not been classified as one of the outstanding problems in mathematical physics. Some mathematicians were unhappy with the standards of proof offered by the physicists, and the quantum Hall resistance was added to a well-known list of mathematical physics unsolved problems. [Sources: 3]

Diagonal resistance R XX and Hall resistivity R XY at 2deg in a stressed Si quantum well at T=30 M K. The micro-physics of the QHE is given by the noncommutative geometry. The QHE is one of the most exciting and beautiful phenomena in all branches of physics. [Sources: 1, 4]

The unconventional QHE in graphene is reviewed, in which the electron dynamics can be treated as relativistic Dirac fermions, and also supersymmetric quantum mechanics plays a crucial role. This third edition is also suited for introductory quantum field theory, with applications described in lively terms. In the 2nd edition, the awesome phenomena associated with interlayer phase coherence in the two-layer system are described in detail. [Sources: 0, 4]

D. K. Tsui and collaborators performed similar coincidence experiments for the odd integer filling factors U=3 and U=5,55, and, in contrast, at the even integer filling factors U=4 and 6.56. In line with earlier experiments, they observed that beyond the coincidence regime for odd integer filling factors, the Zeeman spin split is independent of the magnetic field component in-plane. For electron-electronic interactions, the spin states of the highest level of occupancy are relevant, because the lower two levels are both (N=0,|) states, which differ only by their valley quantum numbers (labeled + and – in Figures 15.5 and 15.6). If under this circumstance, the magnetic order of the system becomes anisotropic along an agnostic axis, the system acts like a Ising ferromagnet.57 In particular, QHF can arise under strong electron-electron interaction regimes, where the two levels of opposite spin states (or quasi-spin states) intersect. [Sources: 1]

QHF may be expected when two energy levels of different quantum indexes become aligned, and competing ground-state configurations are formed. Here, G* and M B are effective G-factors and the Bohr magneton, respectively. Note, however, that the state densities are zero in these regions of the quantized Hall conductance; thus, they cannot generate the plateaus observed in the experiments. [Sources: 1, 6]

##### Sources:

[0]: https://www.worldscientific.com/worldscibooks/10.1142/8210

[1]: https://www.sciencedirect.com/topics/chemistry/quantum-hall-effect

[2]: https://www.ptb.de/cms/en/ptb/fachabteilungen/abt2/fb-26/ag-262/the-quantum-hall-resistance.html

[3]: https://www.nature.com/articles/d41586-020-02230-7

[4]: https://www.amazon.com/QUANTUM-HALL-EFFECTS-EXPERIMENTAL-DEVELOPMENTS/dp/9813203668