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# Fractional quantum Hall effect

The fractional quantum Hall effect (FQHE) is a fascinating manifestation of simple collective behaviour in a two-dimensional system of strongly interacting electrons. At particular magnetic fields, the electron gas condenses into a remarkable state with liquid-like properties. This state is very delicate, requiring high quality material with a low carrier concentration, and extremely low temperatures. As in the integer quantum Hall effect, a series of plateaus forms in the Hall resistance. Each particular values of magnetic field corresponds to a filling factor (the ratio of electrons to magnetic flux quanta)

ν = p / q,

where p and q are integers with no common factors. Here q turns out to be an odd number with the exception of two enigmatic filling factors 5/2 and 7/2. The principal series of such fractions are

${1\over 3}, {2\over 5}, {3\over 7}$ etc.,

and

${2\over3}, {3\over 5}, {4\over 7}$, etc.

There are two main theories of the FQHE.

• Fractionally-charged quasiparticles: this theory, proposed by Laughlin, hides the interactions by constructing a set of quasiparticles with charge $e^*={e\over q}$, where the fraction is ${p\over q}$ as above.
• Composite Fermions: this theory was proposed by Jain, and Halperin, Lee and Read. In order to hide the interactions, it attaches two (or, in general, an even number) flux quanta ${h\over e}$ to each electron, forming integer-charged quasiparticles called composite fermions. The fractional states are mapped to the integer QHE. This makes electrons at a filling factor 1/3, for example, behave in the same way as at filing factor 1. A remarkable result is that filling factor 1/2 corresponds to zero magnetic field. Experiments support this.

The FQHE was experimentally discovered in 1982 by Daniel Tsui and Horst Störmer, in experiments performed on gallium arsenide heterostructures developed by Arthur Gossard. The effect was explained by Robert B. Laughlin in 1983, using a novel quantum liquid phase that accounts for the effects of interactions between electrons. Tsui, Störmer, and Laughlin were awarded the 1998 Nobel Prize for their work. Although it was generally assumed that the discrete resistivity jumps found in the Tsui experiment were due to the presence of fractional charges (i.e., due to the emergence of quasiparticles with charges smaller than an electron charge), it was not until 1997 that R. de-Picciotto, et al., indirectly observed fractional charges through measurements of quantum shot noise.

Fractionally charged quasiparticles are neither bosons nor fermions and exhibit anyonic statistics. The fractional quantum Hall effect continues to be influential in theories about topological order. Certain fractional quantum Hall phases appear to have the right properties for building a topological quantum computer.

## References

• University of Cambridge, Semiconductor Physics Group Research.
• D.C. Tsui, H.L. Stormer, and A.C. Gossard, Phys. Rev. Lett. 48, 1559 (1982) doi:10.1103/PhysRevLett.48.1559
• R.B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983) doi:10.1103/PhysRevLett.50.1395
• R. de-Picciotto, M. Reznikov, M. Heiblum, V. Umansky, G. Bunin and D. Mahalu, Nature 389, 162-164 (1997) doi:10.1038/38241