Fractional quantum Hall effect

The fractional quantum Hall effect (FQHE) is a physical phenomenon in which a certain system behaves as if it were composed of particles with charge smaller than the elementary charge. Its discovery and explanation were recognized by the 1998 Nobel Prize in Physics.

Introduction

The fractional quantum Hall effect (FQHE)is defined as a 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 value of the magnetic field corresponds to a filling factor (the ratio of electrons to magnetic flux quanta)

$\nu =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 filling factors 5/2 and 7/2. The principal series of such fractions are

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

and

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

There were two major steps in the theory of the FQHE.

• Fractionally-charged quasiparticles: this theory, proposed by Laughlin, is based on accurate trial wave functions for the ground state at fraction $1/q$ as well as its quasiparticle and quasihole excitations. The excitations have fractional charge of magnitude $e^{*}={e \over q}$.

• Composite fermions: this theory was proposed by Jain, and further extended by Halperin, Lee and Read. As a result of the repulsive interactions, two (or, in general, an even number) flux quanta ${h \over e}$ are captured by each electron, forming integer-charged quasiparticles called composite fermions. The fractional states of electrons are understood as the integer QHE of composite fermions. This makes electrons at a filling factor 1/3, for example, behave in the same way as at filling factor 1. A remarkable result is that the filling factor 1/2 corresponds to zero magnetic field for composite fermions, resulting in their Fermi sea. Experiments support composite fermion theory.

The FQHE was experimentally discovered in 1982 by Daniel Tsui and Horst Störmer, in experiments performed on gallium arsenide heterostructures developed by Arthur Gossard. Tsui, Störmer, and Laughlin were awarded the 1998 Nobel Prize for their work.

Laughlin's original plasma model was extended to other fractionally charged systems by MacDonald and others. The approach based on composite fermions by Jain has now emerged as a basic paradigm encompassing most of the earlier approaches.

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.

Other evidence for fractionally-charged quasiparticles

Apart from the FQHE itself, further evidence has continued to emerge that specifically supports the understanding that there are fractionally-charged quasiparticles in an electron gas under FQHE conditions.

In 1995, the fractional charge of Laughlin quasiparticles was measured directly in a quantum antidot electrometer at Stony Brook University, New York. In 1997, two groups of physicists at the Weizmann Institute of Science in Rehovot, Israel, and at the Commissariat à l'énergie atomique laboratory near Paris, detected such quasiparticles carrying an electric current, through measuring quantum shot noise.

Impact of fractional quantum Hall effect

The FQH effect shows the limits of Landau's symmetry breaking theory. Previously it was long believed that the symmetry breaking theory could explain all the important concepts and essential properties of all forms of matter. According to this view the only thing to be done is to apply the symmetry breaking theory to all different kinds of phases and phase transitions. From this perspective, we can understand the importance of the FQHE discovered by Tsui, Stormer, and Gossard.

Different FQH states all have the same symmetry and cannot be described by symmetry breaking theory. Thus FQH states represent new states of matter that contain a completely new kind of order—topological order. The existence of FQH liquids indicates that there is a whole new world beyond the paradigm of symmetry breaking, waiting to be explored. The FQH effect opened up a new chapter in condensed matter physics. The new type of orders represented by FQH states greatly enrich our understanding of quantum phases and quantum phase transitions. The associated fractional charge, fractional statistics, non-Abelian statistics, chiral edge states, etc demonstrate the power and the fascination of emergence in many-body systems.

See also

• Laughlin wavefunction
• Hall probe
• Quantum Hall Effect

References

• D.C. Tsui, H.L. Stormer, A.C. Gossard. "Two-Dimensional Magnetotransport in the Extreme Quantum Limit". Physical Review Letters. 48: 1559. doi:10.1103/PhysRevLett.48.1559. {{cite journal}}: Text "year1982" ignored (help)CS1 maint: multiple names: authors list (link)
• H.L. Stormer (1999). "Nobel Lecture: The fractional quantum Hall effect". Reviews of Modern Physics. 71: 875. doi:10.1103/RevModPhys.71.875.
• R.B. Laughlin (1983). "Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations". Physical Review Letters. 50: 1395. doi:10.1103/PhysRevLett.50.1395.

External links

• University of Cambridge, Semiconductor Physics Group Research.
• Fractional quantum Hall effect( List of Authority Articles)