Fractional quantum Hall effect

The fractional quantum Hall effect (FQHE) is a physical phenomenon in which the Hall conductance of 2D electrons shows precisely quantised plateaus at fractional values of e^2/h. It is a property of a collective state in which electrons bind magnetic flux lines to make new quasiparticles, and excitations have a fractional elementary charge and possibly also fractional statistics. Its discovery and explanation were recognized by the 1998 Nobel Prize in Physics.

Contents

Introduction

The fractional quantum Hall effect (FQHE) is a collective behaviour in a two-dimensional 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. As in the integer quantum Hall effect, a series of plateaus form 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\over3}, {3\over 5}, {4\over 7}, \mbox{etc.}

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

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.

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.

Evidence for fractionally-charged quasiparticles

Experiments have reported results that specifically support 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.[3] 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.[4][5] Both of these experiments are somewhat controversial.

A more recent experiment,[6] which measures the quasiparticle charge extremely directly, appears beyond reproach.

The anyonic statistics, however, have neither been confirmed nor disproved by any of the experiments.

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.

The explanation of this effect is open to new relations and interpretations of experiments in light of new theories. It is possible to consider that the FQHE really gives further evidence of the existence of magnetic flux quanta as detected in superconductivity. There are electromagnetic quanta of magnetic flux and electric elementary charge. It was shown that these fundamental quanta are determined by a physical geometric unified theory.[7] Each particle carries an integer charge and an integer flux. Classically, if a particle crosses a line in a plane normal to its flux, there is a fractional relation between the charge and flux crossing the line and a corresponding fractional relation between the current and induced voltage if there are no resistive losses. Quantum mechanically, the Hall effect problem may be treated by filling the quantum Landau energy levels produced by the presence of the magnetic field. The electrons carrying flux quanta may combine to form rotating systems or vortices which act as charge/flux carriers. It has been shown that the Hall electric conductivity is directly proportional to charge/flux ratios of these carriers.[8] Consequently, in accordance with the quantum aspects of the unified theory it is not necessary to assume the existence of fundamental fractional charges.

See also

Notes

  1. ^ M. Greiter (1994). "Microscopic formulation of the hierarchy of quantized Hall states". Phys. Lett. B 336: 48. arXiv:cond-mat/9311062. Bibcode 1994PhLB..336...48G. doi:10.1016/0370-2693(94)00957-0. 
  2. ^ A.H. MacDonald, G.C. Aers, M.W.C. Dharma-wardana (1985). "Hierarchy of plasmas for fractional quantum Hall states". Physical Review B 31 (8): 5529. Bibcode 1985PhRvB..31.5529M. doi:10.1103/PhysRevB.31.5529. 
  3. ^ V.J. Goldman, B. Su (1995). "Resonant Tunneling in the Quantum Hall Regime: Measurement of Fractional Charge". Science 267 (5200): 1010. Bibcode 1995Sci...267.1010G. doi:10.1126/science.267.5200.1010.  See also Description on the researcher's website.
  4. ^ "Fractional charge carriers discovered". Physics World. 24 October 1997. http://physicsworld.com/cws/article/news/3393. Retrieved 2010-02-08. 
  5. ^ R. de-Picciotto, M. Reznikov, M. Heiblum, V. Umansky, G. Bunin, D. Mahalu (1997). "Direct observation of a fractional charge". Nature 389 (6647): 162. Bibcode 1997Natur.389..162D. doi:10.1038/38241. 
  6. ^ J. Martin, S. Ilani, B. Verdene, J. Smet, V. Umansky, D. Mahalu, D. Schuh, G. Abstreiter, A. Yacoby, (2004). "Localization of Fractionally Charged Quasi Particles". Science 305 (5686): 980–3. Bibcode 2004Sci...305..980M. doi:10.1126/science.1099950. PMID 15310895. 
  7. ^ González-Martín, Gustavo R. (1991). "Some electromagnetic consequences of a unified theory of gravitation and electromagnetism". Gen. Rel. And Grav. 23 (7): 827. Bibcode 1991GReGr..23..827G. doi:10.1007/BF00755997. http://www.springerlink.com/content/h4j801591u83h514/. 
  8. ^ González-Martín, Gustavo R. (1997). "Charge to magnetic flux ratios". ArXiv: cond–mat/0009181. http://arxiv.org/ftp/cond-mat/papers/0009/0009181.pdf. 

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