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Advances in Condensed Matter Physics
Volume 2012, Article ID 281371, 13 pages
http://dx.doi.org/10.1155/2012/281371
Research Article

Gap Structure and Gapless Structure in Fractional Quantum Hall Effect

KYOKUGEN (Center for Quantum Science and Technology under Extreme Conditions), Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan

Received 14 November 2011; Revised 9 January 2012; Accepted 24 January 2012

Academic Editor: Roberto Zivieri

Copyright © 2012 Shosuke Sasaki. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Higher-order composite fermion states are correlated with many quasiparticles. The energy calculations are very complicated. We develop the theory of Tao and Thouless to explain them. The total Hamiltonian is ( 𝐻 𝐷 + 𝐻 𝐼 ) , where 𝐻 𝐷 includes Landau energies and classical Coulomb energies. We find the most uniform electron configuration in Landau states which has the minimum energy of 𝐻 𝐷 . At 𝜈 = ( 2 𝑗 1 ) / ( 2 𝑗 ) , all the nearest electron pairs are forbidden to transfer to any empty states because of momentum conservation. Therefore, perturbation energies of the nearest electron pairs are zero in all order of perturbation. At 𝜈 = 𝑗 / ( 2 𝑗 1 ) , 𝑗 / ( 2 𝑗 + 1 ) , all the nearest electron (or hole) pairs can transfer to all hole (or electron) states. At 𝜈 = 4 / 1 1 , 4 / 1 3 , 5 / 1 3 , 5 / 1 7 , 6 / 1 7 , only the specific nearest hole pairs can transfer to all electron states. For example, the nearest-hole-pair energy at 𝜈 = 4 / 1 1 is lower than the limiting energies from both sides (the left side 𝜈 = ( 4 𝑠 + 1 ) / ( 1 1 𝑠 + 3 ) and the right side 𝜈 = ( 4 𝑠 1 ) / ( 1 1 𝑠 3 ) for infinitely large 𝑠 ). Thus, the nearest-hole-pair energy at specific 𝜈 is different from the limiting values from both sides. The property yields energy gap for the specific 𝜈 . Also gapless structure appears at other filling factors (e.g., at 𝜈 = 1 / 2 ).