Analytic formulas for small systems of electrons at odd denominator filling factors of the lowest Landau level

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The fractional quantum Hall effect (or integer quantum Hall effect) is a quantum-mechanical version of the classical Hall effect observed in two-dimensional electron systems at vey low temperatures and subject to strong perpendicular magnetic fields. Under these extreme conditions the Hall conductivity becomes quantized signaling the creation of a novel collective quantum liquid electronic state of matter at absolute zero temperature. Impurities are important for both the integer and fractional quantum Hall effects. While the integer version can be understood even without interactions, the fractional quantum Hall effect originates from strong Coulomb interactions between electrons. This unique fact has stimulated a great deal of research and has contributed to the overall progress of the field of theoretical condensed matter physics. The most pronounced fractional quantum Hall states occur at odd denominator filling factors of the lowest Landau level and are described by the Laughlin wave function. It is well-known that exact closed-form solutions for many-body wave functions, including the Laughlin wave function, are generally very rare and hard to obtain. In this work we present some exact results corresponding to small systems of electrons in the fractional quantum Hall regime at odd denominator filling factors. Use of Jacobi coordinates is the key tool that facilitates the exact calculation of various quantities. Expressions involving integrals over many variables are considerably simplified with the help of Jacobi coordinates allowing us to calculate exactly various quantities corresponding to systems with several electrons. © 2013 by Nova Science Publishers, Inc. All rights reserved.

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