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Energy values versus filling factors ν are examined for fractional quantum Hall states (FQHS). Initially, it is shown that the classical Coulomb energy of nearest electron pairs in FQHS is linearly dependent upon 1/ν. This function value is continuous for variation of ν.
The residual Coulomb interaction produces quantum transitions. Examining the second order transitions, we find a discrepancy between the second order perturbation energy at ν=2/3 and the limiting energy value at ν=(2s+1)/(3s+1) for an infinitely large value of s.
That is to say, the energy ε(ν=2/3) = Z/6, whereas ε(ν=(2s+1)/(3s+1)) → Z/12 for an infinitely large s.
Accordingly, the energy gap appears at ν=2/3. For ν=3/4, the second order perturbation energy ε(ν=3/4) is equal to the limiting energy value at ν=(3s+1)/(4s+1) for an infinitely large value of s; therefore, the energy spectrum is continuous near ν=3/4. It means gapless at ν=3/4. Both the gap mechanism and gapless mechanism are derived from the Pauli exclusion principle. The same mechanisms appear in a higher order perturbation calculation because the number of forbidden transitions in the higher order calculation is equal to that in the second order calculation. That is to say, the gap and gapless mechanisms can be extended to higher order calculations. In fractional filling factors other than ν=2/3 and 3/4, either the gap mechanism or the gapless mechanism appears for each filling factor. Consequently, our results can theoretically explain the precise confinement of Hall resistance at several values of the fractional filling factor. |