Energy gaps in fractional quantum hall states

Energy values versus filling factors v are examined for fractional quantum Hall states (FQHS). First, the classical Coulomb energy of nearest electron pairs in FQHS is shown to be linearly dependent upon 1/v. The residual Coulomb interaction produces quantum transitions. Examination of second order transitions indicates a discrepancy between the second order perturbation energy at v = 2/3 and the limiting energy value at v = (2s + 1)/(3s + 1) for an infinitely large value of s. Accordingly, an energy gap appears at v = 2/3. On the other hand, the second order perturbation energy ε2(v = 3/4) is equal to the limiting energy value at v = (3s + 1)/(4s + 1) for an infinitely large s. Therefore, the energy spectrum is continuous near v = 3/4, meaning that it is gapless at v = 3/4. The same mechanisms appear in higher order perturbation calculations 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 v = 2/3 and 3/4, either a gap mechanism or a gapless mechanism appears for each filling factor. Consequently, our results can theoretically explain the precise confinement of Hall resistance at fractional filling factors with a gap mechanism