The fractional quantum Hall effect: The cases of 5/2 and 12/5

We find that there is a state of zero energy because of a zero value in (1/2)g. When negative sign is used, L = 0, S = 1/2, g = (2J+1)/(2L+1) [2(L-S)+1]/[2L+1] = 0 so that [n+(1/2)][(1/2)g] = 0. For positive sign, L+S, L = 0, g = 2 so that [n+(1/2)][(1/2)g] = 5/2 for n = 2. Hence 0 and 5/2 become particle-hole conjugates. In this definition, the sign of the spin for the particle is different from that for the hole as required by the helicity, p.s. For negative sign, L = 2, (1/2)g = 2/5 and (n-n')[(1/2)g] = 12/5 with n-n' = 6. For the positive sign, (1/2)g = 3/5 for L = 2 and for n-n'' = 4, we get 12/5. Thus 12/5 can arise for up spin as well as for down spin for different Landau levels. On the basis of a product of [n+(1/2)][(1/2)g] we are able to understand all of the fractions found in the experimental data.